Local Fractional Integral Transforms and their Applications

Local Fractional Integral Transforms and TheirApplications

Local Fractional IntegralTransforms and TheirApplications

Xiao-Jun YangDepartment of Mathematics and Mechanics, China University of Mining andTechnology, Xuzhou, China

Dumitru BaleanuDepartment of Mathematics and Computer Sciences, Faculty of Arts and Sciences,Cankaya University, Ankara, TurkeyandInstitute of Space Sciences, Magurele-Bucharest, Romania

H. M. SrivastavaDepartment of Mathematics and Statistics, University of Victoria, Victoria,British Columbia V8W 3R4, Canada

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ISBN: 978-0-12-804002-7

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List of figures

Fig. 1.1 The distance between two points of A and B in a discontinuous space-time 2Fig. 1.2 The curve of ε-dimensional Hausdorff measure with ε = ln 2/ ln 3 3Fig. 1.3 The chart of �(μ) when ω = 1 and ε = ln 2/ ln 3 4Fig. 1.4 The concentration-distance curves for nondifferentiable source (see [23]) 6Fig. 1.5 The comparisons of the nondifferentiable functions (1.89)–(1.93) when

β = 2 and ε = ln 2/ ln 3 16Fig. 1.6 The comparisons of the nondifferentiable functions (1.94) and (1.95) when

ε = ln 2/ ln 3 17Fig. 2.1 The local fractional Fourier series representation of fractal signal ψ (τ)

when ε = ln 2/ ln 3, k = 0, k = 1, k = 3, and k = 5 80Fig. 2.2 The local fractional Fourier series representation of fractal signal ψ (τ)

when ε = ln 2/ ln 3, k = 1, k = 2, k = 3, k = 4, and k = 5 82Fig. 2.3 The plot of fractal signal ψ (τ) is shown when ε = ln 2/ ln 3 82Fig. 2.4 The plot of fractal signal ψ (τ) is shown when ε = ln 2/ ln 3 84

Fig. 3.1 The plots of a family of good kernels: (a) the plot of �ε(

14π , τ

)with

fractal dimension ε = ln 2/ ln 3 and (b) the plot of �ε (1, τ) with fractaldimension ε = ln 2/ ln 3 126

Fig. 3.2 The graphs of analogous rectangular pulse and its local fractional Fouriertransform: (a) the graph of rectε (τ ) and (b) the graph of (� rectε) (ω) 131

Fig. 3.3 The graphs of the analogous triangle function and its local fractional Fouriertransform: (a) the plot of triangε (τ ) and (b) the plot of

(� triangε)(ω) 132

Fig. 3.4 The graphs of �(μ) when ε = ln 2/ ln 3, p = 1, p = 2, and p = 3 140Fig. 3.5 The graph of �(μ) when ε = ln 2/ ln 3 141Fig. 3.6 The graph of �(μ) when ε = ln 2/ ln 3 142Fig. 4.1 The graph of θ (τ ) when ε = ln 2/ ln 3 163Fig. 4.2 The graph of θ (τ ) when ε = ln 2/ ln 3 163Fig. 4.3 The graph of θ (τ ) when ε = ln 2/ ln 3 164Fig. 4.4 The graph of θ (τ ) when ε = ln 2/ ln 3 165Fig. 4.5 The graph of θ (τ ) when ε = ln 2/ ln 3 166Fig. 4.6 The graph of θ (τ ) when ε = ln 2/ ln 3 169Fig. 5.1 The plot of �(μ, τ) in fractal dimension ε = ln 2/ ln 3 182Fig. 5.2 The plot of �(η,μ) in fractal dimension ε = ln 2/ ln 3 184Fig. 5.3 The plot of �(η,μ) in fractal dimension ε = ln 2/ ln 3 190Fig. 5.4 The plot of �(η,μ) in fractal dimension ε = ln 2/ ln 3 192

List of tables

Table 1.1 Basic operations of local fractional derivative of some of nondifferentiablefunctions defined on fractal sets 21

Table 1.2 Basic operations of local fractional integral of some of nondifferentiablefunctions defined on fractal sets 33

Table 1.3 Basic operations of local fractional integral of some of nondifferentiablefunctions via Mittag–Leffler function defined on fractal sets 33

Table E.1 Tables for local fractional Fourier transform operators 223Table F.1 Tables for local fractional Laplace transform operators 230

Preface

The purpose of this book is to give a detailed introduction to the local fractionalintegral transforms and their applications in various fields of science and engineering.The local fractional calculus is utilized to handle various nondifferentiable prob-lems that appear in complex systems of the real-world phenomena. Especially, thenondifferentiability occurring in science and engineering was modeled by the localfractional ordinary or partial differential equations. Thus, these topics are importantand interesting for researchers working in such fields as mathematical physics andapplied sciences.

In light of the above-mentioned avenues of their potential applications, we system-atically present the recent theory of local fractional calculus and its new challengesto describe various phenomena arising in real-world systems. We describe the basicconcepts for fractional derivatives and fractional integrals. We then illustrate the newresults for local fractional calculus. Specifically, we have clearly stated the basic ideasof local fractional integral transforms and their applications.

The book is divided into five chapters with six appendices.Chapter 1 points out the recent concepts involving fractional derivatives. We give

the properties and theorems associated with the local fractional derivatives and thelocal fractional integrals. Some of the local fractional differential equations occurringin mathematical physics are discussed. With the help of the Cantor-type circular coor-dinate system, Cantor-type cylindrical coordinate system, and Cantor-type sphericalcoordinate system, we also present the local fractional partial differential equationsin fractal dimensional space and their forms in the Cantor-type cylindrical symmetryform and in the Cantor-type spherical symmetry form.

In Chapter 2, we address the basic idea of local fractional Fourier series viathe analogous trigonometric functions, which is derived from the complex Mittag–Leffler function defined on the fractal set. The properties and theorems of the localfractional Fourier series are discussed in detail. We mainly focus on the Besselinequality for local fractional Fourier series, the Riemann–Lebesgue theorem forlocal fractional Fourier series, and convergence theorem for local fractional Fourierseries. Some applications to signal analysis, ODEs and PDEs are also presented.We specially discuss the local fractional Fourier solutions of the homogeneous andnonhomogeneous local fractional heat equations in the nondimensional case andthe local fractional Laplace equation and the local fractional wave equation in thenondimensional case.

Chapter 3 is devoted to an introduction of the local fractional Fourier transformoperator via the Mittag–Leffler function defined on the fractal set, which is derivedby approximating the local fractional integral operator of the local fractional Fourierseries. The properties and theorems of the local fractional Fourier transform operator

xii Preface

are discussed. A particular attention is paid to the logical explanation for the theoremsfor the local fractional Fourier transform operator and for another version of the localfractional Fourier transform operator (which is called the generalized local fractionalFourier transform operator). Meanwhile, we consider some application of the localfractional Fourier transform operator to signal processing, ODEs, and PDEs with thehelp of the local fractional differential operator.

Chapter 4 addresses the study of the local fractional Laplace transform operatorbased on the local fractional calculus. Our attentions are focused on the basicproperties and theorems of the local fractional Laplace transform operator and itspotential applications, such as those in signal analysis, ODEs, and PDEs involvingthe local fractional derivative operators. Some typical examples for the PDEs inmathematical physics are also discussed.

Chapter 5 treats the variational iteration and decomposition methods and thecoupling methods of the Laplace transform with them involved in the local fractionaloperators. These techniques are then utilized to solve the local fractional partial dif-ferential equations. Their nondifferentiable solutions with graphs are also discussed.

We take this opportunity to thank many friends and colleagues who helped us inour writing of this book. We would also like to express our appreciation to severalstaff members of Elsevier for their cooperation in the production process of this book.

Xiao-Jun YangDumitru BaleanuH.M. Srivastava

1Introduction to local fractionalderivative and integral operators

1.1 Introduction

1.1.1 Definitions of local fractional derivatives

The concept of local fractional calculus (also called fractal calculus), which was firstproposed by Kolwankar and Gangal [1, 2] based on the Riemann–Liouville fractionalderivative [3–6], was applied to deal with nondifferentiable problems from scienceand engineering [7–16]. Several other points of fractal calculus were presented, suchas the fractal derivative via Hausdorff measure [1, 17, 18], fractal derivative usingfractal geometry [1, 19, 20], and local fractional derivative using the fractal geometry[1, 21–25]. Here, in this chapter, we present the logical extensions of the definitionsto the subject of local derivative on fractals.

Let us recall the basic definitions as follows.Local fractional derivative of �(μ) of order ε (0 < ε ≤ 1) defined in [1, 2, 7–16]

is given by

D(ε)� (μ) = dε� (μ)

dμε

∣∣∣∣ μ=μ0 = limμ→μ0

dε [�(μ)−�(μ0)]

[d (μ− μ0)]ε, (1.1)

where the term dε [�(μ)] / [d (μ− μ0)]ε is the Riemann–Liouville fractional deriva-tive of order ε of �(μ).

Local fractional (fractal) derivative of �(μ) of order ε (0 < ε ≤ 1) via Hausdorffmeasure με defined in [1, 17, 18] is given by

D(ε)� (μ) = dε� (μ)

dμε

∣∣∣∣ μ=μ0 = limμ→μ0

�(μ)−�(μ0)

με − με0, (1.2)

where με is a fractal measure.Local fractional (fractal) derivative using fractal geometry of �(μ) of order

ε (0 < ε ≤ 1) defined in [1, 19, 20] is written as

D(ε)� (μ) = d�(μ)

dμε

∣∣∣∣ μ=μ0 = d�(μ)

dσ= lim�μ→μ0

�(μB)−�(μA)

ϒηε0, (1.3)

where dσ = ϒηε0 with geometric parameter ϒ and measure scale η0 is shown inFigure 1.1.

Local Fractional Integral Transforms and Their Applications. http://dx.doi.org/10.1016/B978-0-12-804002-7.00001-2Copyright © 2016 Xiao-Jun Yang, Dumitru Baleanu and Hari M. Srivastava. Published by Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/B978-0-12-804002-7.00001-2

2 Local Fractional Integral Transforms and Their Applications

B

A

h0

Figure 1.1 The distance between two points of A and B in a discontinuous space-time.

The local fractional derivative using the fractal geometry �(μ) of orderε (0 < ε ≤ 1) defined in [1, 21–25] has the following form:

D(ε)� (μ) = dε� (μ)

dμε

∣∣∣∣ μ=μ0 = limμ→μ0

�ε [�(μ)−�(μ0)]

(μ− μ0)ε , (1.4)

where �ε [�(μ)−�(μ0)] ∼= � (1 + ε) [�(μ)−�(μ0)] with the Euler’s Gammafunction � (1 + ε) = :

∫∞0 με−1 exp (−μ) dμ.

Following (1.4), we define ε (0 < ε ≤ 1)-dimensional Hausdorff measure given by[1–25]

Hε [ ∩ (μ0,μ)] = (μ− μ0)ε , (1.5)

and its plot when ε = ln 2/ ln 3 is the dimension of the fractal set and μ0 = 0 isshown in Figure 1.2.

1.1.2 Comparisons of fractal relaxation equation in fractalkernel functions

The fractal relaxation equation with the help of (1.1) is given as

D(ε)� (μ)+ ω� (μ) = 0, (1.6)

Introduction to local fractional derivative and integral operators 3

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

m

Figure 1.2 The curve of ε-dimensional Hausdorff measure with ε = ln 2/ ln 3.

where �(0) = 1. Its solution is written as follows:

�(μ) = exp (−ωFc (μ)) , (1.7)

where Fc (μ) is a Lebesgue–Cantor function and Fc (μ) ∼ με.The fractal relaxation equation with the help of (1.2) is given as follows [26]:

D(ε)� (μ)+ ω� (μ) = 0, (1.8)

where �(0) = 1, and its solution is given by

�(μ) = exp(−ωμε) . (1.9)

The fractal relaxation equation by using (1.3) (see [19]):

D(ε)� (μ)+ ω� (μ) = 0, (1.10)

with �(0) = 1 that has the solution

�(μ) = exp(−ωϒηε−1

0 μ)

, (1.11)

where σ = ϒηε−10 μ.

The fractal relaxation equation based on (1.4) is given as follows (see [26]):

D(ε)� (μ)+ ω� (μ) = 0, (1.12)

and its solution is presented as


0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

4

4.5

m

Φ(m

)

Figure 1.3 The chart of �(μ) when ω = 1 and ε = ln 2/ ln 3.

�(μ) = Eε(−ωμε) , (1.13)

where Eε (−ωμε) =∑∞i=0

(−1)ωiμεi

�(1+εi) is defined on the Cantor sets. The correspondinggraph for ω = 1 and ε = ln 2/ ln 3 is shown in Figure 1.3.

1.1.3 Comparisons of fractal diffusion equation in fractal kernelfunctions

The fractal diffusion equation based on (1.1) is presented as follows (see [8]):

∂ε� (σ ,μ)

∂με= �∂

2�(σ ,μ)

∂σ 2 , (1.14)

where � = � (1 + ε) χc (μ) /4, and its solution is given by

�(σ ,μ) = 1√πFc (μ)

exp

(− σ 2

Fc (μ)

), (1.15)

where Fc (μ) is a Lebesgue–Cantor function and χc (μ) is the membership functionof a Cantor set.

We mention that the fractal diffusion equation within (1.2) has the form [17]:

∂ε� (σ ,μ)

∂με= �∂

2ς� (σ ,μ)

∂σ 2ς , (1.16)


where 0 < ς ≤ 1 and � is a contact, and its solution is

�(σ ,μ) = 1√4π�με

exp

(− σ 2σ

4�με

). (1.17)

The fractal diffusion equation based on (1.3) has the form

∂ε� (σ ,μ)

∂με= �∂

2ς� (σ ,μ)

∂σ 2ς , (1.18)

where 0 < ς ≤ 1 and � is a contact, and its solution is given by

�(σ ,μ) = 1√4π�ϒη1−ε

0 μ

exp

⎛⎜⎝−

(ιξ

1−ς0 σ

)2

4�ϒη1−ε0 μ

⎞⎟⎠ . (1.19)

We mention below the fractal diffusion equation based on (1.4) [23]

∂ε� (σ ,μ)

∂με= �∂

2ε� (σ ,μ)

∂σ 2ε , (1.20)

where � is a contact. The solution is given by

�(σ ,μ) = �0μβεEε

(− σ 2ε

(4�μ)ε

), (1.21)

where Eε(−ωμ2ε

) = ∑∞i=0

2ε2(−1)iωiμ2εi

�(1+εi) is defined on the Cantor sets and its graph,when ω = 1 and ε = ln 2/ ln 3 [23], is shown in Figure 1.4.

When � = 1, we conclude that

�(σ ,μ) = �0,0μβεEε

(− σ 2ε

(4μ)ε

), (1.22)

such that [21]

�(σ , 0) = δε (σ ) . (1.23)

Below, we present a new definition of the local fractional Dirac function, namely,

δε (σ ) = limμ→0

�0,0μβεEε

(− σ 2ε

(4μ)ε

). (1.24)

Using the reference [27], we have

1

� (1 + ε)∫ ∞

−∞1

(4πμ)ε2

�(1+ε)Eε

(− σ 2ε

(4μ)ε

)(dσ)ε, (1.25)

so that


1

(4πμ)ε2

�(1+ε)Eε

(− σ 2ε

(4μ)ε

). (1.26)


−1 −0.5 0 0.5 10.4

0.5

0.6

0.7

0.8

0.9

1

s

Φ(s

, m)

Figure 1.4 The concentration-distance curves for nondifferentiable source (see [23]).

Hence, with the help of (1.24) and (1.26), we get

�0,0 = 1

(4π)ε2

�(1+ε)(1.27)

and

β = −ε2

. (1.28)

In a similar manner, we obtain

1

� (1 + ε)∫ ∞

−∞1

(4π�μ)ε2

�(1+ε)Eε

(− σ 2ε

(4�μ)ε

)(dσ)ε. (1.29)

Therefore, there is a local fractional Dirac function defined by


1

(4π�μ)ε2

�(1+ε)Eε

(− σ 2ε

(4�μ)ε

), (1.30)

so that

�0 = 1

(4πμ)ε2

�(1+ε). (1.31)


1.1.4 Fractional derivatives via fractional differences

Fractional derivatives via fractional differences were applied to solve the numericalproblems for fractional differential equations in mathematical physics. We present thebasic definitions of them given below:

The Grünwald–Letnikov derivative of the function �(μ) of fractional orderε (0 < ε ≤ 1) [6, 28–34] is a fractional derivative via fractional difference, given by

D(ε)� (μ) = dε� (μ)

dμε

∣∣∣∣ μ=μ0 = limρ→0

�ε� (μ)

ρε, (1.32)

where the fractional difference term is

�ε� (μ) =∞∑

i=0

(−1)i(ε

i

)�(μ− iρ), (1.33)

with

(ε

i

)= �(1+ε)�(1+i)�(1+ε−i) .

The fractional derivative of the function �(μ) of fractional order ε (0 < ε ≤ 1)[35–37] is a fractional derivative via fractional difference, given by

D(ε)� (μ) = dε� (μ)

dμε

∣∣∣∣ μ=μ0 = limρ→0

�ε� (μ)

ρε. (1.34)

Here the fractional difference term is given by

�ε� (μ) =∞∑

i=0

(−1)i(ε

i

)�(μ− (ε − i) ρ). (1.35)

The fractional derivative of the function �(μ) of fractional order ε (0 < ε ≤ 1)introduced in [38] is a fractional derivative via fractional difference, given by

D(ε)� (μ) = dε� (μ)

dμε

∣∣∣∣ μ=μ0 = limρ→0

�ε [�(μ)−�(μ0)]

ρε, (1.36)

where the fractional difference term is

�ε� (μ) =∞∑

i=0

(−1)i(ε

i

)�(μ− (ε − i) ρ). (1.37)

The fractional derivative of the function �(μ) of variational order ε (μ) (0 < ε (μ)≤ 1) [24] is defined as

D(ε(μ))� (μ) = dε(μ)� (μ)

dμε(μ)

∣∣∣∣∣ μ=μ0 = limρ→0

�ε(μ) [�(μ)−�(0)]ρε(μ)

, (1.38)

where the fractional difference term is given by

�ε(μ)� (μ) =∞∑

i=0

(−1)i1

� (i − ε (μ))� (μ− (ε (μ)− i) ρ). (1.39)


The Grünwald–Letnikov–Riesz derivative of the function �(μ) of fractional orderε (0 < ε ≤ 1) via Grünwald–Letnikov derivative introduced in [39] is defined as

D(ε)� (μ) = dε� (μ)

dμε

∣∣∣∣ μ=μ0 = cε limρ→0

[�ε+�(μ)+�ε−�(μ)

]ρε

, (1.40)

where

cε = 1

2 cos(πε2

) , (1.41)

and the fractional difference terms are for ρ > 0 and ρ < 0,

�ε+�(μ) =∞∑

i=0

(−1)|i|(ε

i

)�(μ− iρ), (1.42)

�ε−�(μ) =∞∑

i=0

(−1)|i|(ε

i

)�(μ+ iρ), (1.43)

respectively.

1.1.5 Fractional derivatives with and without singular kernelsand other versions of fractional derivatives

Fractional derivatives with singular kernel [28–69] have found popular applicationsin the fields of science and engineering. We mention some of them, for example,Liouville, Riemann–Liouville, Caputo, Weyl, Marchaud, Hadamard, Chen, Canavati,Riesz, and Cossar. The details on the conformable fractional derivatives were dis-cussed recently in [40, 41]. A tempered fractional derivative was proposed in [42].Generalized Riemann and Caputo versions of fractional derivatives were proposed in[43]. A fractional derivative without singular kernel and some of its properties werediscussed very recently in [44, 45]. Below, we present the definitions of fractionalderivatives with and without singular kernels as well as the conformable and temperedfractional derivatives.

Liouville fractional derivative of the function �(μ) of fractional order ε isdefined as

D(ε)� (μ) = 1

� (1 − ε)d

dμ

∫ μ−∞

�(λ)

(μ− λ)ε dλ, (1.44)

where −∞ < μ <∞ and ε is a real number.Liouville left-sided fractional derivative of the function �(μ) of fractional order ε

is defined by

D(ε)+ �(μ) = 1

� (n − ε)dn

dμn

∫ μ0

�(λ)

(μ− λ)ε+1−ndλ, (1.45)

where 0 < μ, n is integer, and ε denotes a real number.


Liouville right-sided fractional derivative of the function �(μ) of fractional orderε is given by

D(ε)− �(μ) = (−1)n

� (n − ε)dn

dμn

∫ μ−∞

�(λ)

(μ− λ)ε+1−ndλ, (1.46)

where μ <∞, n is integer, and ε is real number.Riemann–Liouville left-sided fractional derivative of a function�(μ) of fractional

order ε is

D(ε)a+�(μ) = 1

� (n − ε)dn

dμn

∫ μa

�(λ)

(μ− λ)ε+1−ndλ, (1.47)

where a ≤ μ, n is integer, and ε is real number.Riemann–Liouville right-sided fractional derivative of the function �(μ) of

fractional order ε is defined as

D(ε)a+�(μ) = (−1)n

� (n − ε)dn

dμn

∫ b

μ

� (λ)

(μ− λ)ε+1−ndλ, (1.48)

where μ ≤ b, n is integer, and ε denotes a real number.Caputo left-sided fractional derivative of the function�(μ) of fractional order ε is

defined as

D(ε)a+�(μ) = 1

� (n − ε)∫ μ

a

1

(μ− λ)ε+1−n

[dn

dλn�(λ)

]dλ. (1.49)

Here a ≤ μ, n denotes an integer, and ε is real number.Caputo right-sided fractional derivative of the function �(μ) of fractional order ε

is defined by

D(ε)a+�(μ) = (−1)n

� (n − ε)∫ b

μ

1

(μ− λ)ε+1−n

[dn

dλn�(λ)

]dλ, (1.50)

where μ ≤ b, n is integer, and ε is real number.Weyl fractional derivative of the function �(μ) of fractional order ε (alternative

definition; see [24]) is defined as

D(ε)μ � (μ) = 1

� (n − ε)dn

dμn

∫ ∞

μ

� (λ)

(μ− λ)ε+1−ndλ. (1.51)

Here n is an integer and ε denotes a real number.Marchaud fractional derivative of the function �(μ) of fractional order ε is

defined as

D(ε)+ �(μ) = {ε}� (1 − {ε})

∫ ∞

μ

[�(μ)−�(λ)](μ− λ){ε}+1

dλ, (1.52)

where ε = [ε] + {ε}.


Marchaud left-sided fractional derivative of the function �(μ) of fractional orderε is written as

D(ε)+ �(μ) = {ε}� (1 − {ε})

∫ ∞

μ

[�([ε]) (μ)−�([ε]) (μ− λ)]

λ{ε}+1dλ, (1.53)

for ε = [ε] + {ε}.Marchaud right-sided fractional derivative of a function �(μ) of fractional order

ε has the form

D(ε)− �(μ) = {ε}� (1 − {ε})

∫ μ0

[�([ε]) (μ)−�([ε]) (μ+ λ)]

λ{ε}+1dλ, (1.54)

where ε = [ε] + {ε}.Below, the Hadamard fractional derivative of a function �(μ) of fractional order

ε is defined as

D(ε)+ �(μ) = ε

� (1 − ε)∫ μ

0

[�(μ)−�(λ)][ln (μ/λ)]ε+1

dλ

λ, (1.55)

where ε is real number.Now, we define the Chen left-sided fractional derivative of the function �(μ) of

fractional order ε has the form

D(ε)a �(μ) = 1

� (1 − ε)d

dμ

∫ μa

�(λ)

(μ− λ)ε dλ, (1.56)

where a ≤ μ and ε is real number.Chen right-sided fractional derivative of the function �(μ) of fractional order ε is

defined as

D(ε)a �(μ) = − 1

� (1 − ε)d

dμ

∫ b

μ

� (λ)

(λ− μ)ε dλ, (1.57)

where μ ≤ b and ε is real number.Canavati fractional derivative of the function �(μ) of fractional order ε is

given by

D(ε)� (μ) = 1

� (1 − ε)d

dμ

∫ μ0

1

(μ− λ)ε−n

[∂n

∂λn�(λ)

]dλ, (1.58)

where 0 ≤ μ, ε is real number, and [ε] = n is integral.Riesz fractional derivative of the function �(μ) of fractional order ε has the form

D(ε)� (μ) = −cε� (ε)

∂n

∂μn

[∫ μ−∞

�(λ)

(μ− λ)ε+1−ndλ+

∫ ∞

μ

� (λ)

(λ− μ)ε+1−ndλ

],

(1.59)

where cε = 1/[2 cos

(πε2

)], ε is real number, and n is integer.


Cossar fractional derivative of the function�(μ) of fractional order ε is defined as

D(ε)� (μ) = −1

� (1 − ε) limN→∞

∂

∂μ

[∫ N

μ

� (λ)

(λ− μ)ε dλ

], (1.60)

where ε is real number.Modified Riemann–Liouville fractional derivative of the function �(μ) of frac-

tional order ε is defined as

D(ε)� (μ) = 1

� (1 − ε)∂

∂μ

∫ μ0

�(λ)−�(0)(λ− μ)ε dλ, (1.61)

where ε is real number.The conformable fractional derivative of the function �(μ) of fractional order

ε [40] is defined as

D(ε)� (μ) = limκ→0

�(μ+ κμ1−ε)−�(μ)

κ, (1.62)

where ε(0 < ε ≤ 1) is real number.The modified conformable left-sided fractional derivative of the function �(μ) of

fractional order ε [41] is defined as

D(ε)� (μ) = limκ→0

�(μ+ κ (μ− a)1−ε)−�(μ)

κ, (1.63)

where ε(0 < ε ≤ 1) is real number.The modified conformable right-sided fractional derivative of the function �(μ)

of fractional order ε [41] is defined as

D(ε)� (μ) = − limκ→0

�(μ+ κ (μ− a)1−ε)−�(μ)

κ, (1.64)

where ε(0 < ε ≤ 1) is real number.Tempered left-sided fractional derivative of the function �(μ) of fractional order

ε introduced in [42] is defined as

D(ε)a �(μ) = ε

� (1 − ε)∫ ∞

0

�(μ)−�(μ− λ)λε+1 exp (−ιλ) dλ, (1.65)

where ε is real number.Tempered left-sided fractional derivative of the function �(μ) of fractional order


D(ε)a �(μ) = ε

� (1 − ε)∫ ∞

0

�(μ)−�(μ+ λ)λε+1

exp (−ιλ) dλ, (1.66)

where ε is real number.Generalized Riemann fractional derivative of the function�(μ) of fractional order


γD(ε)a �(μ) = (1 + γ ) ε� (1 − ε)

d

dμ

∫ μa

λγ� (μ)(μγ+1 − λγ+1

)ε dλ, (1.67)


where a ≤ μ and ε is real number.Generalized Caputo fractional derivative of the function �(μ) of fractional order


γD(ε)0 �(μ) = (1 + γ ) ε� (1 − ε)

d

dμ

∫ μa

λγ� (μ)(μγ+1 − λγ+1

)ε dλ, (1.68)

where 0 ≤ μ and ε is real number.Erdelyi–Kober fractional derivative of the function �(μ) of fractional order ε is

defined as

D(ε)0,ξ ,ζ� (μ) = μ−nζ(

1

ξμζ−1

d

dμ

)n

μ−ζ (n+ζ )In−ε0,ξ ,ξ+ζ� (μ) , (1.69)

where

In−ε0,ξ ,ξ+ζ� (μ) = ξμ−ζ (ζ+ε)

� (ε)

∫ μ0

λξζ+ξ−1�(μ)(μζ − λζ )1−ε dλ, (1.70)

with real number ε.Caputo–Fabrizio fractional derivative of the function �(μ) of fractional order ε

introduced in [44, 45] is defined as

D(ε)� (μ) = 1

1 − ε∫ μ

0exp

(− ε

1 − ε (μ− λ))�(1) (μ) dλ, (1.71)

where 0 < μ and ε is real number.Coimbra fractional derivative of the function �(μ) of fractional order ε (μ) is

defined as

D(ε(μ))� (μ) = 1

� (1 − ε (μ)){∫ μ

a

1

(μ− λ)ε(μ)[∂� (λ)

∂λ

]dλ+�(0) μ−ε(μ)

},

(1.72)

where a < μ and ε (μ) (0 < ε (μ) < 1) is real number related to μ.Left-sided Riemann–Liouville fractional derivative of the function �(μ) of vari-

able fractional order ε (λ,μ) is defined as

D(ε(λ,μ))a+ �(μ) = d

dμ

∫ μa

�(λ)

(μ− λ)ε(λ,μ)

dλ

� [1 − ε (λ,μ)], (1.73)

where a < μ and ε (λ,μ) (0 < ε (λ,μ) < 1) is real number related to μ.Right-sided Riemann–Liouville fractional derivative of the function �(μ) of

variable fractional order ε (λ,μ) is defined as

D(ε(λ,μ))b− �(μ) = d

dμ

∫ b

μ

� (λ)

(λ− μ)ε(λ,μ)

dλ

� [1 − ε (λ,μ)], (1.74)

where μ < b and ε (λ,μ) (0 < ε (λ,μ) < 1) is real number related to μ.


Left-sided Caputo fractional derivative of the function �(μ) of variable fractionalorder ε (λ,μ) is defined as

D(ε(λ,μ))a+ �(μ) =

∫ μa

1

(μ− λ)ε(λ,μ)

[d

dμ�(λ)

]dλ

� [1 − ε (λ,μ)], (1.75)

where a < μ and ε (λ,μ) (0 < ε (λ,μ) < 1) is real number related to μ.Right-sided Caputo fractional derivative of the function�(μ) of variable fractional

order ε (λ,μ) is defined as

D(ε(λ,μ))b− �(μ) =

∫ b

μ

1

(λ− μ)ε(λ,μ)

[d

dμ�(λ)

]dλ

� [1 − ε (λ,μ)], (1.76)

where μ < b and ε (λ,μ) (0 < ε (λ,μ) < 1) is real number related to μ.Caputo fractional derivative of variable fractional order is defined as

D(ε(μ))a+ �(μ) = 1

� [1 − ε (μ)]∫ μ

a

1

(μ− λ)ε(μ)[

d

dμ�(λ)

]dλ, (1.77)

where μ < b and ε (λ,μ) (0 < ε (λ,μ) < 1) is real number related to μ.

1.2 Definitions and properties of local fractionalcontinuity

1.2.1 Definitions and properties

Let ℘ be a fractal set and let d1 and d0 be two metric spaces. Suppose �: (℘, d0) →(ℵ, d1) is a bi-Lipschitz mapping, then, we have

ω1ε (℘) ≤ ε (� (℘)) ≤ ω1ε (℘) (1.78)

such that

ω1 |μ1 − μ2| ≤ |�(μ1)−�(μ2)| ≤ ω1 |μ1 − μ2| , (1.79)

where μ1,μ2 ∈ ℘, ℘ ⊂ R, and ω1,ω2 > 0.Using (1.79), for ∀ρ > 0 and 0 < ε < 1, we have

|�(μ1)−�(μ2)| < ρε, (1.80)

where ε is fractal dimension of the fractal set ℘. This form is analogues of Lipschitzmapping.

Definition 1.1. Let �: ℘ → ℵ be a function defined on a fractal set ℘ of fractaldimension ε(0 < ε < 1). A real number χ is called a generalized limit of �(μ) asμ tends to a, or the limit of �(μ) at a, if to each τ > 0 there corresponds δ > 0such that

|�(μ)− χ | < τε, (1.81)


whenever

0 < |μ− a| < δ. (1.82)

The above statement is expressible in terms of inequalities as follows.Suppose τ > 0. Then, there is each δ > 0 such that |�(μ)− χ | < τε if

0 < |μ− a| < δ.Thus, we write

�(μ)→ χ (1.83)

as μ→ a, or

limμ→a

�(μ) = χ . (1.84)

We say that �(μ) tends to χ as μ tends to a.

Definition 1.2. A function �(μ) is said to be local fractional continuous atμ = μ0 if for each τ > 0, there exists for δ > 0 such that

|�(μ)−�(μ0)| < τε, (1.85)

whenever 0 < |μ− μ0| < δ.It is written as

limμ→μ0

�(μ) = �(μ0) . (1.86)

A function �(μ) is said to be local fractional continuous at μ = μ0 from the rightif for each τ > 0, there exists for δ > 0 such that (1.82) holds whenever μ0 < μ <

δ + μ0.A function �(μ) is said to be local fractional continuous at μ = μ0 from the

left if for each τ > 0, there exists for δ > 0 such that (1.82) holds wheneverδ − μ0 < μ < μ0.

If limμ→μ+0�(μ) = � (μ+

0

), limμ→μ−

0�(μ) = � (μ−

0

), and �

(μ+

0

) = � (μ−0

)exist, then, we have

limμ→μ0

�(μ) = limμ→μ+

0

�(μ) = limμ→μ−

0

�(μ) . (1.87)

Suppose a function�(μ) is local fractional continuous in the domain I = (a, b), then,we write it as

�(μ) ∈ Cε (a, b) . (1.88)

Theorem 1.1. Suppose that limμ→μ0 �(μ) = �(μ0) and limμ→μ0 �(μ) =�(μ0) . Then

(a) limμ→μ0 [�(μ)±�(μ)] = �(μ0)±�(μ0);(b) limμ→μ0 |�(μ)| = |�(μ0)|;(c) limμ→μ0 [�(μ)� (μ)] = �(μ0)� (μ0); and(d) limμ→μ0 [�(μ) /� (μ)] = �(μ0) /� (μ0), provided �(μ0) = 0.


For the details of formal proofs of the validity of these four rules, see [1, 16, 21].Theorem 1.1 is a natural generalized result of those known when the order is a positiveinteger.

1.2.2 Functions defined on fractal sets

Following the definition of ε-dimensional Hausdorff measure, we define the functionsdefined on fractal sets as follows:

Let �: ℘ → ℵ be a function defined on a fractal set ℘ of fractal dimension ε(0 <ε < 1). A real-valued function �(μ) defined on the fractal set ℘ is given by

�(μ) = με, (1.89)

where με ∈ ℘ and 0 < ε < 1.We now notice that (1.89) is a Lebesgue–Cantor function and limε→1�(μ) = μ ∈

R with real number set R.The Mittag–Leffler function defined on the fractal set ℘ is given by

Eε(με) =

∞∑k=0

μkε

� (1 + kε), (1.90)

where μ ∈ R and 0 < ε < 1.An extended version of (1.90) defined on the fractal set ℘ is given as

Eε(β,με

) =∞∑

k=0

μkε

� (β + kε), (1.91)

where β is real number, μ ∈ R, and 0 < ε < 1.The following rules via Mittag–Leffler functions defined on the fractal set ℘ hold:

(a) Eε (με)Eε (νε) = Eε (με + νε);(b) Eε (με)Eε (−νε) = Eε (με − νε);(c) Eε (με)Eε (iενε) = Eε (με + iενε);(d) Eε (iεμε)Eε (iενε) = Eε (iεμε + iενε); and(e) [Eε (με + iεμε)]n = Eε (nεμε + nεiενε), where n is integer and iε is a imaginary unit of a

fractal set ℘.

The sine function defined on the fractal set ℘ is given by

sinε(με) =

∞∑k=0

(−1)k μ(2k+1)ε

� (1 + (2k + 1) ε), (1.92)

where μ ∈ R and 0 < ε < 1.The cosine function defined on the fractal set ℘ is given by

cosε(με) =

∞∑k=0

(−1)k μ2kε

� (1 + 2kε), (1.93)

where μ ∈ R and 0 < ε < 1.


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

m

Ee (mε)

Ee (b,mε)

sine (mε)

cose (mε)

mε

Figure 1.5 The comparisons of the nondifferentiable functions (1.89)–(1.93) when β = 2 andε = ln 2/ ln 3.

Their graphs corresponding to the fractal dimension ε = ln 2/ ln 3 are shown inFigure 1.5.

The following rules via Mittag–Leffler, sine, and cosine functions defined on thefractal set ℘ hold:

(a) Eε (iεμε) = cosε (με)+ iε sinε (με);

(b) sinε (με) = Eε(iεμε)−Eε(−iεμε)2iε ;

(c) cosε (με) = Eε(iεμε)+Eε(−iεμε)2 ;

(d) cosε (−με) = cosε (με);(e) sinε (−με) = − sinε (με);(f) sin2

ε (με)+ cos2

ε (με) = 1; and

(g) 12 +∑n

k=1 cosε (kμε) = sinε((2n+1)με/2)2 sinε(με/2)

, provided sinε (με/2) = 0.

Other properties are listed in Appendix A.The hyperbolic functions via Mittag–Leffler function defined on the fractal set ℘

are given by

sinhε(με) = Eε (με)− Eε (−με)

2=

∞∑k=0

μ(2k+1)ε

� (1 + (2k + 1) ε), (1.94)


coshε(με) = Eε (με)+ Eε (−με)

2=

∞∑k=0

μ2kε

� (1 + 2kε), (1.95)

tanhε(με) = Eε (με)− Eε (−με)

Eε (με)+ Eε (−με) , (1.96)

cothε(με) = Eε (με)+ Eε (−με)

Eε (με)− Eε (−με) , (1.97)

sec hε(με) = 2

Eε (με)+ Eε (−με) , (1.98)

csc hε(με) = 2

Eε (με)− Eε (−με) . (1.99)

The comparison plot of the nondifferentiable functions (1.94) and (1.95) whenε = ln 2/ ln 3 is shown in Figure 1.6.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

m

sinhe (mε)

coshe (mε)

Figure 1.6 The comparisons of the nondifferentiable functions (1.94) and (1.95) whenε = ln 2/ ln 3.


1.3 Definitions and properties of local fractionalderivative

We discuss the definitions and prosperities of local fractional derivative for nondiffer-entiable functions defined on a fractal set.

1.3.1 Definitions of local fractional derivative

Definition 1.3. Suppose that �(μ) ∈ Cε (a, b) and 0 < ε ≤ 1. For σ > 0 and0 < |μ− μ0| < δ, the limit

D(ε)� (μ0) = dε� (μ)

dμε

∣∣∣∣ μ=μ0 = limμ→μ0

�ε [�(μ)−�(μ0)]

(μ− μ0)ε (1.100)

exists and is finite, where�ε [�(μ)−�(μ0)] ∼= � (1 + ε) [�(μ)−�(μ0)]. In thiscase, D(ε)� (μ) is said to be the local fractional derivative of �(μ) of order ε atμ = μ0.

For our purposes, it is convenient to denote the local fractional derivative in the

form D(ε)� (μ0) or dε�(μ)dμε

∣∣∣ μ=μ0 .

If�(μ) is defined on the interval [μ, b), the left-hand local fractional derivative of�(μ) of order ε at μ = μ0 is defined to be

dε� (μ)

dμε

∣∣∣μ=μ−0

= limμ→μ−

0

�ε [�(μ)−�(μ0)]

(μ− μ0)ε , (1.101)

where

�ε [�(μ)−�(μ0)] ∼= � (1 + ε) [�(μ)−�(μ0)] ,

if the limit exists.If �(μ) is defined on (a,μ], the right-hand local fractional derivative of �(μ) of

order ε at μ = μ0 is defined to be

dε� (μ)

dμε

∣∣∣μ=μ+0

= limμ→μ+

0

�ε [�(μ)−�(μ0)]

(μ− μ0)ε , (1.102)

where

�ε [�(μ)−�(μ0)] ∼= � (1 + ε) [�(μ)−�(μ0)] ,

if the generalized limit exists.Suppose that

dε� (μ)

dμε

∣∣∣μ=μ+0


and

dε� (μ)

dμε

∣∣∣μ=μ−0

exist and

dε� (μ)

dμε

∣∣∣μ=μ+0

= dε� (μ)

dμε

∣∣∣μ=μ−0

.

Then, we have

dε� (μ)

dμε

∣∣∣∣ μ=μ0 = dε� (μ)

dμε

∣∣∣∣ μ=μ+0

= dε� (μ)

dμε

∣∣∣∣ μ=μ−0

. (1.103)

For 0 < ε ≤ 1, the fractal increment of �(μ) of order ε at μ = μ0 is defined by

� (1 + ε)�ε� (μ0) = �ε [�(μ)−�(μ0)] = D(ε)� (μ0) (�μ)ε +� (�μ)ε ,

(1.104)

where �μ is increment of μ and� → 0 as �μ→ 0.For 0 < ε ≤ 1, the local fractional differential of �(μ) of order ε at μ = μ0 is

defined by

dε� (μ0) = D(ε)� (μ0) (dμ)ε +� (dμ)ε . (1.105)

Suppose that there exists any point μ ∈ (a, b) such that

�(ε) (μ) = dε� (μ)

dμε= D(ε)� (μ) . (1.106)

In this case, Dε (a, b) is called a ε-local fractional derivative set.Property 1. Suppose that �(μ) ∈ Dε (a, b). Then, �(μ) ∈ Cε (a, b).

Proof. Using the formula (1.104), we arrive at

|�(μ)| =∣∣∣∣D(ε)� (μ0)

(�μ)ε

� (1 + ε) +� (�μ)ε

� (1 + ε) +�(μ0)

∣∣∣∣ . (1.107)

Taking the generalized limit of formula (1.107), we conclude

limμ→μ0

�(μ) = �(μ0) . (1.108)

For any μ0, we get the result.

Property 2. If �(μ) ∈ Dε (a, b) , then �(μ) is local fractional differentiable onthe domain I = (a, b).

Proof. From (1.104), we have the relation

�ε� (μ0) = D(ε)� (μ0) (�μ)ε +� (�μ)ε , (1.109)

where limμ→μ0 � = 0.


If we replace�ε� (μ0) and (�μ)ε by dε� (μ0) and (dμ)ε in (1.109), respectively,this identity yields

dε� (μ0) = D(ε)� (μ0) (dμ)ε +� (dμ)ε . (1.110)

Successively, making use of limμ→μ0 � = 0 in (1.100), we deduce the result.

Suppose that�(μ) ,�(μ) ∈ Dε (a, b). The local fractional differentiation rules ofnondifferentiable functions defined on fractal sets are listed as follows:

(a) D(ε) [�(μ)±�(μ)] = D(ε)� (μ)± D(ε)� (μ);(b) D(ε) [�(μ)� (μ)] = [D(ε)� (μ)]�(μ)+�(μ) [D(ε)� (μ)]; and(c) D(ε) [�(μ) /� (μ)] = {[

D(ε)� (μ)]�(μ)−�(μ) [D(ε)� (μ)]} /�2 (μ), provided

�(μ) = 0.

One observes that the formulas (a), (b), and (c) are presented to generalize thedifferentiation rules of the differentiable functions. These, in the Kolwankar-Gangalsense, are valid (e.g., [1–16]).

Setting �(μ) ∈ Dnε (a, b) and n = 2, the interchanging operator of the order of the

local fractional operators is defined as follows:

(dε

dμε⊕ dε

dμε

)�(μ) = d2ε� (μ)

dμ2ε. (1.111)

There is one mechanism which may indeed be applied to the local fractionalchain rule leading to the generalized chain rule of local fractional-order differentialoperator.

We present the local fractional chain rule via the interchanging operator ofnondifferential functions as follows [1, 16, 21]:

Suppose that �(μ) = (φ ⊗ ϕ) (μ). Then, we have

dε� (μ)

dμε= φ(ε) (ϕ)

[ϕ(1) (μ)

]ε, (1.112)

if φ(ε) (ϕ) and ϕ(1) (μ) exist.

Let C be a constant. The local fractional derivative of some of nondifferentiablefunctions defined on fractal sets are listed in Table 1.1.

The above results devoted to local fractional derivative were listed in [1], and theproofs of them are also found in Appendix B.

In order to derive them, one start with the new series expansion in the form

(φ + ϕ)nε =∞∑

i=0

(nεiε

)φ(n−i)εϕiε =

∞∑i=0

(nεiε

)φiεϕ(n−i)ε, (1.113)

where(nεiε

)= � (1 + nε)

� (1 + iε) � (1 + (n − i) ε). (1.114)


Table 1.1 Basic operations of local fractional derivative of some ofnondifferentiable functions defined on fractal sets

Original function Transformed function

C 0μkε/� (1 + kε) μ(k−1)ε/� (1 + (k − 1) ε)Eε (με) Eε (με)Eε (Cμε) CEε (Cμε)Eε (−με) −Eε (−με)Eε(μ2ε)

(2μ)ε Eε(μ2ε)

Eε(Cμ2ε

)(2μ)ε CEε

(Cμ2ε

)Eε(−μ2ε

) − (2μ)ε Eε(−μ2ε

)sinε (με) cosε (με)sinε (Cμε) C cosε (Cμε)cosε (με) − sinε (με)cosε (Cμε) −C sinε (Cμε)sinhε (με) coshε (με)sinhε (Cμε) C coshε (Cμε)coshε (με) − sinhε (με)coshε (Cμε) −C sinhε (Cμε)

In this case, we present three characters of the series expansion below:

(a) (φ + ϕ)nε = 1, when n = 0;(b) (φ + ϕ)nε = φε + ϕε , when n = 1; and(c) (φ + ϕ)ε = (2ϕ)ε = (2φ)ε , when φ = ϕ.

We notice (b) is true when it is defined on fractal sets [1].When nε = σ is a real number, a fractional series expansion via arbitrary powers

σ is presented as [69]

(φ + ϕ)σ =∞∑

i=0

(σ

i

)φσ−kϕk =

∞∑i=0

(σ

i

)φkϕσ−k, (1.115)

where(σ

i

)= � (1 + σ)� (1 + i) � (1 + σ − i)

. (1.116)

With the help of (1.113), the nondifferential difference takes the form

�ε [�(μ)−�(μ0)] = � (1 + ε)�ε� (μ0) ∼= � (1 + ε) [�(μ)−�(μ0)] ,

(1.117)

where

�ε� (μ0) =∞∑

i=0

(−1)i(ε

iε

)�(μ− iρ) (1.118)

with ρ = μ− μ0.


Adopting (1.117), we present two examples, namely,

dε

dμεμε

� (1 + ε) = lim�μ→0

1

� (1 + ε)� (1 + ε) [(μ+�μ)ε − με]

(�μ)ε= 1. (1.119)

dε

dμεμkε

� (1 + kε)= lim�μ→0

{� (1 + ε)� (1 + kε)

[(μ+�μ)kε − μkε

](�μ)ε

}

= lim�μ→0

⎧⎪⎪⎨⎪⎪⎩� (1 + ε)� (1 + kε)

[μkε + � (1 + kε)

� (1 + ε) � (1 + (k − 1) ε)μ(k−1)ε (�μ)ε + · · · + −μkε

](�μ)ε

⎫⎪⎪⎬⎪⎪⎭

= lim�μ→0

⎧⎪⎪⎨⎪⎪⎩� (1 + ε)� (1 + kε)

[� (1 + kε)

� (1 + ε) � (1 + (k − 1) ε)μ(k−1)ε (�μ)ε

](�μ)ε

⎫⎪⎪⎬⎪⎪⎭

= μ(k−1)ε

� (1 + (k − 1) ε). (1.120)

In this case, from (1.120) we have

dε

dμεEε(με) = dε

dμε

( ∞∑k=0

μkε

� (1 + kε)

)= 1 +

∞∑k=1

μkε

� (1 + kε), (1.121)

which leads to

1 +∞∑

k=1

μkε

� (1 + kε)=

∞∑k=0

μkε

� (1 + kε). (1.122)

Therefore, we conclude that

dε

dμεEε(με) = Eε

(με)

. (1.123)

1.3.2 Properties and theorems of local fractional derivatives

Theorem 1.2 (Local fractional Rolle’s theorem). Suppose that �(μ) ∈ Cε [a, b],�(μ) ∈ Dε (a, b), and �(a) = �(b) . Then, there exists a point μ0 ∈ (a, b) andε ∈ (0, 1] such that

�(ε) (μ0) = 0. (1.124)

Proof.

(a) Let �(μ) = 0 in [a, b]. Then, for all μ0 in (a, b), there is �(ε) (μ0) = 0.(b) Let �(μ) = 0 in [a, b].

Since �(μ) is a local fractional continuous function in the domain Cε [a, b], thereare points at which �(μ) attains its maximum and minimum values, denoted by and T , respectively.

Because �(μ) = 0, at least one of the values , T is not zero.


Suppose, for instance, = 0 and that �(μ0) = . In this case, we consider

�(μ0 +�μ) ≤ �(μ0) . (1.125)

Assuming that �μ > 0, there is

�ε [�(μ0 +�μ)−�(μ0)]

(�μ)ε≤ 0 (1.126)

such that

lim�μ→0

�ε [�(μ0 +�μ)−�(μ0)]

(�μ)ε≤ 0. (1.127)

In similar manner, we consider �μ < 0.Considering �(μ) ∈ Dε (a, b) and applying (1.113), there is �(ε) (μ0) = 0. As

similar argument can be applied in case of = 0 and T = 0.Therefore, there is the formula �(ε) (μ0) = 0.

There is a generalized local fractional Rolle’s theorem devoted to the localfractional derivative in Kolwankar and Gangal sense.

Theorem 1.3. Suppose�(μ) ∈ Cε [a, b] and �(μ) ∈ Dε (a, b) . Then, there existsa point μ0 ∈ (a, b) and ε ∈ (0, 1] such that

�(b)−�(a) = �(ε) (μ0)(b − a)ε

� (1 + ε) . (1.128)

Proof. Let us define the nondifferentiable function, which is given by

�(μ) = � (1 + ε){

[�(μ)−�(a)] − [�(b)−�(a)] (μ− a)ε

(b − a)ε

}(1.129)

with ε ∈ (0, 1].Then, there are �(a) = 0 and �(b) = 0.In this case, for μ0 ∈ (a, b) there is the following identity in the form

�(μ) = � (1 + ε){

[�(μ)−�(a)] − [�(b)−�(a)] (μ− a)ε

(b − a)ε

}. (1.130)

Therefore, we have the result.

Theorem 1.4. Suppose that �(μ) ∈ Cε [a, b] and �(μ) ∈ Dε (a, b) . Then, thereexist limμ→μ0 �(μ) = 0 and limμ→μ0 (μ) = 0, where K denotes either a realnumber or one of the symbols −∞, ∞. Suppose that limμ→μ0

[�(ε) (μ) /(ε) (μ)

] =K. Then,

limμ→μ0

[�(μ) / (μ)] = K. (1.131)


Proof. Let �(μ) ∈ Cε [a, b] and �(μ) ∈ Dε (a, b) . There is μ0 ∈ (a, b) such that�(μ0) = 0 and (μ0) = 0.

There is η ∈ (μ0,μ) such that

�(μ)

(μ)= �(μ)−�(μ0)

(μ)−(μ0)= �(ε) (η)

(ε) (η). (1.132)

As μ→ μ+0 , the identity

limμ→μ+

0

�(μ)

(μ)= limμ→μ+

0

�(μ)−�(μ0)

(μ)−(μ0)= limμ→μ+

0

�(ε) (μ0)

(ε) (μ0)= K (1.133)

holds.

In similar manner, when μ→ μ−0 , there is

limμ→μ−

0

�(μ)

(μ)= limμ→μ−

0

�(ε) (μ0)

(ε) (μ0)= K. (1.134)

Therefore, we get the result.

For more details regarding the proof of (1.121), we recommend to readers refs[1, 21, 70].

In order to demonstrate the above mechanism, we present elementary examples:Using (1.132), for μ→ 0, we have

Eε(με)− 1 ≈ με

� (1 + ε) (1.135)

such that

limμ→0

Eε (με)− 1με

� (1 + ε)= limμ→0

dε

dμε

[με

� (1 + ε)]

dε

dμε

[με

� (1 + ε)] = 1, (1.136)

limμ→0

sinε (με)με

� (1 + ε)= limμ→0

dε

dμε[sinε (με)]

dε

dμε

[με

� (1 + ε)] = lim

μ→0cosε

(με) = 1. (1.137)

Similarly, for μ→ 0 we conclude

1 − cosε(με) ≈ μ2ε

� (1 + 2ε)(1.138)

such that


limμ→0

1 − cosε (με)

μ2ε

� (1 + 2ε)

= limμ→0

dε

dμε

[μ2ε

� (1 + 2ε)

]dε

dμε

[μ2ε

� (1 + 2ε)

] = 1. (1.139)

1.4 Definitions and properties of local fractional integral

1.4.1 Definitions of local fractional integrals

Definition 1.4. Suppose ϕ (μ) ∈ Cε [a, b]. Then, we define the local fractionalintegral of ϕ (μ) of order ε(0 < ε ≤ 1) by

aI(ε)b ϕ (μ) = 1

� (1 + ε)∫ b

aϕ (μ) (dμ)ε = 1

� (1 + ε) lim�μk→0

N−1∑k=0

ϕ (μk) (�μk)ε,

(1.140)

where �μk = μk+1 − μk with μ0 = a < μ1 < · · · < μN−1 < μN = b.

Suppose the local fractional integral of ϕ (μ) on the closed interval [a, b] be equalto �.

For each ρ > 0, there exists 0 < |�μk| < δ such that∣∣∣∣∣�− 1

� (1 + ε) lim�μk→0

N−1∑k=0

ϕ (μk) (�μk)ε

∣∣∣∣∣ < ρε. (1.141)

In fact, we recall the condition of the Riemann integral that suppose ϕ (μ) isbounded on [a, b], then, a necessary and sufficient condition for the existence of∫ b

aϕ (μ) dμ (1.142)

is that ϕ (μ) has a Lebesgue measure zero.Will the proposed procedure lead to Riemann integral on fractal sets? The answer

is yes. The suggested mechanism may indeed be adapted to condition of the Riemannintegral leading to generalized condition of the Riemann integral on fractal sets.

The Riemann integral on fractal sets is stated as follows [1, 16, 21]:Let ϕ: ℘ → ℵ be a function defined on a fractal set ℘ of fractal dimension ε(0 <

ε < 1). Suppose ϕ (μ) is bounded on [a, b] (or ϕ (μ) ∈ Cε [a, b]). Then, a necessaryand sufficient condition for the existence of

1

� (1 + ε)∫ b

aϕ (μ) (dμ)ε (1.143)

is that a fractal set of local fractional continuity of ϕ (μ) has a generalized Lebesguemeasure zero.


We easily get the following result:Suppose ϕ (μ) ∈ Cε [a, b], then, ϕ (μ) is local fractional integral on [a, b].For convenience, we can write the following rules:

(a) aI(ε)b ϕ (μ) = 0 if a = b.

(b) aI(ε)b ϕ (μ) = −bI(ε)a ϕ (μ) if a < b.

(c) aI(ε)b ϕ (μ) = ϕ (μ) if ε = 0.

1.4.2 Properties and theorems of local fractional integrals

Suppose ϕ (μ) ,ϕ1 (μ), and ϕ2 (μ) ∈ Cε [a, b], the local fractional integral rulesof nondifferentiable functions defined on fractal sets are listed as follows[1, 16, 21]:

(a) aI(ε)b [ϕ1 (μ)+ ϕ2 (μ)] = aI(ε)b ϕ1 (μ)+ aI(ε)b ϕ2 (μ);

(b) aI(ε)b [Cϕ (μ)] = CaI(ε)b ϕ (μ), provided a constant C;

(c) aI(ε)b 1 = (b − a)ε /� (1 + ε);(d) aI(ε)b ϕ (μ) ≥ 0, provided ϕ (μ) ≥ 0;

(e)∣∣∣aI(ε)b ϕ (μ)

∣∣∣ ≤ aI(ε)b |ϕ (μ)|;(f) aI(ε)b ϕ (μ) = aI(ε)c ϕ (μ)+ cI(ε)b ϕ (μ), provided a < c < b; and

(g) aI(ε)b ϕ (μ) ∈ [T (b − a)ε /� (1 + ε) , (b − a)ε /� (1 + ε)], provided that the maximumand minimum values of ϕ (μ) are and T , respectively.

Theorem 1.5 (Mean value theorem for local fractional integrals). Suppose thatϕ (μ) ∈ Cε [a, b]. Then, there exists a point ξ in (a, b) such that

aI(ε)b ϕ (μ) = ϕ (ξ) (b − a)ε

� (1 + ε) . (1.144)

Proof. In view of ϕ (μ) ∈ Cε [a, b], we have

aI(ε)b ϕ (μ) ∈ [T (b − a)ε /� (1 + ε) , (b − a)ε /� (1 + ε)] , (1.145)

which leads us to

aI(ε)b ϕ (μ)

(b−a)ε

�(1+ε)∈ [T, ] . (1.146)

Therefore, for ξ ∈ (a, b) , we have

aI(ε)b ϕ (μ)

(b−a)ε

�(1+ε)= ϕ (ξ) , (1.147)

which yields the result.

Theorem 1.6. Suppose that ϕ (μ) ∈ Cε [a, b]. Then, for μ ∈ (a, b) , there exists afunction �(μ) given by


�(μ) = aI(ε)μ ϕ (μ) , (1.148)

with the following local fractional derivative:

∂ε� (μ)

∂με= ϕ (μ) . (1.149)

Proof. Let μ ∈ [a, b]; then, there exists μ+�μ ∈ [a, b] such that

�(μ) = aI(ε)μ+�μϕ (μ) . (1.150)

In this case, we present

�ε [�(μ+�μ)−�(μ)] =∫ μ+�μ

aϕ (μ) (dμ)ε −

∫ μaϕ (μ) (dμ)ε, (1.151)

which leads to

�ε [�(μ+�μ)−�(μ)] =∫ μ+�μ

μ

ϕ (μ) (dμ)ε. (1.152)

From (1.144), for ξ ∈ (a, b), we present the formula

μI(ε)μ+�μϕ (μ) = ϕ (ξ) (�μ)ε

� (1 + ε) , (1.153)

which yields that

μI(ε)μ+�μϕ (μ)(�μ)ε

�(1+ε)= ϕ (ξ) (1.154)

or

�ε [�(μ+�μ)−�(μ)](�μ)ε

= ϕ (ξ) . (1.155)

As �μ→ 0, we present

lim�μ→0

�ε [�(μ+�μ)−�(μ)](�μ)ε

= �(ε) (μ) = ϕ (ξ) . (1.156)

For �μ > 0, there exists a point μ = a such that

�(ε) (μ)

∣∣∣μ=a+ = ϕ (a+) . (1.157)

In a similar manner, for �μ < 0, there exists a point μ = b such that

�(ε) (μ)

∣∣∣μ=b− = ϕ (b−) . (1.158)

Hence, we get the result.


Theorem 1.7 (Newton–Leibniz formula of local fractional integrals). Suppose that

�(ε) (μ) = ϕ (μ) ∈ Cε [a, b] .

Then

aI(ε)b ϕ (μ) = �(b)−�(a) . (1.159)

Proof. Let us define the function �0 (μ) = aI(ε)μ ϕ (μ). Thus, we have

∂ε

∂με(�0 (μ)−�(μ)) = ∂ε

∂με�0 (μ)− ∂ε

∂με� (μ) = ϕ (μ)− ϕ (μ) = 0,

(1.160)

which leads to

�0 (μ)−�(μ) = C, (1.161)

with C be a constant.

Therefore, from (1.160), we have the following identity

aI(ε)b ϕ (μ) = �0 (b)−�0 (a) = �(b)−�(a) . (1.162)

Hence, we obtain the desired result.

Theorem 1.8 (Local fractional integration by parts). Suppose that ϕ1 (μ) ,ϕ2 (μ) ∈Cε [a, b], and ϕ1 (μ) ,ϕ2 (μ) ∈ Dε (a, b) . Then,

aI(ε)b

{[∂ε

∂μεϕ1 (μ)

]ϕ2 (μ)

}= [ϕ1 (μ) ϕ2 (μ)]

ba−aI(ε)b

{ϕ1 (μ)

[∂ε

∂μεϕ2 (μ)

]}.

(1.163)

Proof. We have

[ϕ1 (μ) ϕ2 (μ)]ba = aI(ε)b

{∂ε

∂με[ϕ1 (μ) ϕ2 (μ)]

}. (1.164)

Thus, there is

aI(ε)b

{[∂ε

∂μεϕ1 (μ)

]ϕ2 (μ)

}= [ϕ1 (μ) ϕ2 (μ)]

ba−aI(ε)b

{ϕ1 (μ)

[∂ε

∂μεϕ2 (μ)

]}.

(1.165)

Therefore, we obtain the desired result.Suppose D(kε)ϕ (μ) ∈ Cε (a, b), then, there is

D(kε){μ0 I(kε)μ ϕ (μ)

}= ϕ (μ) , (1.166)


where μ0I(kε)μ ϕ (μ) =k-times︷︸︸︷

μ0I(ε)μ · · · μ0 I(ε)μ ϕ (μ) and D(kε)ϕ (μ) =k-times︷︸︸︷

D(ε) · · · D(ε) ϕ (μ).

Theorem 1.9. Suppose that D(kε)ϕ (μ), D((k+1)ε)ϕ (μ) ∈ Cε (a, b) . Then, for 0 <ε < 1, there is a point μ0 ∈ (a, b) such that

μ0 I(kε)μ

[D(kε)ϕ (μ)

]− μ0I((k+1)ε)

μ

[D((k+1)ε)ϕ (μ)

]= D(kε)ϕ (μ0)

(μ− μ0)kε

� (1 + kε),

(1.167)

where μ0I(kε)μ ϕ (μ) =k-times︷︸︸︷

μ0 I(ε)μ · · · μ0I(ε)μ ϕ (μ) and D(kε)ϕ (μ) =k-times︷︸︸︷

D(ε) · · · D(ε) ϕ (μ) .

Proof. We present the formula

μ0 I((k+1)ε)μ

[D((k+1)ε)ϕ (μ)

]= μ0I(kε)μ

{μ0 I(ε)μ

[D((k+1)ε)ϕ (μ)

]}= μ0I(kε)μ

{D(kε)ϕ (μ)− D(kε)ϕ (μ0)

}= μ0I(kε)μ

[D(kε)ϕ (μ)

]− μ0I(kε)μ

[D(kε)ϕ (μ0)

].

(1.168)

Adopting the formula

μ0 I(kε)μ

[D(kε)ϕ (μ0)

]= D(kε)ϕ (μ0) μ0I(kε)μ 1

= D(kε)ϕ (μ0) μ0I((k−1)ε)μ

(μ− μ0)ε

� (1 + ε)= D(kε)ϕ (μ0)

(μ− μ0)kε

� (1 + kε), (1.169)

there is

μ0 I(kε)μ

[D(kε)ϕ (μ)

]− μ0I((k+1)ε)

μ

[D((k+1)ε)ϕ (μ)

]= D(kε)ϕ (μ0)

(μ− μ0)kε

� (1 + kε).

(1.170)

Therefore, we proved the result.

1.4.3 Local fractional Taylor’s theorem for nondifferentiablefunctions

Theorem 1.10 (Local fractional Taylor’s theorem). Suppose that

D((k+1)ε)ϕ (μ) ∈ Cε (a, b) .


Then, for k = 0, 1, . . . , n,

ϕ (μ) =n∑

k=0

D(kε)ϕ (μ0)

� (1 + kε)(μ− μ0)

kε + D((n+1)ε)ϕ (ξ)

� (1 + (n + 1) ε)(μ− μ0)

(n+1)ε

(1.171)

with a < μ0 < ξ < μ < b, ∀μ ∈ (a, b), where D(kε)ϕ (μ) =k-times︷︸︸︷

D(ε) · · · D(ε) ϕ (μ).

Proof. By making use of

μ0I(kε)μ

[D(kε)ϕ (μ)

]− μ0 I((k+1)ε)

μ

[D((k+1)ε)ϕ (μ)

]= D(kε)ϕ (μ0)

(μ− μ0)kε

� (1 + kε),

(1.172)

we conclude thatn∑

k=0

{μ0I(kε)μ

[D(kε)ϕ (μ)

]− μ0 I((k+1)ε)

μ

[D((k+1)ε)ϕ (μ)

]}

= ϕ (μ)− μ0 I((k+1)ε)μ

[D((k+1)ε)ϕ (μ)

]

=n∑

k=0

{D(kε)ϕ (μ0)

(μ− μ0)kε

� (1 + kε)

}. (1.173)

Thus, we show that

μ0I((k+1)ε)μ

[D((k+1)ε)ϕ (μ)

]= μ0 I(ε)μ

{μ0 I(kε)μ

[D((k+1)ε)ϕ (μ)

]}= D((k+1)ε)ϕ (ξ) μ0I((k+1)ε)

μ 1

= D((k+1)ε)ϕ (ξ)(μ− μ0)

(k+1)ε

� (1 + (k + 1) ε), (1.174)

where μ0 < ξ < μ, ∀μ ∈ (a, b).Therefore, we have proved the result.

Theorem 1.11. Suppose that

D((k+1)ε)ϕ (μ) ∈ Cε (a, b) .

Then, for k = 0, 1, . . . , n, there is

ϕ (μ) =n∑

k=0

D(kε)ϕ (μ0)

� (1 + kε)(μ− μ0)

kε + Rnε (μ− μ0) (1.175)



D(ε) · · · D(ε) ϕ (μ) andRnε (μ− μ0) = O ((μ− μ0)

nε).

Proof. Using (1.171), we can write∣∣∣∣Rnε (μ− μ0)

(μ− μ0)nε

∣∣∣∣ =∣∣∣∣∣ D((n+1)ε)ϕ (ξ)

� (1 + (k + 1) ε)

(μ− μ0)(n+1)ε

(μ− μ0)nε

∣∣∣∣∣ =∣∣∣∣ D((n+1)ε)ϕ (ξ)

� (1 + (k + 1) ε)(μ− μ0)

ε

∣∣∣∣ .(1.176)

Therefore, we conclude that∣∣∣∣Rnε (μ− μ0)

(μ− μ0)nε

∣∣∣∣ =∣∣∣∣ D((n+1)ε)ϕ (ξ)

� (1 + (k + 1) ε)(μ− μ0)

ε

∣∣∣∣ = 0. (1.177)


D((k+1)ε)ϕ (μ) ∈ Cε (a, b) .

Then, for k = 0, 1, . . . , n, there is

ϕ (μ) =n∑

k=0

D(kε)ϕ (0)

� (1 + kε)μkε + D((n+1)ε)ϕ (θμ)

� (1 + (n + 1) ε)μ(n+1)ε (1.178)

with 0 < θ < 1, ∀μ ∈ (a, b), where D(kε)ϕ (μ) =k-times︷︸︸︷

D(ε) · · · D(ε) ϕ (μ).

Proof. For μ0 = 0 and μ ∈ (a, b), from (1.175), we present

ϕ (μ) =n∑

k=0

D(kε)ϕ (0)

� (1 + kε)(μ− μ0)

kε + D((n+1)ε)ϕ (ξ)

� (1 + (n + 1) ε)μ(n+1)ε, (1.179)

where a < μ0 < ξ < μ < b.If ξ = θμ in (1.179), then, there is

D((n+1)ε)ϕ (ξ)

� (1 + (n + 1) ε)μ(n+1)ε = D((n+1)ε)ϕ (θμ)

� (1 + (n + 1) ε)μ(n+1)ε (1.180)

with 0 < θ < 1.

1.4.4 Local fractional Taylor’s series for elementary functions


D((k+1)ε)ϕ (μ) ∈ Cε (a, b) .

Then, for k = 0, 1, . . . , n,


ϕ (μ) =∞∑

k=0

D(kε)ϕ (μ0)

� (1 + kε)(μ− μ0)

kε (1.181)

with a < μ0 < μ < b, ∀μ ∈ (a, b) , where D(kε)ϕ (μ) =k-times︷︸︸︷

D(ε) · · · D(ε) ϕ (μ).

Proof. According to local fractional Taylor’s theorem, from (1.171), there is

ϕ (μ) = limμ→μ0

{n∑

k=0

D(kε)ϕ (μ0)

� (1 + kε)(μ− μ0)

kε + D((n+1)ε)ϕ (ξ)

� (1 + (n + 1) ε)(μ− μ0)

(n+1)ε

}

=∞∑

k=0

D(kε)ϕ (μ0)

� (1 + kε)(μ− μ0)

kε (1.182)


D(ε) · · · D(ε) ϕ (μ).

In this case, we present the following result.Suppose D((k+1)ε)ϕ (μ) ∈ Cε (a, b). Then, for k = 0, 1, . . . , n, there is

ϕ (μ) =∞∑

k=0

D(kε)ϕ (0)

� (1 + kε)μkε (1.183)

with a < 0 < μ < b, ∀μ ∈ (a, b), where D(kε)ϕ (μ) =k-times︷︸︸︷

D(ε) · · · D(ε) ϕ (μ).This series is said to be local fractional MacLaurin’s series of the function ϕ (μ).In this case, we present the following local fractional MacLaurin’s series of

elementary functions:

(a) Eε (με) =∑∞k=0

μkε

� (1 + kε);

(b) Eε (−με) =∑∞k=0

(−1)k μkε

� (1 + kε);

(c) sinε (με) =∑∞k=0

(−1)k μ(2k+1)ε

� (1 + (2k + 1) ε);

(d) cosε (με) =∑∞k=0

(−1)k μ2kε

� (1 + 2kε);

(e) sinhε (με) =∑∞k=0

μ(2k+1)ε

� (1 + (2k + 1) ε); and

(f) coshε (με) =∑∞k=0

μ2kε

� (1 + 2kε).

The proofs of them are listed in Appendix C. Let C be a constant. The localfractional integrals of some of nondifferentiable functions defined on fractal sets arelisted in Table 1.2.

Let m, n (m = n) be integrals. The local fractional integrals of some of nondif-ferentiable functions via Mittag–Leffler function defined on fractal sets are listedin Table 1.3.


Table 1.2 Basic operations of local fractional integral of some ofnondifferentiable functions defined on fractal sets


C Cμε/� (1 + ε)μkε/� (1 + kε) μ(k+1)ε/� (1 + (k + 1) ε)

Eε (με) Eε (με)− 1

Eε (Cμε)Eε(Cμε)−1

C

sinε (με) − [cosε (με)− 1]

sinε (Cμε)− [cosε (με)− 1]

C

cosε (με) sinε (με)

cosε (Cμε)sinε (Cμε)

Cμε

�(1+ε) sinε (Cμε) − 1C

[με

�(1+ε) cosε (Cμε)− 1C sinε (Cμε)

]με

�(1+ε) cosε (Cμε) 1C

{με

�(1+ε) sinε (Cμε)− 1C [cosε (Cμε)− 1]

}Eε (με) sinε (Cμε)

Eε (με) [sinε (Cμε)− C cosε (Cμε)] + C

1 + C2

Eε (με) cosε (Cμε)Eε (με) [cosε (Cμε)+ C sinε (Cμε)] − 1

1 + C2

Table 1.3 Basic operations of local fractional integral of some ofnondifferentiable functions via Mittag–Leffler function definedon fractal sets


sinε (με) 0

cosε (με) 0

sinε (mεμε) 0

cosε (mεμε) 0

sinε (mεμε) cosε (nεμε) 0

sinε (mεμε) cosε (mεμε) 0

sinε (mεμε) sinε (mεμε) πε/� (1 + ε)cosε (nεμε) cosε (nεμε) πε/� (1 + ε)sinε [(2n + 1) μ/2]ε

2ε sinε (μ/2)επε/� (1 + ε)


1.5 Local fractional partial differential equationsin mathematical physics

1.5.1 Local fractional partial derivatives

The general equation of the circle of Cantor type with fractal dimension ε (0 < ε ≤ 1)is given by

μ2ε + η2ε = a2ε, (1.184)

where a is the radius of the circle.Let �: ℘ → ℵ be a function defined on a fractal set ℘ of fractal dimension ε(0 <

ε < 1). A function�(μ, η) is local fractional continuous at the point (μ0, η0) if thereis a number τ > 0 such that

|�(μ, η)−�(μ0, η0)| < τε, (1.185)

where its circular δ neighborhood of (μ0, η0) is

(μ− μ0)2ε + (η − η0)

2ε < δ2ε. (1.186)

It is said to be the local fractional continuous if there is

lim(μ,η)→(μ0,η0)

� (μ, η) = �(μ0, η0) . (1.187)

Let �(μ, η) be defined in the domain ℘ of the μη-plane. The local fractionalpartial derivative operator of �(μ, η) of order ε(0 < ε < 1) with respect to μ inthe domain ℘ is defined as follows:

�(ε) (μ0, η) = ∂ε� (μ, η)

∂με

∣∣∣∣μ=μ0

= limμ→μ0

�ε [�(μ, η)−�(μ0, η)]

(μ0 − μ0)ε , (1.188)

where �ε [�(μ, η)−�(μ0, η)] ∼= � (1 + ε) [�(μ, η)−�(μ0, η)].The local fractional partial derivative operator of �(μ, η) of order ε(0 < ε < 1)

with respect to η in the domain ℘ is defined as follows:

�(ε) (μ, η0) = ∂ε� (μ, η)

∂ηε

∣∣∣∣η=η0

= limη→η0

�ε [�(μ, η)−�(μ, η0)]

(η0 − η0)ε , (1.189)

where �ε [�(μ, η)−�(μ, η0)] ∼= � (1 + ε) [�(μ, η)−�(μ, η0)].The local fractional partial derivative operator of �(μ, η) of higher order

(m + n) ε(0 < ε < 1) with respect to η and μ in the domain ℘ is defined asfollows:

∂ε

∂με· · · ∂

ε

∂με︸︷︷︸n-times

∂ε

∂ηε· · · ∂

ε

∂ηε︸︷︷︸m-times

�(μ, η) = ∂(m+m)ε� (μ, η)

∂με . . . ∂με︸︷︷︸n-times

∂ηε . . . ∂ηε︸︷︷︸m-times

= �(m+n)αηmμn (μ, η) ,

(1.190)


where m and n are positive integers.We have

�(μ, η) ∈ Cm+nε , (1.191)

if (1.190) holds.The local fractional gradient and Laplace operators of a local fractional scalar field

ϕ (μ, η, σ) in 3 fractal dimensional space are presented as

∇εϕ (μ, η, σ) = ∂εϕ (μ, η, σ)

∂μεeε1 + ∂

εϕ (μ, η, σ)

∂ηεeε2 + ∂

εϕ (μ, η, σ)

∂σ εeε3

and

∇2εϕ (μ, η, σ) = ∂2εϕ (μ, η, σ)

∂μ2ε + ∂2εϕ (μ, η, σ)

∂η2ε + ∂2εϕ (μ, η, σ)

∂σ 2ε ,

respectively.The local fractional gradient and Laplace operators of a local fractional scalar field

ϕ (μ, σ) in 2 fractal dimensional space are presented as

∇εϕ (μ, σ) = ∂εϕ (μ, σ)

∂μεeε1 + ∂

εϕ (μ, σ)

∂σ εeε2

and

∇2εϕ (μ, σ) = ∂2εϕ (μ, σ)

∂μ2ε + ∂2εϕ (μ, σ)

∂η2ε ,

respectively.The local fractional gradient and Laplace operators of a local fractional scalar field

ϕ (μ, σ) in 1 fractal dimensional space are presented as

∇εϕ (μ) = ∂εϕ (μ)

∂μεeε1

and

∇2εϕ (μ) = ∂2εϕ (μ, σ)

∂μ2ε ,

respectively.Here, we do not refer to Jacobian and inequality theory via local fractional partial

derivative operator [1, 16, 21, 70–72].

1.5.2 Linear and nonlinear partial differential equations inmathematical physics

In mathematical physics, the partial differential equations describing the physicalphenomena were always derived from the calculus involving the different kernelfunctions of differentiability and nondifferentiability. Theory of local fractionalcalculus was applied to solve the mathematical models from science and engineering,


such as vibrating strings, traffic flow, and mass and heat transfer in fractal dimensionaltime-space. Here, we consider the local fractional partial differential equations insense of the nondifferentiable characteristics [1, 73–88]. Here, we will put ourwork upon linear and nonlinear local fractional partial differential equations in1 + 1 fractal dimensional space and in 1 + 3 fractal dimensional space, suchas heat equation, wave equation, the Laplace equation, the Klein–Gordon equa-tion, the Schrödinger equation, diffusion equation, transport equation, the Pois-son equation, the linear Korteweg–de Vries equation, the Tricomi equation, theFokker–Planck equation, the Lighthill–Whitham–Richards equation, the Helmholtzequation, damped wave equation, dissipative wave equation, the Boussinesq equation,nonlinear wave equation, the Burgers equation, the forced Burgers equation, theinviscid Burgers equation, the nonlinear Korteweg–de Vries equation, the modifiedKorteweg–de Vries equation, the generalized Korteweg–de Vries equation, the non-linear Klein–Gordon equation, Maxwell’s equation, the Navier–Stokes equation, andEuler’s equation involving the local fractional partial derivative operator.

We now present some linear local fractional partial differential equations that areof important concern:

The local fractional heat equation in 1 + 1 fractal dimensional space takes the form

∂ε�(μ, τ)

∂τ ε− κ ∂

2ε�(μ, τ)

∂μ2ε = �(μ, τ), (1.192)

where κ is the thermal conductivity coefficient (a positive constant) and �(μ, τ) is anondifferentiable heat source.

The local fractional wave equation in 1 + 1 fractal dimensional space takes the form

∂2ε�(μ, τ)

∂τ 2ε −� ∂2ε�(μ, τ)

∂μ2ε = 0, (1.193)

where� is a constant.The local fractional Laplace equation in 1 + 1 fractal dimensional space takes

the form

∂2ε�(μ, η)

∂μ2ε + ∂2ε�(μ, η)

∂η2ε = 0. (1.194)

The local fractional Klein–Gordon equation in 1 + 1 fractal dimensional spacetakes the form

∂ε�(μ, τ)

∂τ ε− ∂

2ε�(μ, τ)

∂μ2ε = �(μ, τ). (1.195)

The local fractional Schrödinger equation in 1 + 1 fractal dimensional space takesthe form

iεhε∂ε�(μ, τ)

∂τ ε= − h2

ε

2m

∂2ε�(μ, τ)

∂μ2ε , (1.196)

where m and hε are constants.


Local fractional diffusion equation in 1 + 1 fractal dimensional space takes theform

∂ε�(μ, τ)

∂τ ε− D

∂2ε�(μ, τ)

∂μ2ε = 0, (1.197)

where D is a diffusive coefficient.The linear local fractional transport equation in 1 + 1 fractal dimensional space

takes the form

∂ε�(μ, τ)

∂τ ε+ ∂

ε�(μ, τ)

∂με= 0. (1.198)

The local fractional Poisson equation in 1 fractal dimensional space takes the form

∂2ε�(μ, η)

∂μ2ε + ∂2ε�(μ, η)

∂η2ε = �(μ, η), (1.199)

where �(μ, η) is a nondifferentiable function.The linear local fractional Korteweg–de Vries equation in 1 + 1 fractal dimensional

space takes the form

∂ε�(μ, τ)

∂τ ε+ ∂

ε�(μ, τ)

∂με+ ∂

3ε�(μ, τ)

∂μ3ε = 0. (1.200)

The local fractional wave equation of fractal transverse vibration of a beam takesthe form

∂2ε�(μ, η)

∂μ2ε + ∂4ε�(μ, η)

∂η4ε = 0. (1.201)

The local fractional Tricomi equation in 1 + 1 fractal dimensional space takesthe form

ηε

� (1 + ε)∂2ε�(μ, η)

∂μ2ε + ∂2ε�(μ, η)

∂η2ε = 0. (1.202)

The local fractional Fokker–Planck equation in 1 + 1 fractal dimensional spacetakes the form

∂ε�(μ, τ)

∂τ ε= ∂2ε�(μ, τ)

∂μ2ε − ∂ε�(μ, τ)

∂με. (1.203)

The linear local fractional Lighthill–Whitham–Richards equation on a finite lengthhighway is given by

∂ε�(μ, τ)

∂τ ε+ μ∂

ε�(μ, τ)

∂με= 0, (1.204)

where μ is a constant.


The linear local fractional homogeneous Helmholtz equation in 1 fractal dimen-sional space takes the form

∂2ε�(μ, η)

∂μ2ε + ∂2ε�(μ, η)

∂η2ε +��(μ, η) = 0, (1.205)

where� is a constant.The linear local fractional inhomogeneous Helmholtz equation in 1 fractal dimen-

sional space with nondifferentiable inhomogeneous term takes the form

∂2ε�(μ, η)

∂μ2ε + ∂2ε�(μ, η)

∂η2ε +��(μ, η) = �(μ, η), (1.206)

where� is a constant and �(μ, η) is a differentiable function.The linear local damped wave equation in 1 + 1 fractal dimensional space with

nondifferentiable inhomogeneous term takes the form

∂2ε�(μ, τ)

∂τ 2ε − ∂ε�(μ, τ)

∂τ ε− ∂

2ε�(μ, τ)

∂μ2ε = �(μ, τ), (1.207)

where �(μ, τ) is a nondifferentiable inhomogeneous term.The linear local homogeneous damped wave equation of fractal strings in 1 + 1

fractal dimensional space takes the form

∂2ε�(μ, τ)

∂τ 2ε − ∂ε�(μ, τ)

∂τ ε− ∂

2ε�(μ, τ)

∂μ2ε = 0. (1.208)

The local fractional inhomogeneous dissipative wave equation of fractal strings in1 + 1 fractal dimensional space takes the form

∂2ε�(μ, τ)

∂τ 2ε − ∂ε�(μ, τ)

∂τ ε− ∂

ε�(μ, τ)

∂με− ∂

2ε�(μ, τ)

∂μ2ε = �(μ, τ), (1.209)

where �(μ, τ) is a nondifferentiable inhomogeneous term.The local fractional inhomogeneous dissipative wave equation of fractal strings in

1 + 1 fractal dimensional space takes the form

∂2ε�(μ, τ)

∂τ 2ε − ∂ε�(μ, τ)

∂τ ε− ∂

ε�(μ, τ)

∂με− ∂

2ε�(μ, τ)

∂μ2ε = 0. (1.210)

The linear local fractional Boussinesq equation of fractal long water waves in1 + 1 fractal dimensional space takes the form

∂2ε�(μ, τ)

∂τ 2ε − ∂2ε�(μ, τ)

∂μ2ε − ∂4ε�(μ, τ)

∂μ2ε∂τ 2ε = 0. (1.211)

Here, we present some nonlinear local fractional partial differential equations that areof important concern:


The local fractional nonlinear wave equation for the velocity potential of fluid flowin 1 + 1 fractal dimensional space takes the form

∂2ε�(μ, τ)

∂τ 2ε = ω∂2ε�(μ, τ)

∂μ2ε +�φ∂2ε�(μ, τ)

∂μ2ε , (1.212)

where ω and � are constants.The nonlinear local fractional Burgers equation in 1 + 1 fractal dimensional space

is given by

∂ε�(μ, τ)

∂τ ε+�(μ, τ)

∂ε�(μ, τ)

∂με= κ ∂

2ε�(μ, τ)

∂μ2ε , (1.213)

where κ is a constant.The nonlinear local fractional forced Burgers equation in 1 + 1 fractal dimensional

space is given by

∂ε�(μ, τ)

∂τ ε+�(μ, τ)

∂ε�(μ, τ)

∂με= κ ∂

2ε�(μ, τ)

∂μ2ε +�(μ, τ), (1.214)

where �(μ, τ) is a forced source.The nonlinear local fractional inviscid Burgers equation in 1 + 1 fractal dimen-

sional space is given by

∂ε�(μ, τ)

∂τ ε+�(μ, τ)

∂ε�(μ, τ)

∂με= 0. (1.215)

The nonlinear local fractional transport equation in 1 + 1 fractal dimensional space isgiven by

∂ε�(μ, τ)

∂τ ε+�(μ, τ)

∂ε�(μ, τ)

∂με= �(μ, τ), (1.216)

where �(μ, τ) is a forced source.The nonlinear local fractional Korteweg–de Vries equation in 1 + 1 fractal

dimensional space is given by

∂ε�(μ, τ)

∂τ ε− R�(μ, τ)

∂ε�(μ, τ)

∂με+ ∂

3ε�(μ, τ)

∂μ3ε + ∂ε�(μ, τ)

∂με= 0 (1.217)

or

∂ε�(μ, τ)

∂τ ε+ S�(μ, τ)

∂ε�(μ, τ)

∂με− ∂

3ε�(μ, τ)

∂μ3ε = 0, (1.218)

where R and S are constants.The nonlinear local fractional modified Korteweg–de Vries equation in 1 + 1 fractal

dimensional space is given by

∂ε�(μ, τ)

∂τ ε+ ∂

3ε�(μ, τ)

∂μ3ε ± S�2(μ, τ)∂ε�(μ, τ)

∂με= 0, (1.219)


where S is a constant.The nonlinear local fractional generalized Korteweg–de Vries equation in 1 + 1

fractal dimensional space is given by

∂ε�(μ, τ)

∂τ ε+ S�(μ, τ)

∂ε�(μ, τ)

∂με− ∂

5ε�(μ, τ)

∂μ5ε = 0, (1.220)

where S is a constant.The nonlinear local fractional Klein–Gordon equation in 1 + 1 fractal dimensional

space is given by

∂2ε�(μ, τ)

∂τ 2ε − ∂2ε�(μ, τ)

∂μ2ε = �(�(μ, τ)) , (1.221)

where �(�(μ, τ)) is a nonlinear term related to �(μ, τ).The nonlinear local fractional Lighthill–Whitham–Richards equation on a finite

length highway is given by

∂2ε�(μ, τ)

∂τ 2ε + ξ ∂ε�(μ, τ)

∂με+ η�(μ, τ)

∂ε�(μ, τ)

∂με= 0, (1.222)

where ξ and η are constants.Here, we present some nonlinear local fractional partial differential equations in

1 + 3 fractal dimensional space that are of important concern:The linear local fractional wave equation for the velocity potential of fluid flow in

1 + 3 fractal dimensional space takes the form

ω∇2ε�(μ, η, σ , τ)− ∂2ε�(μ, η, σ , τ)

∂τ 2ε = 0, (1.223)

where ω are a constant.The local fractional Laplace equation arising in fractal electrostatics in 1 + 3 fractal

dimensional space takes the form

∇2ε�(μ, η, σ) = 0. (1.224)


∇2ε�(μ, η, σ) = �(μ, η, σ), (1.225)

where �(μ, η, σ) is a nondifferentiable function.The linear local fractional inhomogeneous Helmholtz equation in 3 fractal dimen-

sional space with nondifferentiable inhomogeneous term takes the form

∇2ε�(μ, η, σ)+��(μ, η, σ) = �(μ, η, σ), (1.226)

where� is a constant and �(μ, η, σ) is a differentiable function.The local fractional heat-conduction equation in 1 + 3 fractal dimensional space

takes the form

∂ε�(μ, η, σ , τ)

∂τ ε−�∇2ε�(μ, η, σ , τ) = H(μ, η, σ , τ), (1.227)


where � is the thermal conductivity coefficient and H(μ, η, σ , τ) is the heatsource.

The linear local homogeneous damped wave equation of fractal strings in 1 + 3fractal dimensional space takes the form

∂2ε�(μ, η, σ , τ)

∂τ 2ε − ∂ε�(μ, η, σ , τ)

∂τ ε− ∇2ε�(μ, η, σ , τ) = 0. (1.228)


∂2ε�(μ, η, σ , τ)

∂τ 2ε − ∂ε�(μ, η, σ , τ)

∂τ ε− ∇2ε�(μ, η, σ , τ)− ∇ε�(μ, η, σ , τ) = 0.

(1.229)

The local fractional diffusion equation in 1 + 3 fractal dimensional space takesthe form

dε�(μ, η, σ , τ)

dτ ε− ∇εD (φ)∇ε�(μ, η, σ , τ)− D (φ)∇2ε�(μ, η, σ , τ) = 0,

(1.230)

where D (φ) is diffusion coefficient related to �(μ, η, σ , τ).The local fractional Schrödinger equation with the nondifferentiable potential

function in 1 + 3 fractal dimensional space takes the form

iαhα∂α�(μ, η, σ , τ)

∂τα= − h2

α

2m∇2ε�(μ, η, σ , τ)+�(μ, η, σ)�(μ, η, σ , τ),

(1.231)

where �(μ, η, σ) is the nondifferentiable potential function.The nonlinear local fractional wave equation for the velocity potential of fluid flow

in 1 + 3 fractal dimensional space takes the form

∂2ε�(μ, η, σ , τ)

∂τ 2ε = ω∇2ε�(μ, η, σ , τ)+��(μ, η, σ , τ)∇2ε�(μ, η, σ , τ),

(1.232)

where ω and � are constants.Systems of local fractional Maxwell’s equations in 1 + 3 fractal dimensional space

take the form

∇ε · D (μ, η, σ , τ) = ρ (μ, η, σ , τ) , (1.233)

∇ε × H (μ, η, σ , τ) = Jε (μ, η, σ , τ)+ ∂εD (μ, η, σ , τ)

∂τ ε, (1.234)

∇ε × E (μ, η, σ , τ) = −∂εB (μ, η, σ , τ)

∂τ ε, (1.235)

∇ε · B (μ, η, σ , τ) = 0, (1.236)


where ρ (μ, η, σ , τ) is the fractal electric charge density, D (μ, η, σ , τ) is electricdisplacement in the fractal electric field, H (μ, η, σ , τ) is the magnetic field strengthin the fractal field, E (μ, η, σ , τ) is the electric field strength in the fractal field,Jε (μ, η, σ , τ) is the conductive current, and B (μ, η, σ , τ) is the magnetic inductionin the fractal field, and the constitutive relationships in fractal electromagnetic can bewritten as

D (μ, η, σ , τ ) = εfE (μ, η, σ , τ) (1.237)

and

H (μ, η, σ , τ ) = μfB (μ, η, σ , τ ) , (1.238)

with the fractal dielectric permittivity εf and the fractal magnetic permeability μf.Systems of the local fractional compressible Navier–Stokes equations in 1 + 3

fractal dimensional space take the form

∂ερ

∂τ ε+ ∇ε · (ρυ) = 0, (1.239)

ρ

(∂ευ

∂τ ε+ υ · ∇αυ

)= −∇εp + 1

3μ∇ε(∇ε · υ

)+ μ∇2ευ + ρb, (1.240)

ρ

[∂ε (θ + φ)∂τ ε

+ υ · ∇ε (θ + φ)]

= −∇ε ·(pυ)+υ ·(∇ε · J)+ρb·υ+K2ε∇ε ·q,

(1.241)

where υ (μ, η, σ , τ ) is the fractal fluid velocity, μ is the fractal shear moduli ofviscosity, p (μ, η, σ , τ) is the thermodynamic pressure, ρ (μ, η, σ , τ ) is the fractalfluid density, φ (μ, η, σ , τ ) is the kinetic energy per unit of mass, b (μ, η, σ , τ) is theexternal force per unit of mass, J (μ, η, σ , τ ) is the fractal Cauchy stress tensor, andθ (μ, η, σ , τ ) is the internal energy per unit of mass.

Systems of the local fractional incompressible Navier–Stokes equations in 1 + 3fractal dimensional space take the form

∇ε · υ = 0, (1.242)

ρ

(∂ευ

∂τ ε+ υ · ∇ευ

)= −∇εp + μ∇2ευ + ρb, (1.243)

ρ

[∂ε (θ + φ)∂τ ε

+ υ · ∇ε (θ + φ)]

= −∇ε ·(pυ)+υ ·(∇ε · J)+ρb·υ+K2ε∇ε ·q,

(1.244)

where υ (μ, η, σ , τ ) is the fractal fluid velocity, μ is the fractal shear moduli ofviscosity, p (μ, η, σ , τ) is the thermodynamic pressure, ρ (μ, η, σ , τ ) is the fractalfluid density, φ (μ, η, σ , τ ) is the kinetic energy per unit of mass, b (μ, η, σ , τ) is the


external force per unit of mass, J (μ, η, σ , τ) is the fractal Cauchy stress tensor, andθ (μ, η, σ , τ) is the internal energy per unit of mass.

Systems of local fractional compressible Euler’s equation in 1 + 3 fractal dimen-sional space take the form

∂ερ

∂τ ε+ ∇ε · (ρυ) = 0, (1.245)

ρ

(∂ευ

∂τ ε+ υ · ∇ευ

)= −∇εp + ρb, (1.246)

ρ

[∂ε (θ + φ)∂τ ε

+ υ · ∇ε (θ + φ)]

= −∇ · (pυ) , (1.247)

where υ (μ, η, σ , τ) is the fractal fluid velocity, p (μ, η, σ , τ) is the thermodynamicpressure, ρ (μ, η, σ , τ) is the fractal fluid density, φ (μ, η, σ , τ) is the kinetic energyper unit of mass, b (μ, η, σ , τ ) is the external force per unit of mass, and θ (μ, η, σ , τ)is the internal energy per unit of mass.

Systems of local fractional incompressible Euler’s equation in 1 + 3 fractaldimensional space take the form

∂ερ

∂τ ε+ υ · ∇ερ = 0, (1.248)

∇ε · υ = 0, (1.249)

ρ

(∂ευ

∂τ ε+ υ · ∇ευ

)= −∇εp + ρb, (1.250)

where υ (μ, η, σ , τ) is the fractal fluid velocity, p (μ, η, σ , τ) is the thermodynamicpressure, ρ (μ, η, σ , τ ) is the fractal fluid density, and b (μ, η, σ , τ ) is the externalforce per unit of mass.

1.5.3 Applications of local fractional partial derivative operatorto coordinate systems

One of interesting things in coordinate systems is that many three-dimensionalcoordinate systems may be used to convert between them. The Cantorian coordinatesystem is an analogous version of the Cartesian coordinate system on fractal sets.Similarity, we may transfer Cantorian coordinate system into Cantor-type cylindricalcoordinates and Cantor-type spherical coordinates via Mittag–Leffler function definedon the fractal sets [22, 88, 89]. Here, we present the basic theory of Cantor-typecircular coordinates, Cantor-type cylindrical coordinates, and Cantor-type sphericalcoordinates as follows.


For R ∈ (0, +∞) and θ ∈ (0, 2π), the Cantor-type circular coordinate system iswritten as{

με = Rε cosε (θε)σ ε = Rε sinε (θε)

, (1.251)

where R > 0 and 0 < θ < 2π .A local fractional vector is written as

r =Rε cosε(θε)

eε1 + Rε sinε(θε)

eε2

=rReεR + rθeεθ . (1.252)

Hence, we have a local fractional vector{eεR = cosε (θε) eε1 + sinε (θε) eε2,eεθ = − sinε (θε) eε1 + cosε (θε) eε2

(1.253)

such that the local fractional gradient operator and local fractional Laplace operatorin the Cantor-type circular coordinate system is presented as

∇εϕ (R, θ) = eεR∂εϕ

∂Rε+ eεθ

1

Rε∂εϕ

∂θε, (1.254)

∇2εϕ (R, θ) = ∂2εϕ

∂R2ε + 1

R2ε

∂2εϕ

∂θ2ε + 1

Rε∂εϕ

∂Rε. (1.255)

The Cantor-type cylindrical coordinates can be written as follows:⎧⎨⎩με = Rε cosε (θε)ηε = Rε sinε (θε)σ ε = σε

(1.256)

with R ∈ (0, +∞), z ∈ (−∞, +∞), θ ∈ (0,π], and μ2ε + η2ε = R2ε.Adopting (1.256), we have

∇ε · r = ∂εrR

∂Rε+ 1

Rε∂εrθ∂θε

+ rR

Rε+ ∂

αrz

∂σ ε(1.257)

and

∇ε×r =(

1

Rε∂εrθ∂θα

− ∂εrθ∂σ ε

)eεR+

(∂εrR

∂σ ε− ∂

εrz

∂Rε

)eεθ+

(∂εrθ∂Rε

+ rR

Rε− 1

Rε∂εrR

∂θε

)eεσ ,

(1.258)

where

r = Rε cosε(θε)

eε1 + Rε sinε(θε)

eε2 + σεeε3= rReεR + rθe

εθ + rσ eεσ . (1.259)

Hence, we get the local fractional gradient operator and local fractional Laplaceoperator in the Cantor-type cylindrical coordinate system


∇εϕ (R, θ , σ) = eεR∂εϕ

∂Rε+ eεθ

1

Rε∂εϕ

∂θε+ eεσ

∂εϕ

∂σ ε, (1.260)

∇2εϕ (R, θ , σ) = ∂2εϕ

∂R2ε + 1

R2ε

∂2εϕ

∂θ2ε + 1

Rε∂εϕ

∂Rε+ ∂2εϕ

∂σ 2ε , (1.261)

where a local fractional vector is given as⎧⎨⎩

eεR = cosε (θε) eε1 + sinα (θε) eε2,

eεθ = − sinε (θε) eε1 + cosα (θε) eε2,

eεσ = eε3.(1.262)

For R ∈ (0, +∞), η ∈ (0,π), θ ∈ (0, 2π), and μ2ε + η2ε + σ 2ε = R2ε, theCantor-type spherical coordinate system is written as⎧⎨

⎩με = Rε cosε (ϑε) cosε (θε) ,ηε = Rε cosε (ϑε) sinε (θε) ,

σε = Rε sinε (ϑε) .(1.263)

A local fractional vector is written as

r = Rε cosε(ϑε)

cosε(θε)

eε1 + Rε cosε(ϑε)

sinε(θε)

eε2 + Rε sinε(ϑε)

eε3= rReεR + rϑeεϑ + rθe

εθ . (1.264)

Hence, we get the local fractional gradient operator and local fractional Laplaceoperator in the Cantor-type spherical coordinate system

∇εϕ (R,ϑ , θ) = eεR∂εϕ

∂Rε+ eεϑ

1

Rε∂εϕ

∂ϑε+ eεθ

1

Rε1

sinε (ϑε)

∂εϕ

∂θε, (1.265)

∇2εϕ (R,ϑ , θ) =∂2εϕ

∂R2ε + 1

R2ε

1

sinε (ϑε)

∂ε

∂ϑε

(sinε

(ϑε) ∂εϕ∂ϑε

)

+ 2

Rε∂εϕ

∂Rε+ 1

R2ε

1

sin2ε (ϑ

ε)

∂2εϕ

∂θ2ε , (1.266)

where a local fractional vector is determined by⎧⎪⎨⎪⎩

eεR = sinε (ϑε) cosε (θε) eε1 + sinε (ϑε) sinε (θε) eε2 + cosε (ϑε) eε3,

eεϑ = cosε (ϑε) cosε (θε) eε1 + cosε (ϑε) sinε (θε) eε2 − sinε (ϑε) eε3,

eεθ = − sinε (θε) eε1 + cosε (θε) eε2.

(1.267)

The local fractional gradient operator of ϕ (R) in the Cantor-type cylindricalsymmetry form is written as

∇εϕ (R) = eεR∂εϕ

∂Rε. (1.268)


Similarly, the local fractional Laplace operator in the Cantor-type cylindricalsymmetry form can be written as

∇2εϕ (R) = ∂2εϕ

∂R2ε + 1

Rε∂εϕ

∂Rε. (1.269)

The local fractional gradient operator of ϕ (R) in the Cantor-type sphericalsymmetry form is written as

∇εϕ (R) = eεR∂εϕ

∂Rε. (1.270)

Similarly, the local fractional Laplace operator takes in the Cantor-type sphericalsymmetry form

∇2εϕ (R) = ∂2εϕ

∂R2ε + 2

Rε∂εϕ

∂Rε. (1.271)

The theory of the Cantor-type circular coordinates, Cantor-type cylindrical coordi-nates, and Cantor-type spherical coordinates is presented in Appendix D.

1.5.4 Alternative observations of local fractional partialdifferential equations

Based upon the basic theory of Cantor-type circular coordinates, Cantor-type cylin-drical coordinates, and Cantor-type spherical coordinates, we will discuss the localfractional partial differential equations via local fractional partial derivative operator.

We now present some applications of the Cantor-type circular coordinates foradopting wave equation, the Laplace equation, the Poisson equation, the Helmholtzequation, heat-conduction equation, damped wave equation, dissipative wave equa-tion, diffusion equation, and Maxwell’s equations in fractal dimensional space.

The linear local fractional wave equation for the velocity potential of fluid flow in1 + 2 fractal dimensional space takes the form

ω∇2ε�(μ, σ , τ)− ∂2ε�(μ, σ , τ)

∂τ 2ε = 0, (1.272)

where ω is a constant.The linear local fractional wave equation for the velocity potential of fluid flow in

Cantor-type circular coordinate system takes the form

ω

(∂2ε�(R, θ , τ)

∂R2ε + 1

R2ε

∂2ε�(R, θ , τ)

∂θ2ε + 1

Rε∂ε�(R, θ , τ)

∂Rε

)− ∂

2ε�(R, θ , τ)

∂τ 2ε = 0,

(1.273)

where ω is a constant.The local fractional Laplace equation arising in fractal electrostatics in 1 + 2 fractal

dimensional space takes the form

∇2ε�(μ, σ) = 0. (1.274)


The linear local fractional Laplace equation in Cantor-type circular coordinatesystem takes the form

∂2ε�(R, θ)

∂R2ε + 1

R2ε

∂2ε�(R, θ)

∂θ2ε + 1

Rε∂ε�(R, θ)

∂Rε= 0. (1.275)


∇2ε�(μ, σ) = �(μ, σ), (1.276)

where �(μ, σ) is a nondifferentiable function.The linear local fractional Poisson equation in Cantor-type circular coordinate

system takes the form

∂2ε�(R, θ)

∂R2ε + 1

R2ε

∂2ε�(R, θ)

∂θ2ε + 1

Rε∂ε�(R, θ)

∂Rε= �(R, θ). (1.277)

The linear local fractional homogeneous Helmholtz equation in 2 fractal dimen-sional space with nondifferentiable inhomogeneous term takes the form

∇2ε�(μ, σ)+��(μ, σ) = 0, (1.278)

where � is a constant and �(μ, σ) is a differentiable function.The linear local fractional homogeneous Helmholtz equation in Cantor-type

circular coordinate system takes the form

∂2ε�(R, θ)

∂R2ε + 1

R2ε

∂2ε�(R, θ)

∂θ2ε + 1

Rε∂ε�(R, θ)

∂Rε+��(R, θ) = 0, (1.279)

where � is a constant.The local fractional heat-conduction equation in 1 + 2 fractal dimensional space

takes the form

∂ε�(μ, σ , τ)

∂τ ε−�∇2ε�(μ, σ , τ) = H(μ, σ , τ), (1.280)

where � is the thermal conductivity coefficient and H(μ, σ , τ) is the heat source.The local fractional heat-conduction equation in Cantor-type circular coordinate

system takes the form

∂ε�(R, θ , τ)

∂τ ε−�

(∂2ε�(R, θ , τ)

∂R2ε + 1

R2ε

∂2ε�(R, θ , τ)

∂θ2ε + 1


∂Rε

)= H(R, θ , τ), (1.281)

where� is the thermal conductivity coefficient and H(R, θ , τ) is the heat source.The linear local homogeneous damped wave equation of fractal strings in 1 + 2

fractal dimensional space takes the form

∂2ε�(R, θ , τ)

∂τ 2ε − ∂ε�(R, θ , τ)

∂τ ε− ∇2ε�(μ, σ , τ) = 0. (1.282)

The linear local homogeneous damped wave equation of fractal strings in Cantor-typecircular coordinate system takes the form


∂2ε�(μ, σ , τ)

∂τ 2ε − ∂ε�(μ, σ , τ)

∂τ ε

−(∂2ε�(R, θ , τ)

∂R2ε + 1

R2ε

∂2ε�(R, θ , τ)

∂θ2ε + 1


∂Rε

)= 0.

(1.283)


∂2ε�(μ, σ , τ)

∂τ 2ε − ∂ε�(μ, σ , τ)

∂τ ε− ∇2ε�(μ, σ , τ)− ∇ε�(μ, σ , τ) = 0. (1.284)

The local fractional inhomogeneous dissipative wave equation of fractal strings inCantor-type circular coordinate system takes the form

∂2ε�(R, θ , τ)

∂τ 2ε − ∂ε�(R, θ , τ)

∂τ ε

−(∂2ε�(R, θ , τ)

∂R2ε + 1

R2ε

∂2ε�(R, θ , τ)

∂θ2ε + 1


∂Rε

)

−(

eεR∂ε�(R, θ , τ)

∂Rε+ eεθ

1


∂θε

)= 0. (1.285)

The local fractional diffusion equation in 1 + 2 fractal dimensional space has thefollowing form:

dε�(μ, σ , τ)

dτ ε− D∇2ε�(μ, σ , τ) = 0, (1.286)

where D is the diffusive coefficient.The local fractional diffusion equation in Cantor-type circular coordinate system

takes the following form:

dε�(R, θ , τ)

dτ ε−D

(∂2ε�(R, θ , τ)

∂R2ε + 1

R2ε

∂2ε�(R, θ , τ)

∂θ2ε + 1


∂Rε

)= 0,

(1.287)

where D is the diffusive coefficient.Systems of local fractional Maxwell’s equations in 1 + 2 fractal dimensional space

take the following form:

∇ε · D (μ, σ , τ ) = ρ (μ, σ , τ) , (1.288)

∇ε × H (μ, σ , τ ) = Jε (μ, σ , τ )+ ∂εD (μ, σ , τ)

∂τ ε, (1.289)

∇ε × E (μ, σ , τ) = −∂εB (μ, σ , τ)

∂τ ε, (1.290)

∇ε · B (μ, σ , τ) = 0, (1.291)


where ρ (μ, σ , τ) is the fractal electric charge density, D (μ, σ , τ) is electric displace-ment in the fractal electric field, H (μ, σ , τ) is the magnetic field strength in thefractal field, E (μ, σ , τ ) is the electric field strength in the fractal field, Jε (μ, σ , τ)is the conductive current, and B (μ, σ , τ) is the magnetic induction in the fractalfield, and the constitutive relationships in fractal electromagnetic can be writtenas follows:

D (μ, σ , τ ) = εfE (μ, σ , τ) (1.292)

and

H (μ, σ , τ ) = μfB (μ, σ , τ ) , (1.293)

with the fractal dielectric permittivity εf and the fractal magnetic permeability μf.We can write the systems of local fractional Maxwell’s equations in Cantor-type

circular coordinate system as follows:(eεR∂ε

∂Rε+ eεθ

1

Rε∂ε

∂θε

)· D(R, θ , τ) = ρ(R, θ , τ), (1.294)

(eεR∂ε

∂Rε+ eεθ

1

Rε∂ε

∂θε

)× H (R, θ , τ) = Jε (R, θ , τ)+ ∂

εD (R, θ , τ)

∂τ ε, (1.295)

(eεR∂ε

∂Rε+ eεθ

1

Rε∂ε

∂θε

)× E (R, θ , τ) = −∂

εB (R, θ , τ)

∂τ ε, (1.296)

(eεR∂ε

∂Rε+ eεθ

1

Rε∂ε

∂θε

)· B (R, θ , τ) = 0, (1.297)

where ρ (R, θ , τ) is the fractal electric charge density, D (R, θ , τ) is the electricdisplacement in the fractal electric field, H (R, θ , τ) is the magnetic field strength inthe fractal field, E (R, θ , τ) is the electric field strength in the fractal field, Jε (R, θ , τ)is the conductive current, and B (R, θ , τ) is the magnetic induction in the fractal field,and the constitutive relationships in fractal electromagnetic can be written as

D (R, θ , τ) = εfE (R, θ , τ) (1.298)

and

H (R, θ , τ) = μfB (R, θ , τ) , (1.299)

with the fractal dielectric permittivity εf and the fractal magnetic permeability μf.The form of the linear local fractional wave equation for the velocity potential of

fluid flow in Cantor-type cylindrical coordinate system is given by

ω

(∂2ε�

∂R2ε+ 1

R2ε

∂2ε�

∂θ2ε+ 1

Rε∂ε�

∂Rε+ ∂

2ε�

∂σ 2ε

)− ∂

2ε�

∂τ 2ε= 0, (1.300)


where ω is a constant and � = �(R, θ , σ , τ).The linear local fractional wave equation for the velocity potential of fluid flow in

Cantor-type spherical coordinate system takes the following form:

∂2ε�

∂R2ε + 1

R2ε sinε (ϑε)

∂ε

∂ϑε

(sinε

(ϑε) ∂ε�∂ϑε

)

+ 2

Rε∂ε�

∂Rε+ 1

R2ε sin2ε (ϑ

ε)

∂2ε�

∂θ2ε − ∂2ε�

∂τ 2ε = 0, (1.301)

where ω is a constant and � = �(μ, θ ,ϑ , τ).The local fractional Laplace equation in Cantor-type cylindrical coordinate system


∂2ε�

∂R2ε + 1

R2ε

∂2ε�

∂θ2ε + 1

Rε∂ε�

∂Rε+ ∂

2ε�

∂σ 2ε = 0, (1.302)

where � = �(R, θ , σ , τ).The linear local fractional Laplace equation in Cantor-type spherical coordinate

system takes the following form:

∂2ε�

∂R2ε + 1

R2ε sinε (ϑε)

∂ε

∂ϑε

(sinε


)+ 2

Rε∂ε�

∂Rε+ 1

R2ε sin2ε (ϑ

ε)

∂2ε�

∂θ2ε = 0,

(1.303)

where � = �(μ, θ ,ϑ , τ).The local fractional Poisson equation in Cantor-type cylindrical coordinate system


∂2ε�

∂R2ε + 1

R2ε

∂2ε�

∂θ2ε + 1

Rε∂ε�

∂Rε+ ∂

2ε�

∂σ 2ε = �(R, θ , σ) , (1.304)

where � = �(R, θ , σ) and is a nondifferentiable function.The linear local fractional Poisson equation in Cantor-type spherical coordinate

system takes the following form:

∂2ε�

∂R2ε + 1

R2ε sinε (ϑε)

∂ε

∂ϑε

(sinε


)+ 2

Rε∂ε�

∂Rε+ 1

R2ε sin2ε (ϑ

ε)

∂2ε�

∂θ2ε = �,

(1.305)

where � = �(μ, θ ,ϑ).The linear local fractional homogeneous Helmholtz equation in Cantor-type

cylindrical coordinate system takes the following form:

∂2ε�

∂R2ε + 1

R2ε

∂2ε�

∂θ2ε + 1

Rε∂ε�

∂Rε+ ∂

2ε�

∂σ 2ε +�� = 0, (1.306)

where� is a constant and � = �(R, θ , σ).


The linear local fractional homogeneous Helmholtz equation in Cantor-typespherical coordinate system takes the following form:

∂2ε�

∂R2ε + 1

R2ε sinε (ϑε)

∂ε

∂ϑε

(sinε


)

+ 2

Rε∂ε�

∂Rε+ 1

R2ε sin2ε (ϑ

ε)

∂2ε�

∂θ2ε +�� = 0, (1.307)

where � is a constant and � = �(μ, θ ,ϑ).We recall that the local fractional heat-conduction equation in Cantor-type cylin-

drical coordinate system takes the following form:

∂ε�

∂τε−�

(∂2ε�

∂R2ε + 1

R2ε

∂2ε�

∂θ2ε + 1

Rε∂ε�

∂Rε+ ∂

2ε�

∂σ 2ε

)= H, (1.308)

where� is the thermal conductivity coefficient� = �(R, θ , σ , τ) and H (R, θ , σ , τ)

is the heat source.The local fractional heat-conduction equation in Cantor-type spherical coordinate

system is given by

∂ε�

∂τε−�

(∂2ε�

∂R2ε + 1

R2ε sinε (ϑε)

∂ε

∂ϑε

(sinε


)

+ 2

Rε∂ε�

∂Rε+ 1

R2ε sin2ε (ϑ

ε)

∂2ε�

∂θ2ε

)= H, (1.309)

where� is the thermal conductivity coefficient,� = �(μ, θ ,ϑ , τ), and H (μ, θ ,ϑ , τ)is the heat source.

The linear local homogeneous damped wave equation of fractal strings in Cantor-type cylindrical coordinate system can be written as follows:

∂2ε�

∂τ 2ε − ∂ε�

∂τε−(∂2ε�

∂R2ε + 1

R2ε

∂2ε�

∂θ2ε + 1

Rε∂ε�

∂Rε+ ∂

2ε�

∂σ 2ε

)= 0 (1.310)

where � = �(R, θ , σ , τ).The linear local homogeneous damped wave equation of fractal strings in Cantor-

type spherical coordinate system has the following form:

∂2ε�

∂τ 2ε − ∂ε�

∂τε−(∂2ε�

∂R2ε + 1

R2ε sinε (ϑε)

∂ε

∂ϑε

(sinε


)

+ 2

Rε∂ε�

∂Rε+ 1

R2ε sin2ε (ϑ

ε)

∂2ε�

∂θ2ε

)= 0, (1.311)

where � = �(μ, θ ,ϑ , τ).


The local fractional inhomogeneous dissipative wave equation of fractal strings inCantor-type cylindrical coordinate system takes the following form:

∂2ε�

∂τ 2ε − ∂ε�

∂τε−(∂2ε�

∂R2ε + 1

R2ε

∂2ε�

∂θ2ε + 1

Rε∂ε�

∂Rε+ ∂

2ε�

∂σ 2ε

)

−(

eεR∂ε�

∂Rε+ eεθ

1

Rε∂ε�

∂θε+ eεσ

∂ε�

∂σε

)= 0, (1.312)

where � = �(R, θ , σ , τ).The local fractional inhomogeneous dissipative wave equation of fractal strings in

Cantor-type spherical coordinate system takes the following form:

∂2ε�

∂τ2ε− ∂ε�

∂τε

−(∂2ε�

∂R2ε+ 1

R2ε sinε (ϑε)

∂ε

∂ϑε

(sinε


)+ 2

Rε∂ε�

∂Rε+ 1

R2ε sin2ε (ϑ

ε)

∂2ε�

∂θ2ε

)

−(

eεR∂ε�

∂Rε+ eεϑ

1

Rε∂ε�

∂ϑε+ eεθ

1

Rε1

sinε (ϑε)

∂ε�

∂θε

)= 0, (1.313)

where � = �(μ, θ ,ϑ , τ).The local fractional diffusion equation in Cantor-type cylindrical coordinate system


dε�

dτ ε− D

(∂2ε�

∂R2ε + 1

R2ε

∂2ε�

∂θ2ε + 1

Rε∂ε�

∂Rε+ ∂

2ε�

∂σ 2ε

)= 0, (1.314)

where D is the diffusive coefficient and � = �(R, θ , σ , τ).The local fractional diffusion equation in Cantor-type spherical coordinate system

has the following form:

dε�

dτ ε− D

(∂2ε�

∂R2ε + 1

R2ε sinε (ϑε)

∂ε

∂ϑε

(sinε


)

+ 2

Rε∂ε�

∂Rε+ 1

R2ε sin2ε (ϑ

ε)

∂2ε�

∂θ2ε

)= 0, (1.315)

where D is the diffusive coefficient and � = �(μ, θ ,ϑ , τ).Systems of local fractional Maxwell’s equations in Cantor-type cylindrical coordi-

nate system is given by(eεR∂ε

∂Rε+ eεθ

1

Rε∂ε

∂θε+ eεσ

∂ε

∂σ ε

)· D = ρ, (1.316)

(eεR∂ε

∂Rε+ eεθ

1

Rε∂ε

∂θε+ eεσ

∂ε

∂σ ε

)× H = Jε + ∂

εD

∂τ ε, (1.317)


(eεR∂ε

∂Rε+ eεθ

1

Rε∂ε

∂θε+ eεσ

∂ε

∂σ ε

)× E = −∂

εB

∂τ ε, (1.318)

(eεR∂ε

∂Rε+ eεθ

1

Rε∂ε

∂θε+ eεσ

∂ε

∂σ ε

)· B = 0, (1.319)

where ρ = ρ (μ, θ , σ , τ) is the fractal electric charge density, D = D (μ, θ , σ , τ ) iselectric displacement in the fractal electric field, H = H (μ, θ , σ , τ ) is the magneticfield strength in the fractal field, E = E (μ, θ , σ , τ) is the electric field strength in thefractal field, Jε = Jε (μ, θ , σ , τ) is the conductive current, and B = B (μ, θ , σ , τ) isthe magnetic induction in the fractal field, and the constitutive relationships in fractalelectromagnetic can be written as

D (μ, θ , σ , τ) = εfE (μ, θ , σ , τ) (1.320)

and

H (μ, θ , σ , τ) = μfB (μ, θ , σ , τ) , (1.321)

with the fractal dielectric permittivity εf and the fractal magnetic permeability μf.Systems of local fractional Maxwell’s equations in Cantor-type circular coordinate

system take the following form:(eεR∂ε

∂Rε+ eεϑ

1

Rε∂ε

∂ϑε+ eεθ

1

Rε1

sinε (ϑε)

∂ε

∂θε

)· D = ρ, (1.322)

(eεR∂ε

∂Rε+ eεϑ

1

Rε∂ε

∂ϑε+ eεθ

1

Rε1

sinε (ϑε)

∂ε

∂θε

)× H = Jε + ∂

εD

∂τ ε, (1.323)

(eεR∂ε

∂Rε+ eεθ

1

Rε∂ε

∂θε

)× E (R, θ , τ) = −∂

εB (R, θ , τ)

∂τ ε, (1.324)

(eεR∂ε

∂Rε+ eεθ

1

Rε∂ε

∂θε

)· B (R, θ , τ) = 0, (1.325)

where ρ = ρ (R, θ ,ϑ , τ) is the fractal electric charge density, D = D (R, θ ,ϑ , τ) iselectric displacement in the fractal electric field, H = H (R, θ ,ϑ , τ) is the magneticfield strength in the fractal field, E = E (R, θ ,ϑ , τ) is the electric field strength in thefractal field, Jε = Jε (R, θ ,ϑ , τ) is the conductive current, and B = B (R, θ ,ϑ , τ) isthe magnetic induction in the fractal field, and the constitutive relationships in fractalelectromagnetic can be written as

D (R, θ ,ϑ , τ) = εfE (R, θ ,ϑ , τ) (1.326)

and

H (R, θ ,ϑ , τ) = μfB (R, θ ,ϑ , τ) , (1.327)

with the fractal dielectric permittivity εf and the fractal magnetic permeability μf.


We present the wave equation, heat-conduction equation, damped wave equation,and diffusion equation in the Cantor-type cylindrical symmetry form and in theCantor-type spherical symmetry form.

The linear local fractional wave equation for the velocity potential of fluid flow inthe Cantor-type cylindrical symmetry form is presented as follows:

ω

(∂2ε�(R, τ)

∂R2ε + 1

Rε∂ε�(R, τ)

∂Rε

)− ∂

2ε�(R, τ)

∂τ 2ε = 0, (1.328)

where ω is a constant.The linear local fractional wave equation for the velocity potential of fluid flow in

the Cantor-type spherical symmetry form is

ω

(∂2ε�(R, τ)

∂R2ε + 2

Rε∂ε�(R, τ)

∂Rε

)− ∂

2ε�(R, τ)

∂τ 2ε = 0, (1.329)

where ω is a constant.

The local fractional heat-conduction equation in the Cantor-type cylindrical sym-metry form can be written as

∂ε�(R, τ)

∂τ ε−�

(∂2ε�(R, τ)

∂R2ε + 1

Rε∂ε�(R, τ)

∂Rε

)= H(R, τ), (1.330)

where� is the thermal conductivity coefficient and H(R, τ) is the heat source.The local fractional heat-conduction equation in the Cantor-type spherical symme-

try form has the following form:

∂ε�(R, τ)

∂τ ε−�

(∂2ε�(R, τ)

∂R2ε + 2

Rε∂ε�(R, τ)

∂Rε

)= H(R, τ), (1.331)

where� is the thermal conductivity coefficient and H(R, τ) is the heat source.The linear local homogeneous damped wave equation of fractal strings in the

Cantor-type cylindrical symmetry form is given by

∂2ε�(R, τ)

∂τ 2ε− ∂

ε�(R, τ)

∂τ ε−(∂2ε�(R, τ)

∂R2ε+ 1

Rε∂ε�(R, τ)

∂Rε

)= 0. (1.332)

The linear local homogeneous damped wave equation of fractal strings in the Cantor-type spherical symmetry form is written as

∂2ε�(R, τ)

∂τ 2ε − ∂ε�(R, τ)

∂τ ε−(∂2ε�(R, τ)

∂R2ε + 2

Rε∂ε�(R, τ)

∂Rε

)= 0. (1.333)

Next, the local fractional diffusion equation in the Cantor-type cylindrical symmetryform is given by

dε�(R, τ)

dτ ε− D

(∂2ε�(R, τ)

∂R2ε + 1

Rε∂ε�(R, τ)

∂Rε

)= 0, (1.334)

where D is the diffusive coefficient.


The local fractional diffusion equation in the Cantor-type spherical symmetry formis given by

dε�(R, τ)

dτ ε− D

(∂2ε�(R, τ)

∂R2ε + 2

Rε∂ε�(R, τ)

∂Rε

)= 0, (1.335)

where D is the diffusive coefficient.

2Local fractional Fourier series

2.1 Introduction

In the Euclidean dimensional space, the sum of the special trigonometric functionsis called the Fourier series in honor of the French mathematician, Jean BaptisteJoseph Fourier (1768–1830). The expansions of functions as trigonometric series playimportant roles in the analysis of periodic functions, which are studied in science andengineering [90–93].

In fractal dimensional space, there are the following special series via the Mittag–Leffler function defined on the fractal set (e.g., [16, 21, 94–98]):

�(μ) = A0

2+

∞∑k=1

(Ak cosε (kτ)

ε + Bk sinε (kτ)ε)

(2.1)

and

�(μ) =∞∑

k=−∞ϕkEε

(iε (kμ)ε

), (2.2)

where A0, Ak, Bk, and ϕk are local fractional Fourier coefficients of (2.1) and (2.2),respectively.

Just as in the classical mechanism of the Fourier series, we need to answer thefollowing questions:

(a) In fractal dimensional space, how do we present the complex number defined on the fractalset �?

(b) Is there a generalized Hilbert space interpretation of local fractional Fourier series vianondifferentiable functions?

(c) How do we get the local fractional Fourier coefficients of the special series via theMittag–Leffler function defined on the fractal set?

For μ, η ∈ R, and 0 < ε ≤ 1, the complex number defined on the fractal set � isdefined as follows [1, 16, 21, 94–104]:

Zε = με + iεηε, (2.3)

where Iε ∈ �.The conjugate of (2.3) is defined by

Zε = με − iεηε, (2.4)

where Zε ∈ �.


http://dx.doi.org/10.1016/B978-0-12-804002-7.00002-4


The modulus of the complex number defined on the fractal set � is given by [16,21, 94]∣∣Zε∣∣ = √Z

εZε =

√ZεZ

ε =√μ2ε + η2ε, (2.5)

where Eε (iε (2π)ε) = 1.For Zε ∈ � and 0 < ε ≤ 1, the complex Mittag–Leffler function defined on the

fractal set � is given by

Eε(Zε) =

∞∑i=0

Ziε

� (1 + iε). (2.6)

For Zε1, Zε2 ∈ �, and 0 < ε ≤ 1, we present the following properties of the complexMittag–Leffler function defined on the fractal set:

Eε(Zε1 + Zε2

) = Eε(Zε1)

Eε(Zε2)

, (2.7)

Eε(Zε1 − Zε2

) = Eε(Zε1)

Eε(−Zε2

), (2.8)

and

Eε(iεZε1 + iεZε2

) = Eε(iεZε1

)Eε(iεZε2

). (2.9)

For Zε ∈ � and 0 < ε ≤ 1, we have

Eε(iεZε

) = cosε(Zε)+ iε sinε

(Zε)

, (2.10)

where

sinε(Zε) = Eε (iεZε)− Eε (−iεZε)

2iε(2.11)

and

cosε(Zε) = Eε (iεZε)+ Eε (−iεZε)

2. (2.12)

We call (2.11) and (2.12) the analogues of trigonometric functions, which are derivedfrom the complex Mittag–Leffler function defined on the fractal set.

Let us consider a set of functions given by [16–94]:

�(Eε) = {Eε (iε (2π)ε (kτ)ε) , kε ∈ �} (2.13)

is an orthonormal basis of the generalized Hilbert space of functions in the interval[−π ,π].

In the generalized Hilbert space, the dot product of ϕ (τ) and φ (τ) with the 2π -period via local fractional integral is defined as follows [16, 21, 94]:

〈ϕ,φ〉ε =∫ π

−πϕ (τ) φ (τ) (dτ)ε. (2.14)

For a given generalization of the Hilbert space Hε, we have the following formula[16, 21, 94]:

Local fractional Fourier series 59

n∑k=1

∣∣ϕεk ∣∣2 = ‖φ‖2ε , (2.15)

where all functions φ ∈ Hε,∣∣ϕεk ∣∣2 = ⟨ϕεk ,ϕεk⟩ε

, (2.16)

‖φ‖2ε = 〈φ,φ〉ε , (2.17)

and

ϕεk = ⟨φ, eεk⟩ε

. (2.18)

We also have

φ =n∑

k=1

ϕεk eεk (2.19)

with sum convergent in the generalized Hilbert space Hε for all φ ∈ Hε, where{eεk}

is a basis of the generalized Hilbert space Hε.For k = j, the orthogonality condition of ϕk and ϕj is defined as follows:

⟨ϕk,ϕj

⟩ε

= 1

πε

∫ π

−πϕk (τ ) ϕj (τ ) (dτ)

ε = 0. (2.20)

For k = j, the normalized condition of ϕk is defined as follows:

〈ϕk,ϕk〉ε = 1

πε

∫ π

−πϕ2

k (τ ) (dτ)ε = 1. (2.21)

Based upon the space given by

Hε = span{eε1, . . . , eεn

}= span

{1, Eε

(πεiετ ε

Lε

), Eε

(πεiε (2τ)ε

Lε

), . . . , Eε

(πεiε (nτ)ε

Lε

)}, (2.22)

we have

ϕε0 = 〈φ, 1〉ε (2.23)

and

ϕεk = ⟨φ, Eε(iε (kτ)ε

)⟩ε

, (2.24)

where k ∈ Z.One also gets the inverse relations as follows:

φ =n∑

k=1

ϕεk eεk , (2.25)

where k ∈ N.


Hence, the local fractional Fourier series from the generalized Hilbert space Hε ispresented as follows:

φ =n∑

k=1

⟨φ, Eε

(iε (kτ)ε

)⟩ε

eεk , (2.26)

where{eεk}∞

k=1 is a complete orthonormal set of functions.Let φ (τ) be 2L-periodic. For k ∈ Z, the local fractional Fourier series of the

nondifferentiable function φ (τ) is defined as

φ (τ) =∞∑

k=−∞ϕεk Eε

(πεiε (kτ)ε

Lε

), (2.27)

where the local fractional Fourier coefficients are written as follows:

ϕεk = 1

(2L)ε

∫ L

−Lφ (τ)Eε

(−π

εiε (kτ)ε

Lε

)(dτ)ε. (2.28)

Therefore, for L = π , we get the following pair:

φ (τ) =n∑

k=1

ϕεk Eε(iεkετ ε

)(2.29)

and

ϕεk = 1

(2π)ε

∫ π

−πφ (τ)Eε

(−iεkετ ε)(dτ)ε. (2.30)

For k ∈ N, a set of functions given by

1

2, sinε (τ )

ε, cosε (τ )ε, . . . , sinε (kτ)

ε, cosε (kτ)ε (2.31)

are orthogonal. In this case, we have

�(τ) = A0

2+

∞∑k=1

(Ak cosε (kτ)

ε + Bk sinε (kτ)ε), (2.32)

where

A0 = 1

πε

∫ π

−πφ (τ) (dτ)ε, (2.33)

Ak = 1

πε

∫ π

−πφ (τ) cosε (kτ)

ε (dτ)ε, (2.34)

and

Bk = 1

πε

∫ π

−πφ (τ) sinε (kτ)

ε (dτ)ε. (2.35)


Hence, the local fractional Fourier series from the generalized Hilbert space Hε ispresented as follows:

φ =n∑

k=1

⟨φ, eεk

⟩ε

eεk , (2.36)

where{eεk}∞

k=1 is a complete orthonormal set of functions.In different bases, we can obtain a class of different Fourier series from the

generalized Hilbert space. In this case, we find the local fractional Fourier coefficientsof the special series via the Mittag–Leffler function defined on the fractal set.

2.2 Definitions and properties

2.2.1 Analogous trigonometric form of local fractionalFourier series

Let φ (τ) be 2L-periodic. For k ∈ N, a generalized local fractional Fourier series ofψ (τ) is defined as follows:

ψ (τ) ∼ A (0, ε)

2+

∞∑k=1

(A (k, ε) cosε

(πkτ

L

)ε+ B (k, ε) sinε

(πkτ

L

)ε),

(2.37)

where

A (0, ε) = 1

Lε

∫ L

−Lφ (τ) (dτ)ε, (2.38)

A (k, ε) = 1

Lε

∫ L

−Lψ (τ) cosε

(πkτ

L

)ε(dτ)ε, (2.39)

and

B (k, ε) = 1

Lε

∫ L

−Lψ (τ) sinε

(πkτ

L

)ε(dτ)ε (2.40)

are the local fractional Fourier coefficients of a generalized local fractional Fourierseries of ψ (τ).

Let φ (τ) be 2π-periodic. For k ∈ N, a local fractional Fourier series of φ (τ) isdefined as follows:

φ (τ) ∼ A (0, ε)

2+

∞∑k=1

(A (k, ε) cosε (kτ)

ε + B (k, ε) sinε (kτ)ε), (2.41)


where

A (0, ε) = 1

πε

∫ π

−πφ (τ) (dτ)ε, (2.42)

A (k, ε) = 1

πε

∫ π


ε (dτ)ε, (2.43)

and

B (k, ε) = 1

πε

∫ π


ε (dτ)ε (2.44)

are the local fractional Fourier coefficients of a local fractional Fourier series of φ (τ)in (2.39).

2.2.2 Complex Mittag–Leffler form of local fractionalFourier series

Let φ (τ) be 2L-periodic. For k ∈ Z, a generalized local fractional Fourier series ofψ (τ) is defined as follows:

ψ (τ) ∼∞∑

k=−∞ϕ (k, ε)Eε

(πεiε (kτ)ε

Lε

), (2.45)

where

ϕ (k, ε) = 1

(2L)ε

∫ L

−Lψ (τ)Eε

(−π

εiε (kτ)ε

Lε

)(dτ)ε (2.46)

is the local fractional Fourier coefficient of a generalized local fractional Fourier seriesof ψ (τ).

Let φ (τ) be 2π-periodic. For k ∈ Z, a local fractional Fourier series of φ (τ) isdefined as follows:

φ (τ) ∼∞∑

k=−∞ϕ (k, ε)Eε

(iε (kτ)ε

), (2.47)

where

ϕ (k, ε) = 1

(2L)ε

∫ L

−Lφ (τ)Eε

(−iε (kτ)ε)(dτ)ε (2.48)

is the local fractional Fourier coefficient of a local fractional Fourier series of φ (τ).Adopting (2.11) and (2.12), we easily obtain the following relationships:

ϕ (0, ε) = A (0, ε)

2ε, (2.49)

ϕ (k, ε) = A (k, ε)− iεB (k, ε)

2ε, (2.50)


and

ϕ (−k, ε) = A (k, ε)+ iεB (k, ε)

2ε. (2.51)

In this case, we also write transformation pairs in the following forms:

ψ (τ) ↔ ϕ (k, ε) . (2.52)

Other forms of local fractional Fourier series were presented in the literature (e.g.,[1, 16, 21, 94–104]).

2.2.3 Properties of local fractional Fourier series

Property 3 (Linearity of local fractional Fourier series). Suppose that

ψ1 (τ ) ↔ ϕ1 (k, ε)

and

ψ2 (τ ) ↔ ϕ2 (k, ε) .

Then, for two constants a and b, we have

aψ1 (τ )+ bψ2 (τ ) ↔ aϕ1 (k, ε)+ bϕ2 (k, ε) . (2.53)

Proof. The proof of this property is a straightforward application of the linearityproperty of integration.

Property 4 (Conjugation of local fractional Fourier series). Suppose that ϕ (k, ε)is the local fractional Fourier coefficient of ψ (τ). Then, we get

ψ (τ) ↔ ϕ (−k, ε). (2.54)

Proof. Since

Eε (−iε (kτ)ε) = Eε(iε (kτ)ε

), (2.55)

it follows by direct calculation of the local fractional Fourier coefficient of ψ (τ) that

1

(2π)ε

∫ π

−πψ (τ)Eε

(−iε (kτ)ε)(dτ)ε = 1

(2π)ε

∫ π

−πψ (τ)Eε (iε (kτ)ε) (dτ)ε

= 1

(2π)ε

∫ π

−πψ (τ)Eε (−iε (−kτ)ε) (dτ)ε

= ϕ (−k, ε). (2.56)

The proof of this property is thus completed.

Property 5 (Shift in fractal time of local fractional Fourier series). Suppose thatϕ (k, ε) is the local fractional Fourier coefficient of ψ (τ). Then, we have


ψ (τ − τ0) ↔ Eε(−iε (kτ0)

ε)ϕ (k, ε) . (2.57)

Proof. Adopting the definition of local fractional Fourier series, we find that

1

(2π)ε

∫ π

−πψ (τ − τ0)Eε

(−iε (kτ)ε)(dτ)ε

= Eε(−iε (kτ0)

ε) 1

(2π)ε

∫ π

−πψ (τ − τ0)Eε

(−iε [k (τ − τ0)]ε)(dτ)ε

= Eε(−iε (kτ0)

ε) 1

(2π)ε

∫ π

−πψ (τ)Eε

(−iε (kτ)ε)(dτ)ε

= Eε(−iε (kτ0)

ε)ϕ (k, ε) . (2.58)

The proof of this property is evidently completed.

Property 6 (Fractal time reversal of local fractional Fourier series). Suppose thatϕ (k, ε) is the local fractional Fourier coefficient of ψ (τ). Then, we obtain

ψ (−τ ) ↔ ϕ (−k, ε) . (2.59)

Proof. With the help of the definition of local fractional Fourier series, we have

1

(2π)ε

∫ π

−πψ (−τ)Eε

(−iε (kτ)ε)(dτ)ε = 1

(2π)ε

∫ π

−πψ (−τ)Eε

(iε (−kτ)ε

)(dτ)ε

= 1

(2π)ε

∫ π

−πψ (τ)Eε

(iε (kτ)ε

)(dτ)ε

= 1

(2π)ε

∫ π

−πψ (τ)Eε

(−iε (−kτ)ε)(dτ)ε

= ϕ (−k, ε) . (2.60)

The proof of this property is thus completed.

2.2.4 Theorems of local fractional Fourier series

Theorem 2.1 (Bessel inequality for local fractional Fourier series). Suppose thatψ (τ) is 2π -periodic, bounded, and locally fractional integrable on [−π ,π ]. Then,the following inequality holds true:

A2 (0, ε)

2+

n∑k=1

(A2 (k, ε)+ B2 (k, ε)

)≤ 1

πε

∫ π

−πψ2 (τ ) (dτ)ε, (2.61)

provided that A (0, ε), A (k, ε), and B (k, ε) are the local fractional Fourier coefficientsof ψ (τ).

Proof. Let us consider the sum of the local fractional Fourier series, namely,


Sn,ε (τ ) = A (0, ε)

2+

n∑k=1


ε + B (k, ε) sinε (kτ)ε). (2.62)

Following (2.62), we calculate

1

πε

∫ π

−π[ψ (τ)− Sn,ε (τ )

]2(dτ)ε = 1

πε

∫ π

−π

[ψ2 (τ )− 2ψ (τ) Sn,ε (τ )

+S2n,ε (τ )

](dτ)ε , (2.63)

where

1

πε

∫ π

−πS2

n,ε (τ ) (dτ)ε

= 1

πε

∫ π

−π

[ ∞∑k=0


ε + B (k, ε) sinε (kτ)ε)]2

(dτ)ε

= A2 (0, ε)

2+ 1

πε

∫ π

−π

[ ∞∑k=1


ε + B (k, ε) sinε (kτ)ε)]2

(dτ)ε

= A2 (0, ε)

2+

n∑k=1

(A2 (k, ε)+ B2 (k, ε)

)(2.64)

and

1

πε

∫ π

−π[ψ (τ) Sn,ε (τ )

](dτ)ε

= 1

πε

∫ π

−π

{[A (0, ε)+

n∑k=1


ε + B (k, ε) sinε (kτ)ε)]

Sk,ε (τ )

}(dτ)ε

= A2 (0, ε)

2+

n∑k=1

(A2 (k, ε)+ B2 (k, ε)

)

= A2 (0, ε)

2+

n∑k=1

(A2 (k, ε)+ B2 (k, ε)

)(2.65)

with

ψ (τ) ∼ A (0, ε)+∞∑

k=1


ε + B (k, ε) sinε (kτ)ε). (2.66)

Therefore, this theorem is proved.


Theorem 2.2 (Riemann-Lebesgue theorem for local fractional Fourier series).Suppose that ψ (τ) is 2π -periodic, bounded, and locally fractional integrable on[−π ,π]. Then,

limk→∞

1

πε

∫ π

−πψ (τ) sinε (kτ)

ε (dτ)ε = 0 (2.67)

and

limk→∞

1

πε

∫ π

−πψ (τ) cosε (kτ)

ε (dτ)ε = 0. (2.68)

Proof. Considering the integration

�(k) = 1

πε

∫ π


ε (dτ)ε (2.69)

and changing the variable in (2.66) with

τ = t + π

k, (2.70)

we find that

sinε (kτ)ε = − sinε (kt)ε (2.71)

and

�(k) = − 1

πε

∫ π(

1− 1k

)−π(

1− 1k

) ψ(

t + π

k

)sinε (kt)ε (dt)ε. (2.72)

Therefore, from (2.66) and (2.69), we have

2�(k) = 1

πε

∫ π

−πψ (t) sinε (kt)ε (dt)ε

− 1

πε

∫ π(

1− 1k

)−π(

1− 1k

) ψ(

t + π

k

)sinε (kt)ε (dt)ε

= − 1

πε

∫ −π

−π(

1− 1k

) ψ(

t + π

k


+ 1

πε

∫ π

π(

1− 1k

) ψ(

t + π

k


+ 1

πε

∫ π

−π

[ψ (t)− ψ

(t + π

k

)]sinε (kt)ε (dt)ε. (2.73)

Since ψ (τ) is 2π -periodic, bounded, and locally fractional integrable on [−π ,π],there exists M such that

|ψ (t)| ≤ M (2.74)

∀t ∈ [−π ,π].In this case, we get


∣∣∣∣∣ 1

πε

∫ −π

−π(

1− 1k

) ψ(

t + π

k


∣∣∣∣∣ ≤ 1

πε

∫ −π

−π(

1− 1k

)∣∣∣ψ (t + π

k

)sinε (kt)ε

∣∣∣ (dt)ε

≤ 1

πε

∫ −π

−π(

1− 1k

)∣∣∣ψ (t + π

k

)∣∣∣ (dt)ε

≤ M

πε

(πk

)ε(2.75)

and∣∣∣∣∣ 1

πε

∫ π

π(

1− 1k

) ψ(

t + π

k


∣∣∣∣∣ ≤ 1

πε

∫ π

π(

1− 1k

)∣∣∣ψ (t + π

k

)sinε (kt)ε

∣∣∣ (dt)ε

≤ 1

πε

∫ π

π(

1− 1k

)∣∣∣ψ (t + π

k

)∣∣∣ (dt)ε

≤ M

πε

(πk

)ε. (2.76)

It then follows that

|2�(k)| ≤ M

πε

(πk

)ε+ M

πε

(πk

)ε+∣∣∣∣ 1

πε

∫ π

−π

[ψ (t)− ψ

(t + π

k

)]sinε (kt)ε (dt)ε

∣∣∣∣ .(2.77)

Therefore, for ρ > 0, we can find K such that (for k > K)∣∣∣ψ (t)− ψ(

t + π

k

)∣∣∣ ≤ (ρ2

)ε. (2.78)

In this case, we also choose K large enough such that (2.74) can be written as follows:

M

πε

(πk

)ε ≤(ρ

4

)ε. (2.79)

Then, for k > K, we get

|�(k)| ≤(ρ

4

)ε +(ρ

4

)ε +(ρ

2

)ε = ρε, (2.80)

which implies (2.67).In a similar manner, we also obtain (2.68). Therefore, this theorem is proved.

We also generalize (2.67) and (2.68). In this case, the following theorem holds true:

Theorem 2.3. Suppose that ψ (τ) is 2π -periodic, bounded, and locally fractionalintegrable on [−π ,π]. Then,

limk→∞

1

πε

∫ π

0ψ (τ) sinε

(2k + 1

2τ

)ε(dτ)ε = 0, (2.81)

limk→∞

1

πε

∫ 0

−πψ (τ) sinε

(2k + 1

2τ

)ε(dτ)ε = 0, (2.82)


limk→∞

1

πε

∫ π

0ψ (τ) cosε

(2k + 1

2τ

)ε(dτ)ε = 0, (2.83)

and

limk→∞

1

πε

∫ 0

−πψ (τ) cosε

(2k + 1

2τ

)ε(dτ)ε = 0. (2.84)

We next state the following theorem:


Sn,ε (τ ) ∼ A (0, ε)

2+

n∑k=1


ε + B (k, ε) sinε (kτ)ε).

Then,

Sn,ε (μ) = 1

πε

∫ π

−πSn,ε (μ+ τ)Dn,ε (τ ) (dτ)

ε, (2.85)

where

Dn,ε (t) = 1

2+

n∑k=1

cosε (kτ)ε. (2.86)

Proof. We can expand Sn,ε (μ+ τ) in the following form:

Sn,ε (μ+ τ) = A (0, ε)

2+

n∑k=1

(A (k, ε) cosε [k (μ+ τ)]ε + B (k, ε) sinε [k (μ+ τ)]ε

)

= A (0, ε)

2+

n∑k=1

A (k, ε)[cosε (kμ)

ε cosε (kτ)ε− sinε (kμ)

ε sinε(kτ)ε]

+n∑

k=1

B (k, ε)[sinε (kμ)

ε cosε (kτ)ε + sinε (kτ)

ε cosε (kμ)ε]

= A (0, ε)

2+

n∑k=0

A (k, ε) cosε (kμ)ε cosε (kτ)

ε

−n∑

k=1

A (k, ε) sinε (kμ)ε sinε (kτ)

ε

+n∑

k=1

B (k, ε) sinε (kμ)ε cosε (kτ)

ε

+n∑

k=1

B (k, ε) cosε (kμ)ε sinε (kτ)

ε . (2.87)


In this case, we consider the following integral:

1

πε

∫ π


ε

= 1

πε

∫ π

−π

{[n∑

k=1


ε

](1

2+

n∑k=1

cosε (kτ)ε

)}(dτ)ε

− 1

πε

∫ π

−π

{[n∑

k=1


ε

](1

2+

n∑k=1

cosε (kτ)ε

)}(dτ)ε

+ 1

πε

∫ π

−π

{[n∑

k=1


ε

](1

2+

n∑k=1

cosε (kτ)ε

)}(dτ)ε

+ 1

πε

∫ π

−π

{[n∑

k=1


ε

](1

2+

n∑k=1

cosε (kτ)ε

)}(dτ)ε

+ 1

πε

∫ π

−π

{A (0, ε)

2

(1

2+

n∑k=1

cosε (kτ)ε

)}(dτ)ε

= A (0, ε)

2+

n∑k=0

A (k, ε) cosε (kμ)ε +

n∑k=0

B (k, ε) sinε (kμ)ε, (2.88)

wheren∑

k=0

A (k, ε) cosε (kμ)ε

=∫ π

−π

{[n∑

k=0


ε

](1

2+

n∑k=1

cosε (kτ)ε

)}(dτ)ε,

(2.89)

1

πε

∫ π

−π

{[n∑

k=0


ε

](1

2+

n∑k=1

cosε (kτ)ε

)}(dτ)ε = 0,

(2.90)

n∑k=0

B (k, ε) sinε (kμ)ε

= 1

πε

∫ π

−π

{[n∑

k=0


ε

](1

2+

n∑k=1

cosε (kτ)ε

)}(dτ)ε,

(2.91)


and

1

πε

∫ π

−π

{[n∑

k=0


ε

](1

2+

n∑k=1

cosε (kτ)ε

)}(dτ)ε = 0.

(2.92)

Therefore, we obtain

1

πε

∫ π


ε =n∑

k=0

A (k, ε) cosε (kμ)ε +

n∑k=0

B (k, ε) sinε(kμ)ε

=n∑

k=0

(A (k, ε) cosε (kμ)

ε + B (k, ε) sinε (kμ)ε).

(2.93)

Clearly, this theorem is proved.


1

πε

∫ π

−πψ2 (τ ) (dτ)ε = A2 (0, ε)

2+

∞∑k=1

(A2 (k, ε)+ B2 (k, ε)

), (2.94)

provided that

ψ (τ) ∼ A (0, ε)

2+

n∑k=1


ε + B (k, ε) sinε (kτ)ε), (2.95)

where

A (0, ε) = 1

πε

∫ π

−πφ (τ) (dτ)ε, (2.96)

A (k, ε) = 1

πε

∫ π


ε (dτ)ε, (2.97)

and

B (k, ε) = 1

πε

∫ π


ε (dτ)ε. (2.98)

Proof. Following (2.94), we consider following integral:

1

πε

∫ π

−πψ2 (τ ) (dτ)ε

= 1

πε

∫ π

−π

{ψ (τ)

(A (0, ε)

2+

n∑k=1


ε + B (k, ε) sinε (kτ)ε))}

(dτ)ε


= 1

πε

∫ π

−π

[ψ (τ)

A (0, ε)

2

](dτ)ε

+ 1

πε

∫ π

−π

{ψ (τ)

[n∑

k=1


ε + B (k, ε) sinε (kτ)ε)]}

(dτ)ε , (2.99)

as well as the following relations:

1

πε

∫ π

−π[ψ (τ)A (0, ε)]

(dτ)ε

= 1

πε

∫ π

−π

⎧⎨⎩⎡⎣A (0, ε)

2+

n∑k=1


ε + B (k, ε) sinε (kτ)ε)⎤⎦ A (0, ε)

2

⎫⎬⎭(dτ)ε

= A2 (0, ε)

2(2.100)

and∞∑

k=1

(A2 (k, ε)+ B2 (k, ε)

)= 1

πε

∫ π

−π

{ψ (τ)

[n∑

k=1


ε

+B (k, ε) sinε (kτ)ε) ]}

(dτ)ε . (2.101)

Making use of (2.99), we obtain the result asserted by this theorem. Therefore, thistheorem is proved.

Theorem 2.6 (Convergence theorem for local fractional Fourier series). Supposethat ψ (τ) is 2π -periodic, bounded, and locally fractional integrable on [−π ,π ].Then, the local fractional Fourier series of ψ (τ) converges to ψ (τ) at τ ∈ [−π ,π]and

ψ (τ + 0)+ ψ (τ − 0)

2= A (0, ε)

2+

∞∑k=1


ε + B (k, ε) sinε (kτ)ε),

(2.102)

where

A (0, ε) = 1

πε

∫ π

−πφ (τ) (dτ)ε, (2.103)

A (k, ε) = 1

πε

∫ π


ε (dτ)ε, (2.104)

and

B (k, ε) = 1

πε

∫ π


ε (dτ)ε. (2.105)


Proof. Let us define the sum of the local fractional Fourier series in the form

Sn,ε (τ ) = A (0, ε)

2+

n∑k=1


ε + B (k, ε) sinε (kτ)ε). (2.106)

In this case, we transform (2.95) into the following equation:

limn→∞

{(ψ (τ + 0)+ ψ (τ − 0)

2

)− Sn,ε (τ )

}= 0. (2.107)

Adopting the formula (2.85), we find that

Sn,ε (τ ) = 1

πε

∫ π

−πψ (τ + t)Dn,ε (t) (dt)ε, (2.108)

where

Dn,ε (t) = 1

2+

n∑k=1

cosε (kt)ε =sinε

[((k + 1

2

)t)ε]

2 sinε[( t

2

)ε] . (2.109)

In this case, starting from (2.109), we are led to the following formulas:

1

πε

∫ π

0ψ (τ + 0)Dn,ε (t) (dt)ε = ψ (τ + 0) (2.110)

and

1

πε

∫ 0

−πψ (τ − 0)Dn,ε (t) (dt)ε = ψ (τ − 0) . (2.111)

From (2.109) to (2.111), we expand (2.107) as follows:(ψ (τ + 0)+ ψ (τ − 0)

2

)− Sn,ε (τ )

=(ψ (τ + 0)+ ψ (τ − 0)

2

)− 1

πε

∫ π

−πψ (τ + t)Dn,ε (t) (dt)ε

=[ψ (τ + 0)

2− 1

πε

∫ π

0ψ (τ + t)Dn,ε (t) (dt)ε

]

+[ψ (τ − 0)

2− 1

πε

∫ 0

−πψ (τ − t)Dn,ε (t) (dt)ε

]

= 1

πε

∫ π

0(ψ (τ + 0)− ψ (τ + t))Dn,ε (t) (dt)ε

+ 1

πε

∫ 0

−π(ψ (τ − 0)− ψ (τ − t))Dn,ε (t) (dt)ε

= 1

πε

∫ π

0(ψ (τ + 0)− ψ (τ + t))

sinε[((

k + 12

)t)ε]

2 sinε[( t

2

)ε] (dt)ε


+ 1

πε

∫ 0

−π(ψ (τ − 0)− ψ (τ − t))

sinε[((

k + 12

)t)ε]

2 sinε[( t

2

)ε] (dt)ε

= 1

πε

∫ π

0

[ψ (τ + 0)− ψ (τ + t)

tε�(1+ε)

]{ tε�(1+ε)

2 sinε[( t

2

)ε]}

sinε

[((k + 1

2

)t

)ε](dt)ε

+ 1

πε

∫ 0

−π

[ψ (τ − 0)− ψ (τ − t)

tε�(1+ε)

]{ tε�(1+ε)

2 sinε[( t

2

)ε]}

sinε

[((k + 1

2

)t

)ε](dt)ε.

(2.112)

For τ ∈ [0, 1], we simulate the following formula:

limt→0

[ψ (τ + 0)− ψ (τ + t)

tε�(1+ε)

]{ tε�(1+ε)

2 sinε[( t

2

)ε]}

= 2ε−1ψ(ε) (τ + 0) , (2.113)

where

limt→0

[ψ (τ + 0)− ψ (τ + t)

tε�(1+ε)

]= ψ(ε) (τ + 0) (2.114)

and

limt→0

tε�(1+ε)

2 sinε[( t

2

)ε] =∂ε

∂tε

[tε

�(1+ε)]

∂ε

∂tε{2 sinε

[( t2

)ε]} = limt→0

1

21−ε cosε[( t

2

)ε] = 2ε−1.

(2.115)

In a similar way, we have

limt→0

[ψ (τ − 0)− ψ (τ − t)

tε�(1+ε)

]{ tε�(1+ε)

2 sinε[( t

2

)ε]}

= 2ε−1ψ(ε) (τ − 0) , (2.116)

where

limt→0

[ψ (τ − 0)− ψ (τ − t)

tε�(1+ε)

]= ψ(ε) (τ − 0) (2.117)

and

limt→0

tε�(1+ε)

2 sinε[( t

2

)ε] =∂ε

∂tε

[tε

�(1+ε)]

∂ε

∂tε{2 sinε

[( t2

)ε]} = limt→0

1

21−ε cosε[( t

2

)ε] = 2ε−1.

(2.118)


Making use of the formulas (2.115) and (2.118), we obtain

limn→∞

[(ψ (τ + 0)+ ψ (τ − 0)

2

)− Sn,ε (τ )

]

= 1

πε

∫ π

02ε−1ψ(ε) (τ + 0) sinε

[((k + 1

2

)t

)ε](dt)ε

+ 1

πε

∫ 0

−π2ε−1ψ(ε) (τ − 0) sinε

[((k + 1

2

)t

)ε](dt)ε. (2.119)

In this case, using the Riemann–Lebesgue theorem for local fractional Fourier series,we have

1

πε

∫ π

0

{[ψ (τ + 0)− ψ (τ + t)

tε�(1+ε)

] tε�(1+ε)

2 sinε[( t

2

)ε]}

sinε

[((k + 1

2

)t

)ε](dt)ε = 0

(2.120)

and

1

πε

∫ π

0

{[ψ (τ − 0)− ψ (τ − t)

tε�(1+ε)

] tε�(1+ε)

2 sinε[( t

2

)ε]}

sinε

[((k + 1

2

)t

)ε](dt)ε = 0.

(2.121)

Considering the above case, we obtain

limn→∞

[(ψ (τ + 0)+ ψ (τ − 0)

2

)− Sn,ε (τ )

]= 0. (2.122)

Therefore, the proof of this theorem is completed.

As a direct application, we present the following result:

Theorem 2.7. Suppose that ψ (τ) is 2π -periodic, locally fractional continuous,and locally fractional integrable on [−π ,π]. Then,

ψ (τ) = A (0, ε)

2+

∞∑k=1


ε + B (k, ε) sinε (kτ)ε), (2.123)

where

A (0, ε) = 1

πε

∫ π

−πφ (τ) (dτ)ε, (2.124)

A (k, ε) = 1

πε

∫ π


ε (dτ)ε, (2.125)


and

B (k, ε) = 1

πε

∫ π


ε (dτ)ε. (2.126)

Proof. Since ψ (τ) is locally fractional continuous on [−π ,π], we have

ψ (τ + 0)+ ψ (τ − 0)

2= ψ (τ) . (2.127)

Now, by using (2.102) and (2.127), we obtain the result asserted by the theorem.Hence, we have completed the proof of this theorem.


ψ (τ) = A (0, ε)

2+

∞∑k=1

A (k, ε) cosε (kτ)ε, (2.128)

where

A (0, ε) = 1

πε

∫ π

−πφ (τ) (dτ)ε (2.129)

and

A (k, ε) = 1

πε

∫ π


ε (dτ)ε, (2.130)

provided that

ψ (τ) = ψ (−τ) . (2.131)

Proof. Since ψ (τ) = ψ (−τ), we obtain∞∑

k=1

B (k, ε) sinε (kτ)ε = 0. (2.132)

Therefore, we obtain the asserted result, and it completes the proof of this theorem.


ψ (τ) =∞∑

k=1

B (k, ε) sinε (kτ)ε, (2.133)

where

B (k, ε) = 1

πε

∫ π


ε (dτ)ε, (2.134)


provided that

ψ (τ) = −ψ (τ) . (2.135)

Proof. Since ψ (τ) = −ψ (τ), we obtain

A (0, ε)

2+

∞∑k=1

A (k, ε) cosε (kτ)ε = 0. (2.136)

Therefore, we obtain the asserted result, and the proof of this theorem is completed.

Theorem 2.10. Suppose that ψ (τ) is 2π -periodic, locally fractional continuous,and locally fractional integrable on [−L, L]. Then,

ψ (τ) = A (0, ε)

2+

∞∑k=1

(A (k, ε) cosε

(πkτ

L

)ε+ B (k, ε) sinε

(πkτ

L

)ε),

(2.137)

where

A (0, ε) = 1

Lε

∫ L

−Lφ (τ) (dτ)ε, (2.138)

A (k, ε) = 1

Lε

∫ L

−Lψ (τ) cosε

(πkτ

L

)ε(dτ)ε, (2.139)

and

B (k, ε) = 1

Lε

∫ L

−Lψ (τ) sinε

(πkτ

L

)ε(dτ)ε. (2.140)

Proof. Suppose that �(η) is 2π -periodic and locally fractional continuous on theinterval [−π ,π]. Then, by defining the variable η by

η =(π

L

)τ , (2.141)

we get

�(η) = �((π

L

)τ)

= ψ (τ) . (2.142)

Using (2.102), we obtain the asserted result. Therefore, the proof of this theorem iscompleted.


Theorem 2.11. Suppose that ψ (τ) is 2π -periodic and locally fractional continu-ous on the interval [−π ,π ]. Then,

ψ (τ) = A (0, ε)

2+

∞∑k=1

A (k, ε) cosε

(πkτ

L

)ε, (2.143)

where

A (0, ε) = 1

Lε

∫ L

−Lφ (τ) (dτ)ε (2.144)

and

A (k, ε) = 1

Lε

∫ L

−Lψ (τ) cosε

(πkτ

L

)ε(dτ)ε, (2.145)

provided that

ψ (τ) = ψ (−τ) . (2.146)

Proof. Since ψ (τ) = ψ (−τ), we obtain∞∑

k=1

B (k, ε) sinε

(πkτ

L

)ε= 0. (2.147)

Therefore, we obtain the asserted result and complete the proof of this theorem.


ψ (τ) =∞∑

k=1

B (k, ε) sinε

(πkτ

L

)ε, (2.148)

where

B (k, ε) = 1

Lε

∫ L

−Lψ (τ) sinε

(πkτ

L

)ε(dτ)ε, (2.149)

provided that

ψ (τ) = −ψ (τ) . (2.150)

Proof. Since ψ (τ) = −ψ (τ), we obtain

A (0, ε)

2+

∞∑k=1

A (k, ε) cosε

(πkτ

L

)ε= 0. (2.151)

Therefore, we obtain the asserted result and complete the proof of this theorem.


2.3 Applications to signal analysis

In this section, we consider some applications of local fractional Fourier series insignal analysis defined on Cantor sets. The aim of this part is to investigate the fractalsignal processes with help of the local fractional Fourier series. The technique is apowerful tool to process fractal signals to applied scientists and engineers. We willgive some examples (see also [94–98]).

We consider the 2L-periodic fractal signal given by

ψ (τ) = C (2.152)

on the interval 0 ≤ τ ≤ L in local fractional Fourier series, where ε = ln 2/ ln 3 andC is a constant.

The local fractional Fourier coefficients take the following form:

A (0, ε) = 1

Lε

∫ L

−Lψ (τ) (dτ)ε

= 1

Lε

∫ L

−LC (dτ)ε

= 1

Lε

∫ L

0C (dτ)ε

= C, (2.153)

A (k, ε) = 1

Lε

∫ L

−Lψ (τ) cosε

(πkτ

L

)ε(dτ)ε

= 1

Lε

∫ L

0C cosε

(πkτ

L

)ε(dτ)ε

= � (1 + ε) sinε(πkL

L

)ε(πk)ε

= 0, (2.154)

and

B (k, ε) = 1

Lε

∫ L

−Lψ (τ) sinε

(πkτ

L

)ε(dτ)ε

= 1

Lε

∫ L

0C sinε

(πkτ

L

)ε(dτ)ε

= 2C� (1 + ε)(1 − (−1)k

)(πk)ε

. (2.155)

For 0 ≤ τ ≤ L, the fractal signal ψ (τ) is represented as follows:

ψ (τ) = C

2+

∞∑k=1

(2C� (1 + ε)

(1 − (−1)k

)(πk)ε

)sinε

(πkτ

L

)ε. (2.156)


When k = 0, the fractal signal ψ (τ) is expanded as follows:

ψ (τ) = C

2. (2.157)


ψ (τ) = C

2+ 4C� (1 + ε)

πεsinε

(πτL

)ε. (2.158)

When k = 2, we have

2C� (1 + ε)(1 − (−1)2

)(2π)ε

sinε

(2πτ

L

)ε= 0. (2.159)

When k = 3, we expand the fractal signal ψ (τ) as given below:

ψ (τ) = C

2+ 4C� (1 + ε)

[1

πεsinε

(πτL

)ε + 1

(3π)εsinε

(3πτ

L

)ε]. (2.160)

When k = 4, we have

2C� (1 + ε)(1 − (−1)4

)(πk)ε

sinε

(4πτ

L

)ε= 0. (2.161)

When k = 5, the fractal signalψ (τ) in local fractional Fourier series can be writtenas follows:

ψ (τ) = C

2+ 4C� (1 + ε)

[1

πεsinε

(πτL

)ε + 1

(3π)εsinε

(3πτ

L

)ε

+ 1

(5π)εsinε

(5πτ

L

)ε]. (2.162)

The local fractional Fourier series of the fractal signal ψ (τ) when k = 0, k = 1,k = 3, and k = 5 is shown in Figure 2.1.

In this way, we can expand the fractal signal ψ (τ) into the local fractional Fourierseries representation.

Expand the 2L-periodic fractal signal

ψ (τ) = τ ε

� (1 + ε)(2.163)

on the interval −L < τ ≤ L in local fractional Fourier series. In this case, we presentthe local fractional Fourier coefficients as follows:

A (0, ε) = 1

Lε

∫ L

−Lψ (τ) (dτ)ε

= 1

Lε

∫ L

−L

τ ε

� (1 + ε)(dτ)ε

= 0, (2.164)


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

t

ψ0(t)

ψ1(t)

ψ3(t)

ψ5(t)

Figure 2.1 The local fractional Fourier series representation of fractal signal ψ (τ) whenε = ln 2/ ln 3, k = 0, k = 1, k = 3, and k = 5.

A (k, ε) = 1

Lε

∫ L

−Lψ (τ) cosε

(πkτ

L

)ε(dτ)ε

= 1

Lε

∫ L

−L

τ ε

� (1 + ε)cosε

(πkτ

L

)ε(dτ)ε

= 0, (2.165)

and

B (k, ε) = 1

Lε

∫ L

−Lψ (τ) sinε

(πkτ

L

)ε(dτ)ε

= 1

Lε

∫ L

−L

τ ε

� (1 + ε)sinε

(πkτ

L

)ε(dτ)ε

= 2

Lε

∫ L

0

τ ε

� (1 + ε)sinε

(πkτ

L

)ε(dτ)ε

= 2Lε (−1)k+1

(kπ)ε. (2.166)


For −L < τ ≤ L, the local fractional Fourier series representation of fractal signalfinally can be written as follows:

ψ (τ) =∞∑

k=1

[2Lε (−1)k+1

(kπ)εsinε

(πkτ

L

)ε]. (2.167)


ψ (τ) = 2Lε

πεsinε

(πτL

)ε. (2.168)

When k = 2, we expand the fractal signal ψ (τ) as given below:

ψ (τ) = 2Lε

πεsinε

(πτL

)ε − 2Lε

(2π)εsinε

(2πτ

L

)ε. (2.169)

When k = 3, we expand the fractal signal ψ (τ) as follows:

ψ (τ) = 2Lε

πεsinε

(πτL

)ε − 2Lε

(2π)εsinε

(2πτ

L

)ε+ 2Lε

(3π)εsinε

(3πτ

L

)ε.

(2.170)

When k = 4, we have

ψ (τ) =2Lε

πεsinε

(πτL

)ε − 2Lε

(2π)εsinε

(2πτ

L

)ε

+ 2Lε

(3π)εsinε

(3πτ

L

)ε− 2Lε

(4π)εsinε

(4πτ

L

)ε. (2.171)

When k = 5, the fractal signalψ (τ) in local fractional Fourier series can be writtenas follows:

ψ (τ) =2Lε

πεsinε

(πτL

)ε − 2Lε

(2π)εsinε

(2πτ

L

)ε

+ 2Lε

(3π)εsinε

(3πτ

L

)ε− 2Lε

(4π)εsinε

(4πτ

L

)ε+ 2Lε

(5π)εsinε

(5πτ

L

)ε.

(2.172)

The local fractional Fourier series of the fractal signal ψ (τ) when k = 1, k = 2,k = 3, k = 4, and k = 5 is shown in Figure 2.2.

We notice that the expression (2.148) is also applied to find the local fractionalFourier series representation of the fractal signal ψ (τ).

Expand the 2π -periodic fractal signal,

ψ (τ) = τ 2ε

� (1 + 2ε), (2.173)

on the interval −π < τ ≤ π in local fractional Fourier series, and its graph is shownin Figure 2.3.


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35−1.5

−1

−0.5

0

0.5

1

1.5

t

ψ2(t )

ψ1(t )

ψ3(t )

ψ4(t )

ψ5(t )

Figure 2.2 The local fractional Fourier series representation of fractal signal ψ (τ) whenε = ln 2/ ln 3, k = 1, k = 2, k = 3, k = 4, and k = 5.

−4 −3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t

ψ(t

)

Figure 2.3 The plot of fractal signal ψ (τ) is shown when ε = ln 2/ ln 3.


Making use of the expression in (2.128) to find the local fractional Fourier seriesrepresentation of fractal signal ψ (τ), we find that

A (0, ε) = 1

πε

∫ π

−πψ (τ) (dτ)ε

= 1

πε

∫ π

−πτ 2ε

� (1 + 2ε)(dτ)ε

= 2π2ε� (1 + ε)

� (1 + 3ε)(2.174)

and

A (k, ε) = 1

πε

∫ π

−πψ (τ) cosε (kτ)

ε (dτ)ε

= 1

πε

∫ π

−πτ 2ε

� (1 + 2ε)cosε (kτ)

ε (dτ)ε

= 2

πε

∫ π

0

τ 2ε

� (1 + 2ε)cosε (kτ)

ε (dτ)ε

= − 2

k2α . (2.175)

Therefore, the 2π -periodic fractal signal ψ (τ) on the interval 0 < τ ≤ 2π isexpressed as follows:

ψ (τ) = 2π2ε� (1 + ε)

� (1 + 3ε)−

∞∑k=1

[2

k2ε cosε (kτ)ε

]. (2.176)

Let us consider the 2π -periodic fractal signal

ψ (τ) = τ 3ε

� (1 + 3ε)(2.177)

on the interval −π < τ ≤ π in local fractional Fourier series, and its graph is shownin Figure 2.4.

Adopting the expression in (2.133) to find the local fractional Fourier seriesrepresentation of the fractal signal ψ (τ), we present the following local fractionalFourier series coefficient:

B (k, ε) = 1

πε

∫ π


ε (dτ)ε

= 1

πε

∫ π

−π

[τ 3ε

� (1 + 3ε)sinε (kτ)

ε

](dτ)ε

= 2

πε

∫ π

0

[τ 3ε

� (1 + 3ε)sinε (kτ)

ε

](dτ)ε

= 1

kεπ3ε

� (1 + εα)− 1

k3ε

πε

� (1 + ε). (2.178)


−4 −3 −2 −1 0 1 2 3 4−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

t

ψ(t

)

Figure 2.4 The plot of fractal signal ψ (τ) is shown when ε = ln 2/ ln 3.

Therefore, the nondifferentiable signal is expressed as follows:

ψ (τ) =∞∑

k=1

(1

kεπ3ε

� (1 + εα)− 1

k3ε

πε

� (1 + ε)

)sinε (kτ)

ε. (2.179)

2.4 Solving local fractional differential equations

We now consider some applications of local fractional Fourier series to handle thelocal fractional differential equations (see [99–104]).

2.4.1 Applications of local fractional ordinary differentialequations

We now consider the following local fractional ordinary differential equation

d2εψ (τ)

dτ 2ε + ψ (τ) = τ ε

� (1 + ε), τ ∈ (0,π) (2.180)

subject to initial value conditions given by

ψ (0) = 0, ψ (π) = 0. (2.181)


We can write the function

τ ε/� (1 + ε)

as follows:

τ ε

� (1 + ε)=

∞∑k=1

B (k, ε) sinε (kτ)ε, τ ∈ (0,π) , (2.182)

where

B (k, ε) = 1

πε

∫ 2π

0

τ ε

� (1 + ε)

[sinε (kx)ε

](dx)ε

= −(

2πk

)ε� (1 + ε)

. (2.183)

In this case, we can rewrite ψ (τ) as follows:

ψ (τ) =∞∑

k=1

B (k, ε) sinε (kτ)ε, τ ∈ (0,π) , (2.184)

which leads us to

d2εψ (τ)

dτ 2ε = −∞∑

k=1

B (k, ε) k2ε sinε (kτ)ε, τ ∈ (0,π) . (2.185)

Using (2.180), (2.184), and (2.185), we obtain

−∞∑

n=1

B (k, ε) k2ε sinε (kτ)ε +

∞∑n=1

B (k, ε) sinε (kτ)ε = −

∞∑k=1

−(

2πk

)ε� (1 + ε)

sinε (kτ)ε.

(2.186)

Matching the coefficients for the same terms in the two series, we find from (2.186)that

(1 − k2ε

)B (k, ε) = −

(2πk

)ε� (1 + ε)

, k ∈ N, (2.187)

which leads us to

B (k, ε) =(

2πk

)ε(k2ε − 1

)� (1 + ε)

. (2.188)

Therefore, the local fractional Fourier solution for the local fractional differentialequation is given as follows:

ψ (τ) =∞∑

k=1

(2πk

)ε(k2ε − 1

)� (1 + ε)

sinε (kτ)ε. (2.189)


2.4.2 Applications of local fractional partial differentialequations

We now present some examples for solving the local fractional PDEs in mathematicalphysics using the local fractional Fourier series.

We consider the following homogeneous local fractional heat equation in thenondimensional case:

∂εψ (μ, τ)

∂τ ε− ∂2εψ (μ, τ)

∂μ2ε = 0, (2.190)


ψ (0, τ) = 0, (2.191)

ψ (L, τ) = 0, (2.192)

and

ψ (μ, 0) = ϑ (μ) . (2.193)

Suppose that

ψ (μ, τ) = �(μ)� (τ) . (2.194)

In this case, we transform (2.190) into the following formula:

γ = �(ε) (τ )

� (τ)= �(2ε) (μ)

� (μ). (2.195)

We set γ = −λ2ε such that

�(ε) (τ )+ λ2ε� (τ) = 0 (2.196)

and

�(2ε) (μ)+ λ2ε� (μ) = 0. (2.197)

We now present the following terms:

�(τ) = C (λ, ε)Eε(−λ2ετ ε

)(2.198)

and

�(μ) = A (λ, ε) cosε(λεμε

)+ B (λ, ε) sinε(λεμε

), (2.199)

where A (λ, ε) , B (λ, ε), and C (λ, ε) are constants to be determined.In this case, we give the general solution as follows:

ψ (μ, τ) = C (λ, ε)Eε(−λ2ετ ε

) [A (λ, ε) cosε

(λεμε


)].

(2.200)


Using (2.191), (2.200) can be rewritten in the form

ψ (μ, τ) = A (λ, ε)C (λ, ε)Eε(−λ2ετ ε

)sinε

(λεμε

)(2.201)

or, equivalently,

B (λ, ε) = 0. (2.202)

Thus, by using (3.192), we have

A (λ, ε) = 0 (2.203)

such that

λ = kπ

L, k ∈ N0. (2.204)

Therefore, we rewrite (2.201) as follows:

ψ (μ, τ) = A

(kπ

L, ε

)C

(kπ

L, ε

)Eε

(−(

kπ

L

)2ε

τ ε

)sinε

((kπ

L

)εμε)

.

(2.205)

In this case, we set the local fractional Fourier series as given below:

ψ (μ, τ) = θ (0, ε)

2+

∞∑k=0

θ (k, ε)Eε

(−(

kπ

L

)2ε

τ ε

)sinε

((kπ

L

)εμε)

,

(2.206)

which leads us to

ψ (μ, 0) = θ (0, ε)

2+

∞∑k=0

θ (k, ε) sinε

((kπ

L

)εμε)

, (2.207)

where the local fractional Fourier coefficients are confirmed by

θ (k, ε) = 1

Lε

∫ L

−L

[ϑ (μ) sinε

((kπ

L

)εμε)](dμ)ε. (2.208)

Therefore, we obtain the local fractional Fourier solution in the form

ψ (μ, τ) = θ (0, ε)

2+

∞∑k=0

θ (k, ε)Eε

(−(

kπ

L

)2ε

τ ε

)sinε

((kπ

L

)εμε)

,

(2.209)

where the local fractional Fourier series can be written in the form

θ (k, ε) = 1

Lε

∫ L

0

[ϑ (μ) sinε

((kπ

L

)εμε)](dμ)ε, k ∈ N0. (2.210)


We solve the following nonhomogeneous local fractional heat equation in thenondimensional case:

dεψ (μ, τ)

dτ ε− d2εψ (μ, τ)

dμ2ε= G (μ, τ) , (2.211)

subject to the initial value conditions given by

ψ (0, τ) = 0, (2.212)

ψ (L, τ) = 0, (2.213)

and

ψ (μ, 0) = ϑ (μ) . (2.214)

We present the local fractional Fourier solution given by

ψ (μ, τ) =∞∑

k=1

ψk (τ ) sinε

(kπμ

L

)ε. (2.215)

We suppose that there are {Gk (τ )}∞k=1 and {ψk (0)}∞k=1 such that

G (μ, τ) =∞∑

k=1

Gk (τ ) sinε

(kπμ

L

)ε(2.216)

and

ψ (μ, 0) =∞∑

k=1

ψk (0) sinε

(kπμ

L

)ε, (2.217)

where

Gk (τ ) = 1

Lε

∫ L

0

[G (μ, τ) sinε

((kπ

L

)εμε)](dμ)ε (2.218)

and

ψk (0) = 1

Lε

∫ L

0

[ψ (μ, 0) sinε

((kπ

L

)εμε)](dμ)ε. (2.219)

Hence, by adopting (2.215) and (2.216), (2.211) becomes

∞∑k=1

[dεψk (τ )

dτ ε−(πk

L

)εψk (τ )− Gk (τ )

]sinε

(kπμ

L

)ε= 0. (2.220)

For any value sinε(

kπμL

)ε, we present

dεψk (τ )

dτ ε−(πk

L

)εψk (τ )− Gk (τ ) = 0, (2.221)


where

ψk (0) = 1

Lε

∫ L

0

[ψ (μ, 0) sinε

((kπ

L

)εμε)](dμ)ε

= 1

Lε

∫ L

0

[ϑ (μ) sinε

((kπ

L

)εμε)](dμ)ε. (2.222)

Hence, we obtain

ψk (τ ) = ψk (0)Eε

(kπτ

L

)ε+ 1

� (1 + ε)

∫ τ

0Eε

(kπ (τ − χ)

L

)εGk (χ) (dχ)

ε.

(2.223)

From (2.217) and (2.223), we have the local fractional Fourier series solutiongiven by

ψ (μ, τ) =∞∑

k=1

ψk (τ ) sinε

(kπμ

L

)ε

=∞∑

k=1

[ψk (0)Eε

(kπτ

L

)ε]sinε

(kπμ

L

)ε

+∞∑

k=1

[1

� (1 + ε)

∫ τ

0Eε

(kπ (τ − χ)

L

)εGk (χ) (dχ)

ε

]sinε

(kπμ

L

)ε.

(2.224)

Let us now consider the local fractional Laplace equation in the form

∂2εψ (μ, η)

∂μ2ε + ∂2εψ (μ, η)

∂η2ε = 0, (2.225)


ψ (μ, 0) = ϑ (μ) , (2.226)

ψ (μ, ρ) = 0, (2.227)

ψ (0, η) = 0, (2.228)

and

ψ (L, η) = 0. (2.229)

Suppose that we construct a special solution in the form

ψ (μ, η) = �(μ)� (η) . (2.230)


γ = �(2ε) (μ)

� (μ)= −�

(2ε) (η)

� (η). (2.231)

Then, by separating the variables, we obtain

γ = −λ2ε (2.232)


such that

�(2ε) (μ)+ λ2ε� (μ) = 0 (2.233)

and

�(2ε) (η)− λ2ε� (η) = 0. (2.234)

Using (2.227) and (2.228), we find that

�(0) = �(L) = 0. (2.235)

From (3.234), we get the general solution in the form



), (2.236)

where A (λ, ε) and B (λ, ε) are the coefficients involved.Using (2.235), (2.236) becomes

�(μ) = B (λ, ε) sinε(λεμε

), (2.237)

A (λ, ε) = 0, (2.238)

and

sinε(λεLε

) = 0. (2.239)

Hence, we find from (2.239) that

λε =(

kπ

L

)ε, k ∈ N0, (2.240)

such that

�(μ) = B (λ, ε) sinε

((kπ

L

)εμε)

. (2.241)

From (2.240) and (2.241), we have

�(η) = C (λ, ε) coshε

((kπ

L

)ε(η − ρ)ε

)+D (λ, ε) sinhε

((kπ

L

)ε(η − ρ)ε

).

(2.242)

Adopting (2.226), we get

C (λ, ε) = 0 (2.243)

such that (2.242) becomes

�(η) = D (λ, ε) sinhε

((kπ

L

)ε(η − ρ)ε

). (2.244)

Therefore, we have

ψ (μ, η) = D (λ, ε)B (λ, ε) sinε

((kπ

L

)εμε)

sinhε

((kπ

L

)ε(η − ρ)ε

).

(2.245)


The local fractional Fourier solution reads as follows:

ψ (μ, η) =∞∑

k=1

{D (λ, ε)B (λ, ε) sinε

((kπ

L

)εμε)

sinhε

((kπ

L

)ε(η − ρ)ε

)}.

(2.246)

With the help of (2.246), we get

ψ (μ, 0) =∞∑

k=1

{D (λ, ε)B (λ, ε) sinhε

((kπ

L

)ε(0 − ρ)ε

)sinε

((kπ

L

)εμε)}

=∞∑

k=1

{ψk (μ, 0) sinε

((kπ

L

)εμε)}

= ϑ (μ) , (2.247)

where

ψk (μ, 0) = −D (λ, ε)B (λ, ε) sinhε

((kπ

L

)ερε)

. (2.248)

Consequently, the local fractional Fourier coefficients are given by

ψk (μ, 0) = 1

Lε

∫ L

0

[ψ (μ, 0) sinε

((kπ

L

)εμε)](dμ)ε

= 1

Lε

∫ L

0

[ϑ (μ) sinε

((kπ

L

)εμε)](dμ)ε. (2.249)

In this case, making use of (2.248), we obtain

D (λ, ε)B (λ, ε) = −1

Lε∫ L

0

[ϑ (μ) sinε

(( kπL

)εμε)](dμ)ε

sinhε(( kπ

L

)ερε) . (2.250)

Hence, the local fractional Fourier solution of the local fractional Laplace equationis given by

ψ (μ, η) =∞∑

k=1

{D

(kπ

L, ε

)B

(kπ

L, ε

)sinε

((kπ

L

)εμε)

sinhε

((kπ

L

)ε(η − ρ)ε

)}

=∞∑

k=1

⎧⎨⎩−

1Lε∫ L

0

[ϑ (μ) sinε

(( kπL

)εμε)](dμ)ε

sinhε(( kπ

L

)ερε) sinε

((kπ

L

)εμε)

sinhε

((kπ

L

)ε(η − ρ)ε

)⎫⎬⎭ . (2.251)


The local fractional wave equation in the nondimensional case takes the followingform:

∂2εψ (μ, τ)

∂μ2ε − ∂2εψ (μ, τ)

∂τ 2ε = 0, (2.252)


ψ (μ, 0) = ϑ (μ) , (2.253)

ψ (μ, ρ) = 0, (2.254)

ψ (0, τ) = 0, (2.255)

and

ψ (L, τ) = 0. (2.256)

Suppose now that we construct a special solution in the form

ψ (μ, η) = �(μ)� (τ) . (2.257)


γ = �(2ε) (μ)

� (μ)= �(2ε) (τ )

� (τ). (2.258)

Thus, by separating the variables, we have

γ = −λ2ε (2.259)

such that

�(2ε) (μ)+ λ2ε� (μ) = 0 (2.260)

and

�(2ε) (τ )+ λ2ε� (τ) = 0. (2.261)

Using (2.255) and (2.256), we obtain

�(0) = �(L) = 0. (2.262)

From (2.260), we get the general solution in the form



), (2.263)

where A (λ, ε) and B (λ, ε) are coefficients.From (2.261), the general solution reads as follows:

�(τ) = C (λ, ε) cosε(λε (τ − ρ)ε

)+ D (λ, ε) sinε(λε (τ − ρ)ε

), (2.264)

where C (λ, ε) and D (λ, ε) are coefficients.Using (2.262), (2.263) can be written as follows:

�(μ) = B (λ, ε) sinε(λεμε

), (2.265)


A (λ, ε) = 0, (2.266)

and

sinε(λεLε

) = 0. (2.267)

Hence, with the help of (2.267), we get

λε =(

kπ

L

)ε, k ∈ N0, (2.268)

such that

�(μ) = B (λ, ε) sinε

((kπ

L

)εμε)

. (2.269)

In view of (2.264), (2.265), and (2.268), we refer to the general solution in the form

ψ (μ, τ) = �(μ)� (τ)

= B (λ, ε) sinε(λεμε

) [C (λ, ε) cosε

(λε (τ − ρ)ε

)+D (λ, ε) sinε

(λε (τ − ρ)ε

)]. (2.270)

Hence, we have

ψ (μ, τ) =∞∑

k=1

B (λ, ε) sinε(λεμε

) [C (λ, ε) cosε

(λε (τ − ρ)ε

)+D (λ, ε) sinε

(λε (τ − ρ)ε

)], (2.271)

which, in light of (2.254), leads us to

C (λ, ε) = 0 (2.272)

and

ψ (μ, τ) =∞∑

k=1

B (λ, ε)D (λ, ε) sinε(λε (τ − ρ)ε

)sinε

(λεμε

). (2.273)

Utilizing (2.253) and (2.268), we get

ψ (μ, 0) =∞∑

k=1

B (λ, ε)D (λ, ε) sinε(λε (0 − ρ)ε

)sinε

(λεμε

)

= −∞∑

k=1

B

(kπ

L, ε

)D

(kπ

L, ε

)sinε

((kπ

L

)ερε)

sinε

((kπ

L

)εμε)

=∞∑

k=1

ψk (μ, 0) sinε

((kπ

L

)εμε)

= ϑ (μ) , (2.274)


where

ψk (μ, 0) = −B

(kπ

L, ε

)D

(kπ

L, ε

)sinε

((kπ

L

)ερε)

(2.275)

and

ψk (μ, 0) = 1

Lε

∫ L

0

[ψ (μ, 0) sinε

((kπ

L

)εμε)](dμ)ε

= 1

Lε

∫ L

0

[ϑ (μ) sinε

((kπ

L

)εμε)](dμ)ε. (2.276)

From (2.275) and (2.276), we easily obtain

B

(kπ

L, ε

)D

(kπ

L, ε

)= −

1Lε∫ L

0

[ϑ (μ) sinε

(( kπL

)εμε)](dμ)ε

sinε(( kπ

L

)ερε) . (2.277)

Therefore, the local fractional Fourier solution of the local fractional wave equationin the nondimensional case has the form

ψ (μ, η) =∞∑

k=1

B

(kπ

L, ε

)D

(kπ

L, ε

)sinε

((kπ

L

)ε(τ − ρ)ε

)sinε

((kπ

L

)εμε)

=∞∑

k=1

⎧⎨⎩−

1Lε∫ L

0

[ϑ (μ) sinε

(( kπL

)εμε)](dμ)ε

sinε(( kπ

L

)ερε) sinε

((kπ

L

)εμε)

sinε

((kπ

L

)ε(τ − ρ)ε

)⎫⎬⎭ . (2.278)

3Local fractional Fourier transformand applications

3.1 Introduction

Fourier transforms play an important role in the theoretical analysis of mathematicalmodels for the problems that appear in mathematical physics, engineering applica-tions, and theoretical and applied physics, as in quantum mechanics, signal analysis,control theory, and both pure and applied mathematics. Especially, the Fourier theoryis utilized to analyze the nonperiodic phenomena of heat conduction.

Recently, it was found that a nondifferentiable function in fractal time is de-composed into the local fractional Fourier series via the Mittag–Leffler functionsor the analogous sine functions and the analogous cosine functions defined on thefractal set because there are analogous trigonometric functions defined on Cantorsets. We expand the idea of the Fourier transform operator to the local fractionalintegral transform in the case of the Mittag–Leffler functions defined on Cantor sets.In order to understand the concept, the following problems should be consideredfirst:

Problem 1. What mechanism does the local fractional Fourier transform operatorpossess?

Problem 2. Does the local fractional Fourier transform preserve the energy of theoriginal quantity of nondifferentiable signals?

We give the answers to the above problems in this chapter. We start by presentingthe basic mechanism of local fractional Fourier transform operator.


3.2.1 Mathematical mechanism is the local fractional Fouriertransform operator

Suppose that φ (τ) is 2L-periodic and locally fractional continuous on the interval[−L, L]. Then, φ (τ) can be decomposed into the local fractional Fourier series inthe form

φ (τ) =∞∑

k=−∞ϕεk Eε

(πεiε (kτ)ε

Lε

), k ∈ Z, (3.1)

where the local fractional Fourier coefficients can determined by

ϕεk = 1

(2L)ε

∫ L

−Lφ (τ)Eε

(−π

εiε (kτ)ε

Lε

)(dτ)ε. (3.2)


http://dx.doi.org/10.1016/B978-0-12-804002-7.00003-6


Now, let us define the local fractional Fourier coefficients as follows:

ϕεk = � (1 + ε)

(2L)ε�εk , k ∈ Z. (3.3)

Therefore, (3.1) and (3.2) can be rewritten in the form

φ (τ) = 1

(2L)ε

∞∑k=−∞

�εkEε

(πεiε (kτ)ε

Lε

)(3.4)

and

�εk = 1

� (1 + ε)

∫ L

−Lφ (τ)Eε

(−π

εiε (kτ)ε

Lε

)(dτ)ε, (3.5)

where k ∈ Z.Defining

θεk =(πk

L

)ε, (3.6)

we have

(θk)ε = (θk+1 − θk)

ε =(π

L

)ε. (3.7)

Therefore, (3.4) can be rewritten as follows:

φ (τ) = 1

(2L)ε

∞∑k=−∞

�εkEε

(iετ ε

(πk

L

)ε)

= 1

(2π)ε

∞∑k=−∞

�εkEε(iετ εθεk

)(θk)

ε, (3.8)

which leads us to

limL→∞φ (τ) = lim

L→∞

[1

(2L)ε

∞∑k=−∞

�εkEε

(iετ ε

(πk

L

)ε)]

= limL→∞

[1

(2π)ε

∞∑k=−∞

�εkEε(iετ εθεk

)(θk)

ε

]

= 1

(2π)ε

∫ ∞

−∞�εkEε

(iετ εθεk

)(dθk)

ε (3.9)

and

limL→∞�

εk = lim

L→∞

[1

� (1 + ε)

∫ L

−Lφ (τ)Eε

(−iετ ε

(πk

L

)ε)(dτ)ε

]= lim

L→∞

[1

� (1 + ε)

∫ ∞

−∞φ (τ)Eε

(iετ εθεk

)(dτ)ε

], (3.10)

where k ∈ Z.

Local fractional Fourier transform and applications 97

Thus, (3.9) and (3.10) can be written as follows:

φ (τ) = 1

(2π)ε

∫ ∞

−∞�(ω)Eε

(iετ εωε

)(dω)ε

= � (1 + ε)

(2π)ε−∞I(ε)∞

[�(ω)Eε

(iετ εωε

)](3.11)

and

�(ω) = 1

� (1 + ε)

∫ ∞

−∞φ (τ)Eε

(−iετ εωε)(dτ)ε

= −∞I(ε)∞[φ (τ)Eε

(−iετ εωε)]

, (3.12)

where τ ,� ∈ R,

ωε = limL→∞

(πk

L

)ε,

(dω)ε = limL→∞

(πL

)ε,

and

�(ω) = limL→∞�

εk

and they converge.Let us define the variable ω by

ω = 2π� . (3.13)

Then, we rewrite (3.11) and (3.12) in the form

φ (τ) =∫ ∞

−∞�(�)Eε

[(2π i)ε τ ε�ε

](d�)ε

= � (1 + ε)−∞I(ε)∞ �(�)Eε[(2π i)ε τ ε�ε

](3.14)

and

�(�) = 1

� (1 + ε)

∫ ∞

−∞φ (τ)Eε

[− (2π i)ε τ ε�ε](dτ)ε

= −∞I(ε)∞{φ (τ)Eε

[− (2π i)ε τ ε�ε]}

, (3.15)

where τ ,� ∈ R.The next step is to define the variable ωε by

ωε = σεh0, (3.16)

where

h0 = (2π)ε

� (1 + ε),


and (3.11) and (3.12) are restructured as follows:

φ (τ) = 1

� (1 + ε)

∫ ∞

−∞[�(σ)Eε

(iεh0τ

εσ ε)](dσ)ε

= −∞I(ε)∞[�(σ)Eε

(iεh0τ

εσ ε)]

(3.17)

and

�(σ) = 1

� (1 + ε)

∫ ∞

−∞[φ (τ)Eε

(−iεh0τεσ ε

)](dτ)ε

= −∞I(ε)∞[φ (τ)Eε

(−iεh0τεσ ε

)], (3.18)

where τ , σ ∈ R.The alternative forms of (3.17) and (3.18) are written as follows:

φ (τ) = 1

� (1 + ε)

∫ ∞

−∞{�(σ)Eε

[(2π i)ε ζ (ε) τ εσ ε

]}(dσ)ε

= −∞I(ε)∞[�(σ)Eε

[(2π i)ε ζ (ε) τ εσ ε

]](3.19)

and

�(σ) = 1

� (1 + ε)

∫ ∞

−∞φ (τ)Eε

(− (2π i)ε ζ (ε) τ εσ ε)(dτ)ε

= −∞I(ε)∞[φ (τ)Eε

(− (2π i)ε ζ (ε) τ εσ ε)]

, (3.20)

where τ , σ ∈ R and

ζ (ε) = 1

� (1 + ε). (3.21)

In a similar manner, we have the following result.Suppose that φ (τ) is 2L-periodic, bounded, and locally fractional integrable on the

interval [−L, L]. In a similar way, we conclude that

(Iφ) (τ) ≡ φ (τ + 0)+ φ (τ − 0)

2= φ (τ + 0)+ φ (τ − 0)

2

= 1

(2π)ε

∫ ∞

−∞�(ω)Eε

(iετ εωε

)(dω)ε (3.22)

and

�(ω) = 1

� (1 + ε)

∫ ∞

−∞(Iφ) (τ)Eε


= −∞I(ε)∞[( Iφ) (τ)Eε

(−iετ εωε)]

, (3.23)

respectively.This is a transfer pair referring to the nondifferentiable function, which is periodic,

bounded, and locally fractional integrable on the interval [−L, L].


By using (3.22) and (3.23), the local fractional Fourier formula for the nondiffer-entiable function becomes

φ (τ + 0)+ φ (τ − 0)

2= 1

(2π)ε

∫ ∞

−∞

{[1

� (1 + ε)

∫ ∞

−∞φ (τ)Eε


]Eε

(iετ εωε

)}(dω)ε

= � (1 + ε)

(2π)ε−∞I(ε)∞

{[−∞I(ε)∞

(φ (τ)Eε

(−iετ εωε))]

Eε(iετ εωε

)}. (3.24)

Suppose that φ (τ) is 2L-periodic and the locally fractional continuous on theinterval [−L, L]. Thus, the local fractional Fourier formula for the nondifferentiablefunction can be written in the form

φ (τ) = 1

(2π)ε

∫ ∞

−∞

{[1

� (1 + ε)

∫ ∞

−∞φ (τ)Eε


]Eε

(iετ εωε

)}(dω)ε

= � (1 + ε)

(2π)ε−∞I(ε)∞

{[−∞I(ε)∞

(φ (τ)Eε

(−iετ εωε))]

Eε(iετ εωε

)}. (3.25)

This is called the local fractional Fourier formula for the nondifferentiable function.Thus, we have completed the derivation of the local fractional Fourier transformoperators.

3.2.2 Definitions of the local fractional Fourier transformoperators

We now give the basic definitions of the local fractional Fourier transform operatorsvia the local fractional integral operator (see [16, 21, 27, 76, 96, 97, 105–107]). Inorder to study the local fractional Fourier transforms, we define the generalized spaceLν,ε [R] under the ν-norm given by

‖θ‖ν,ε =(

1

� (1 + ε)

∫ ∞

−∞|θ (τ )|ν (dτ)ε

)1/ν

(3.26)

for 1 ≤ ε < ∞ and 1 ≤ ν < ∞.

Definition 3.1. By setting

θ ∈ L1,ε [R] and ‖θ‖1,ε < ∞,

the local fractional Fourier transform operator, denoted by

� [θ (τ )] = �(ω)


is defined by

� [θ (τ )] = �(ω) = 1

� (1 + ε)

∫ ∞

−∞θ (τ )Eε

(−iετ εωε)(dτ)ε, (3.27)

where � is called the local fractional Fourier transform operator.

Definition 3.2. The inverse local fractional Fourier transform operator, denoted by

�−1 [�(ω)] = θ (τ )

is defined by

�−1 [�(ω)] = θ (τ ) = 1

(2π)ε

∫ ∞

−∞�(ω)Eε

(iετ εωε

)(dω)ε, (3.28)

where �−1 is called the inverse local fractional Fourier transform operator.

Making use of (3.14) and (3.15), we present the local fractional Fourier formula inthe following form:

φ (τ) = 1

� (1 + ε)

∫ ∞

−∞

(∫ ∞

−∞φ (τ)Eε


)Eε


](d�)ε

= � (1 + ε)−∞I(ε)∞{−∞I(ε)∞ φ (τ)Eε

[− (2π i)ε τ ε�ε]}

Eε[(2π i)ε τ ε�ε

]= 1√

� (1 + ε)

∫ ∞

−∞

(∫ ∞

−∞φ (τ)√� (1 + ε)

Eε[− (2π i)ε τ ε�ε

](dτ)ε

)Eε


](d�)ε . (3.29)

In this case, we find the following quantity:

H (�) =∫ ∞

−∞φ (τ)√� (1 + ε)


](dτ)ε. (3.30)

We now revise (3.30) and we obtain the following alternative definition of localfractional Fourier transform operator.

Definition 3.3. Upon setting g ∈ L1,ε [R] and ‖g‖1,ε < ∞, the generalized localfractional Fourier transform operator, denoted by

� [g (τ )] = G (�) ,

is defined by

G (�) = √� (1 + ε)−∞I(ε)∞

{g (τ )Eε

[− (2π i)ε τ ε�ε]}

= 1√� (1 + ε)

∫ ∞

−∞g (τ )Eε

[− (2π i)ε τ ε�ε](dτ)ε, (3.31)


where τ ,� ∈ R, � is called the generalized local fractional Fourier transformoperator.

Definition 3.4. The inverse generalized local fractional Fourier transform operator,denoted by

�−1 [G (�)] = g (τ ) ,

is defined by

g (τ ) = √� (1 + ε)−∞I(ε)∞

{G (�)Eε


]}= 1√

� (1 + ε)

∫ ∞

−∞G (�)Eε


](d�)ε, (3.32)

where τ ,� ∈ R and � is called the inverse generalized local fractional Fouriertransform operator.

In this case, we find that ω = 2π� .For various definitions of the local fractional Fourier transform operators, we refer

the reader to the earlier works [16, 21, 27, 76, 96, 97, 105–107]. Equation (3.17)provides the common definition of the local fractional Fourier transform operator to beapplied to find the solutions of partial differential equations [27–105]. Equation (3.29)was utilized to handle the nondifferentiable problems in mathematical physics [100].Therefore, we consider the expressions (3.27) and (3.31) throughout this chapter inorder to present some examples.

3.2.3 Properties and theorems of local fractional Fouriertransform operator

Theorem 3.1 (Fourier integral theorem for local fractional Fourier transformoperator). Suppose that θ (τ ) is local fractional continuous on the interval [−∞, ∞](or θ (τ ) ,�(ω) ∈ L1,ε [R] ). Thus, we have

θ (τ ) = � (1 + ε)

(2π)ε−∞I(ε)∞

{[−∞I(ε)∞

(θ (τ )Eε

(−iετ εωε))]

Eε(iετ εωε

)}.

(3.33)

This is the Fourier integral theorem for the local fractional Fourier transformoperator.

Proof. Since θ (τ ) is locally fractional continuous on the interval [−∞, ∞], from(3.24), we conclude that

φ (τ + 0) = φ (τ − 0) = φ (τ) , (3.34)


so that

θ (τ ) = 1

(2π)ε

∫ ∞

−∞

{[1

� (1+ε)∫ ∞

−∞θ (τ )Eε


]Eε

(iετ εωε

)}(dω)ε

= � (1 + ε)

(2π)ε−∞I(ε)∞

{[−∞I(ε)∞

(θ (τ )Eε

(−iετ εωε))]

Eε(iετ εωε

)}. (3.35)

The asserted claim is thus proved.

Property 7 (Linearity for the local fractional Fourier transform operator). Sup-pose that

θ1 (τ ) , θ2 (τ ) ,�1 (ω) ,�2 (ω) ∈ L1,ε [R] ,

� [θ1 (τ )] = �1 (ω)

and

� [θ2 (τ )] = �2 (ω) .

Then,

� [aθ1 (τ )± bθ2 (τ )] = a�1 (ω)± b�2 (ω) , (3.36)

where a and b are constants.

Proof. By using the definition of the local fractional Fourier transform operator,we have

� [aθ1 (τ )± bθ2 (τ )] = 1

� (1 + ε)

∫ ∞

−∞[aθ1 (τ )± bθ2 (τ )] Eε


= a1

� (1 + ε)

∫ ∞

−∞[θ1 (τ )Eε

(−iετ εωε)](dτ)ε

± b1

� (1 + ε)

∫ ∞

−∞[θ2 (τ )Eε

(−iετ εωε)](dτ)ε

= a�1 (ω)± b�2 (ω) . (3.37)

Thus, the asserted claim is proved.

Property 8 (Shifting time for the local fractional Fourier transform operator).Suppose that

θ (τ ) ,�(ω) ∈ L1,ε [R] ,

� [θ (τ )] = �(ω) ,

and a is a constant. Then,

� [θ (τ − a)] = Eε(−iεaεωε

) � [θ (τ )] . (3.38)


Proof. The definition of the local fractional Fourier transform operator leads us to

� [θ (τ − a)] = 1

� (1 + ε)

∫ ∞

−∞θ (τ − a)Eε


= Eε(−iεaεωε

) 1

� (1+ε)∫ ∞

−∞θ (τ−a)Eε

(−iε (τ−a)ε ωε)(dτ)ε

= Eε(−iεaεωε

)�(ω)

= Eε(−iεaεωε

) � [θ (τ )] , (3.39)

which evidently completes the proof.

Property 9 (Scaling time for local fractional Fourier transform operator). Supposeθ (τ ) ,�(ω) ∈ L1,ε [R], � [θ (τ )] = �(ω), and a (a > 0) is a constant, then, there is

� [θ (aτ)] = 1

aε�

(ωa

). (3.40)

Proof. Using the definition of the local fractional Fourier transform operator

� [θ (aτ)] = 1

� (1 + ε)

∫ ∞

−∞θ (aτ)Eε


= 1

aε1

� (1 + ε)

∫ ∞

−∞θ (aτ)Eε

(−iε

(aτ

a

)εωε

)(daτ)ε

= 1

aε�

(ωa

). (3.41)

Therefore, the proof is completed.

Property 10 (Conjugate for local fractional Fourier transform operator). Supposethat

θ (τ ) ,�(ω) ∈ L1,ε [R]

and

� [θ (τ )] = �(ω) .

Then,

�[θ (−τ)

]= � [θ (τ )]. (3.42)

Proof. By using the definition of the local fractional Fourier transform operator,we get

�[θ (−τ)

]= 1

� (1 + ε)

∫ ∞

−∞θ (−τ)Eε


= 1

� (1 + ε)

∫ ∞

−∞θ (−τ)Eε (iετ εωε) (dτ)

ε


= 1

� (1 + ε)

∫ ∞

−∞θ (−τ)Eε (−iε (−τ )ε ωε) (dτ)ε

= 1

� (1 + ε)

∫ ∞

−∞θ (τ )Eε (−iετ εωε) (dτ)ε

= � [θ (τ )]. (3.43)

Thus, we have completed the proof.

Property 11 (Translation for local fractional Fourier transform operator). Supposethat

θ (τ ) ,�(ω) ∈ L1,ε [R] ,

� [θ (τ )] = �(ω) ,


� [Eε

(iεaετ ε

)θ (τ )

] = �(ω − a) . (3.44)

Proof. Once again, by using the definition of the local fractional Fourier transformoperator, we conclude that

� [Eε

(−iεaετ ε)θ (τ )

]= 1

� (1+ε)∫ ∞

−∞[θ (τ )Eε

(iεaετ ε

)]Eε


= 1

� (1 + ε)

∫ ∞

−∞θ (τ )Eε

(−iε (ω − a)ε τ ε)(dτ)ε

= �(ω − a) . (3.45)

Therefore, we have proved the claim.

Property 12 (Duality for local fractional Fourier transform operator). Supposethat

θ (τ ) ,�(ω) ∈ L1,ε [R]

and

� [θ (τ )] = �(ω) .

Then,

� [θ (−τ)] = � (1 + ε)

(2π)ε� [�(τ)] . (3.46)

Proof. It follows directly from the definition of the local fractional Fouriertransform operator that

θ (τ ) = 1

(2π)ε

∫ ∞

−∞�(ω)Eε

(iετ εωε

)(dω)ε = �−1 [�(ω)] . (3.47)


Upon interchanging τ and ω, from (3.47), we have

θ (ω) = 1

(2π)ε

∫ ∞

−∞�(τ)Eε

(iεωετ ε

)(dτ)ε. (3.48)

By replacing ω by −ω, we obtain

θ (−ω) = � (1 + ε)

(2π)ε

[1

� (1 + ε)

∫ ∞

−∞�(τ)Eε

(−iεωετ ε)(dτ)ε

]= � (1 + ε)

(2π)ε� [�(τ)] . (3.49)

This proves the result (3.46).

Property 13 (Composition for the local fractional Fourier transform operator).Suppose that

θ1 (τ ) , θ2 (τ ) ,�1 (ω) ,�2 (ω) ∈ L1,ε [R] ,

� [θ1 (τ )] = �1 (ω) ,

and

� [θ2 (τ )] = �2 (ω) .

Then,∫ ∞

−∞θ1 (ω)�2 (ω)Eε

(iετ εωε

)(dω)ε =

∫ ∞

−∞θ2 (η)�1 (η − τ) (dη)ε. (3.50)

Proof. We write the left-hand side of (3.50) in the following form:∫ ∞

−∞θ1 (ω)�2 (ω)Eε

(iετ εωε

)(dω)ε

=∫ ∞

−∞θ1 (ω)Eε

(iετ εωε

) [1

� (1+ε)∫ ∞

−∞θ2 (η)Eε

(−iεηεωε)(dη)ε

](dω)ε

=∫ ∞

−∞θ2 (η) (dη)

ε

[1

� (1 + ε)

∫ ∞

−∞θ1 (ω)Eε

(−iε (η − τ)ε ωε)(dω)ε

]=

∫ ∞

−∞θ2 (η)�1 (η − τ) (dη)ε. (3.51)

Therefore, we have given the proof of (3.50).

It easily follows from (3.51) that∫ ∞

−∞θ1 (ω)�2 (ω) (dω)

ε =∫ ∞

−∞�1 (η) θ2 (η) (dη)

ε, (3.52)

where τ = 0.



θ (τ ) ,�(ω) ∈ L1,ε [R] ,

� [θ (τ )] = �(ω) ,

and

lim|τ |→∞ θ (τ ) = 0.

Then,

�[θ(ε) (τ )

]= (iω)ε � [θ (τ )] = iεωε� [θ (τ )] . (3.53)

Proof. The local fractional Fourier transform definition leads us to the operatorgiven by

�[θ(ε) (τ )

]= 1

� (1 + ε)

∫ ∞

−∞θ(ε) (τ )Eε

(−iετ εωε)(dτ)ε. (3.54)

Now, upon integration by part for the local fractional integral operator in (3.54), weconclude that

�[θ(ε) (τ )

]= 1

� (1 + ε)

∫ ∞

−∞θ(ε) (τ )Eε


=[θ (τ )Eε

(−iετ εωε)]∞

−∞ − iεωε

� (1+ε)∫ ∞

−∞θ (τ )Eε

(−iετ εωε)(dτ)ε,

(3.55)

which, by using

lim|τ |→∞ θ (τ ) = 0,

yields

�[θ(ε) (τ )

]= iεωε

[1

� (1 + ε)

∫ ∞

−∞θ (τ )Eε


]= iεωε� [θ (τ )] . (3.56)

Therefore, the result (3.53) follows.

In a similar manner, by repeating this process, we have

�[θ(kε) (τ )

]= (iω)kε � [θ (τ )] , k ∈ N, (3.57)

where

θ((k−1)ε) (0) = · · · = θ(ε) (0) = θ (0) = 0. (3.58)



θ (τ ) ,�(ω) ∈ L1,ε [R] ,

� [θ (τ )] = �(ω) ,

and

limτ→∞ −∞I(ε)τ θ (τ ) = 0.

Then,

�[−∞I(ε)τ θ (τ )

]= 1

(iω)ε� [θ (τ )] . (3.59)

Proof. In accordance with the definition of the local fractional Fourier transformoperator, we have

�[−∞I(ε)τ θ (τ )

]= 1

� (1 + ε)

∫ ∞

−∞

(−∞I(ε)τ θ (τ )

)Eε

(−iετ εωε)(dτ)ε, (3.60)

which, by using integration by part for the local fractional integral operator, yields

�[−∞I(ε)τ θ (τ )

]=

[−∞I(ε)τ θ (τ )Eε

(−iετ εωε)] ∣∣∣∞−∞

+[

(iω)ε

� (1 + ε)

∫ ∞

−∞θ (τ )Eε


]. (3.61)

Now, by taking

limτ→∞ −∞I(ε)τ θ (τ ) = 0

into account in (3.61), we conclude that

�[−∞I(ε)τ θ (τ )

]=

[(iω)ε

� (1 + ε)

∫ ∞

−∞θ (τ )Eε


]= (iω)ε � [θ (τ )] . (3.62)

Thus, the claim has been proved.

Continuing in the same manner, repeating the above methodology, we obtain

�[−∞I(kε)τ θ (τ )

]= 1

(iω)kε� [θ (τ )] , (3.63)

where

limτ→∞ −∞I(kε)τ θ (τ ) = 0. (3.64)


θ (τ ) ,�(ω) ∈ L1,ε [R]


and

� [θ (τ )] = �(ω) .

Then,

� [τ εθ (τ )

] = iε�(ε) (ω) . (3.65)

Proof. We consider

�−1[iε�(ε) (ω)

]= 1

(2π)ε

∫ ∞

−∞

[iε�(ε) (ω)

]Eε

(iετ εωε

)(dω)ε, (3.66)

which, by integrating by part for local fractional integral operator, leads us to

�−1[iε�(ε) (ω)

]= � (1 + ε)

(2π)ε[iε� (ω)Eε

(iετ εωε

)]∞−∞

+ τ ε[

1

(2π)ε

∫ ∞

−∞�(ω)Eε

(iετ εωε

)(dω)ε

]. (3.67)

Taking

lim|ω|→∞�(ω) = 0

into account in (3.65), we get

�−1[iε�(ε) (ω)

]= τ ε

[1

(2π)ε

∫ ∞

−∞�(ω)Eε

(iετ εωε

)(dω)ε

]= τ εθ (τ ) , (3.68)


In a similar manner, by repeating this process, we get

�[τ kεθ (τ )

]= ikε�(kε) (ω) , (3.69)

where

lim|ω|→∞�(kε) (ω) = 0, k ∈ N. (3.70)

Definition 3.5. The local fractional convolution of two functions θ1 (τ ) and θ2 (τ )

via the local fractional integral operator, denoted by

(θ1 ∗ θ2) (τ ) = θ1 (τ ) ∗ θ2 (τ ) ,

is defined by

(θ1 ∗ θ2) (τ ) = θ1 (τ ) ∗ θ2 (τ )

= −∞I(ε)∞ [θ1 (t) θ2 (τ − t)]

= 1

� (1 + ε)

∫ ∞

−∞θ1 (t) θ2 (τ − t) (dt)ε. (3.71)


Definition 3.6. The local fractional convolution of the local fractional Fouriertransform operators �1 (ω) and �2 (ω), denoted by

(�1 ∗�2) (ω) = �1 (ω) ∗�2 (ω) ,

is defined by

(�1 ∗�2) (ω) = �1 (ω) ∗�2 (ω)

= � (1 + ε)

(2π)ε−∞I(ε)∞ [�1 (�)�2 (ω −�)]

= 1

(2π)ε

∫ ∞

−∞�1 (�)�2 (ω −�) (d�)ε. (3.72)

From the definition of the local fractional convolution of two functions, we havethe following properties:(a) θ1 (τ ) ∗ θ2 (τ ) = θ2 (τ ) ∗ θ1 (τ ) and(b) θ1 (τ ) ∗ (θ2 (τ )+ θ3 (τ )) = θ1 (τ ) ∗ θ2 (τ )+ θ1 (τ ) ∗ θ3 (τ ).


θ1 (τ ) , θ2 (τ ) ,�1 (ω) ,�2 (ω) ∈ L1,ε [R] ,

� [θ1 (τ )] = �1 (ω) ,

and

� [θ2 (τ )] = �2 (ω) .

Then,

� [θ1 (τ ) ∗ θ2 (τ )] = �1 (ω)�2 (ω) (3.73)

or

θ1 (τ ) ∗ θ2 (τ ) = �−1 [�1 (ω)�2 (ω)] (3.74)

or, equivalently,

1

� (1 + ε)

∫ ∞

−∞θ1 (t) θ2 (τ − t) (dt)ε= 1

(2π)ε

∫ ∞

−∞�1 (ω)�2 (ω)Eε

(iετ εωε

)(dω)ε.

(3.75)

Proof. With the help of the definition of the local fractional Fourier transformoperator, we have

� [θ1 (τ ) ∗ θ2 (τ )] = 1

� (1 + ε)

∫ ∞

−∞Eε


×[

1

� (1 + ε)

∫ ∞

−∞θ1 (η) θ2 (τ − η) (dη)ε

]= 1

� (1 + ε)

∫ ∞

−∞θ1 (η)Eε

(−iεηεωε)

×[

1

� (1+ε)∫ ∞

−∞Eε

(−iε (τ−η)ε ωε) θ2 (τ−η) (dτ)ε](dη)ε ,

(3.76)


where

�2 (ω) = 1

� (1 + ε)

∫ ∞

−∞Eε

(−iε (τ − η)ε ωε)θ2 (τ − η) (dτ)ε (3.77)

and

� [θ1 (τ ) ∗ θ2 (τ )] = 1

� (1 + ε)

∫ ∞

−∞θ1 (η)Eε

(−iεηεωε)�2 (ω) (dη)

ε

= �2 (ω)1

� (1 + ε)

∫ ∞

−∞θ1 (η)Eε

(−iεηεωε)(dη)ε

= �1 (ω)�2 (ω) . (3.78)

Thus, we have proved the result.


θ1 (τ ) , θ2 (τ ) ,�1 (ω) ,�2 (ω) ∈ L1,ε [R] ,

� [θ1 (τ )] = �1 (ω) ,

and

� [θ2 (τ )] = �2 (ω) .

Then,

� [θ1 (τ ) θ2 (τ )] = �1 (ω) ∗�2 (ω) (3.79)

or

θ1 (τ ) θ2 (τ ) = �−1 [�1 (ω) ∗�2 (ω)] (3.80)

or, equivalently,

1

� (1 + ε)

∫ ∞

−∞θ1 (τ ) θ2 (τ )Eε

(−iετ εωε)(dτ)ε= 1

(2π)ε

∫ ∞

−∞�1 (t)�2 (ω−t) (dt)ε.

(3.81)

Proof. By directly using the definition of the local fractional Fourier transformoperator, we conclude that

�−1 [�1 (ω) ∗�2 (ω)] = 1

(2π)ε

∫ ∞

−∞

[1

(2π)ε

∫ ∞

−∞�1 (�)�2 (ω−�) (d�)ε

]× Eε

(iετ εωε

)(dω)ε

= 1

(2π)ε

∫ ∞

−∞Eε

(iετ ε�ε

)�1 (�)

×[

1

(2π)ε

∫ ∞

−∞�2 (ω −�)Eε

(iετ ε (ω −�)ε

)(dω)ε

](d�)ε ,

(3.82)


which yields

θ2 (τ ) = 1

(2π)ε

∫ ∞

−∞�2 (ω −�)Eε

(iετ ε (ω −�)ε

)(dω)ε (3.83)

and

�−1 [�1 (ω) ∗�2 (ω)] = 1

(2π)ε

∫ ∞

−∞Eε

(iετ ε�ε

)�1 (�) θ2 (τ ) (d�)

ε

=[

1

(2π)ε

∫ ∞

−∞Eε

(iετ ε�ε

)�1 (�) (d�)

ε

]θ2 (τ )

= θ1 (τ ) θ2 (τ ) . (3.84)

Therefore, we have completed the proof.

Theorem 3.7 (Convolution theorem for the local fractional Fourier transformoperator). Suppose that

θ1 (τ ) , θ2 (τ ) ,�1 (ω) ,�2 (ω) ∈ L1,ε [R] ,

� [θ1 (τ )] = �1 (ω) ,

and

� [θ2 (τ )] = �2 (ω) .

Then,

1

� (1 + ε)

∫ ∞

−∞θ1 (τ ) θ2 (τ ) (dτ)

ε = 1

(2π)ε

∫ ∞

−∞�1 (ω)�2 (ω) (dω)

ε. (3.85)

Proof. We consider

θ2 (τ ) = 1

(2π)ε

∫ ∞

−∞�2 (ω)Eε (iετ εωε) (dω)ε

= 1

(2π)ε

∫ ∞

−∞�2 (ω)Eε

(−iετ εωε)(dω)ε. (3.86)

In this case, from (3.86), we observe that

1

� (1 + ε)

∫ ∞

−∞θ1 (τ ) θ2 (τ ) (dτ)

ε = 1

� (1 + ε)

∫ ∞

−∞θ1 (τ )

×[

1

(2π)ε

∫ ∞

−∞�2 (ω)Eε

(−iετ εωε)(dω)ε

](dτ)ε

=[

1

(2π)ε

∫ ∞

−∞�2 (ω)

×[

1

� (1 + ε)

∫ ∞

−∞θ1 (τ )Eε

(−iετ εωε)(dτ )ε

](dω)ε

],

(3.87)


which yields

�1 (ω) = 1

� (1 + ε)

∫ ∞

−∞θ1 (τ )Eε

(−iετ εωε)(dτ)ε (3.88)

and

1

� (1 + ε)

∫ ∞

−∞θ1 (τ ) θ2 (τ ) (dτ)

ε = 1

(2π)ε

∫ ∞

−∞�1 (ω)�2 (ω) (dω)

ε. (3.89)


Theorem 3.8 (Parseval’s theorem for local fractional Fourier transform operator).Suppose that

θ (τ ) ∈ L1,ε [R]

and

� [θ (τ )] = �(ω) .

Then,

1

� (1 + ε)

∫ ∞

−∞|θ (τ )|2 (dτ)ε = 1

(2π)ε

∫ ∞

−∞|�(ω)|2 (dω)ε. (3.90)

Proof. Considering

θ1 (τ ) = θ2 (τ ) = θ (τ )

in (3.85), we have

θ1 (τ ) θ2 (τ ) = θ (τ ) θ (τ ) = |θ (τ )|2 (3.91)

and

�1 (ω)�2 (ω) = �(ω)� (ω) = |�(ω)|2 , (3.92)

which lead us to the result (3.90).

3.2.4 Properties and theorems of the generalized local fractionalFourier transform operator

Theorem 3.9 (Fourier integral theorem for generalized local fractional Fouriertransform operator). Suppose that g (τ ) is local fractional continuous on the interval[−∞, ∞] (or g (τ ) , G (�) ∈ L1,ε [R]). Then,

θ (τ ) = � (1 + ε)−∞I(ε)∞{[

−∞I(ε)∞(θ (τ )Eε

(− (2π i)ε τ εωε))]

Eε((2π i)ε τ εωε

)}. (3.93)


This is the Fourier integral theorem for the local fractional Fourier transformoperator.

Proof. When ω = 2π� , we can transform (3.33) into

θ (τ0) = � (1 + ε)

(2π)ε−∞I(ε)∞

{[−∞I(ε)∞

(θ (τ0)Eε

(−iετ ε0 (2π�)ε))]

Eε(iετ ε0 (2π�)

ε)}

, (3.94)

which leads us to

g (τ ) = � (1 + ε)−∞I(ε)∞{[

−∞I(ε)∞(g (τ )Eε

(− (2π i)ε τ ε�ε))]

Eε((2π i)ε τ ε�ε

)}, (3.95)

where

τ0 = τ

2π,

G (�) = 1√� (1 + ε)

∫ ∞

−∞g (τ )Eε


= √� (1 + ε)−∞I(ε)∞

{g (τ )Eε

[− (2π i)ε τ ε�ε]}

, (3.96)

and

g (τ ) = 1√� (1 + ε)

∫ ∞

−∞G (�)Eε


](d�)ε

= √� (1 + ε)−∞I(ε)∞

{G (�)Eε


]}. (3.97)

Hence, we have completed the proof.

Property 14 (Linearity for generalized local fractional Fourier transform operator).Suppose that

g1 (τ ) , g2 (τ ) , G1 (�) , G2 (�) ∈ L1,ε [R] ,

� [g1 (τ )] = G1 (�) ,

and

� [g2 (τ )] = G2 (�) .

Then,

� [ag1 (τ )± bg2 (τ )] = aG1 (�)± bG2 (�) , (3.98)



Proof. We observe that

� [ag1 (τ )± bg2 (τ )] = 1√� (1 + ε)

∫ ∞

−∞[ag1 (τ )± bg2 (τ )]

Eε(− (2π i)ε τ ε�ε

)(dτ)ε

= a1√

� (1 + ε)

∫ ∞

−∞[g1 (τ )Eε

(− (2π i)ε τ ε�ε)](dτ)ε

± b1√

� (1 + ε)

∫ ∞

−∞[g2 (τ )Eε

(− (2π i)ε τ ε�ε)](dτ)ε

= aG1 (�)± bG2 (�) . (3.99)


Property 15 (Shifting time for the generalized local fractional Fourier transformoperator). Suppose that

g (τ ) , G (�) ∈ L1,ε [R] ,

� [g (τ )] = G (�) ,


� [g (τ − a)] = Eε(− (2π i)ε aε�ε

) � [g (τ )] . (3.100)

Proof. By making use of the definition of the generalized local fractional Fouriertransform operator, we have

� [g (τ − a)] = 1√� (1 + ε)

∫ ∞

−∞g (τ − a)Eε

(− (2π i)ε τ ε�ε)(dτ)ε

= Eε(− (2π i)ε aε�ε

) 1√� (1 + ε)

∫ ∞

−∞g (τ − a)

Eε(− (2π i)ε (τ − a)ε �ε

)(dτ)ε


)G (�)


) � [g (τ )] . (3.101)

Thus, the claim has been proved.

Property 16 (Scaling time for the local fractional Fourier transform operator).Suppose that

g (τ ) , G (�) ∈ L1,ε [R] ,

� [g (τ )] = G (�) ,


and a is a positive constant. Then

� [g (aτ)] = 1

aεG

(�a

). (3.102)

Proof. By the definition of the local fractional Fourier transform operator, weobtain

� [g (aτ)] = 1√� (1 + ε)

∫ ∞

−∞g (aτ)Eε


= 1

aε1√

� (1 + ε)

∫ ∞

−∞θ (aτ)Eε

(− (2π i)ε

(aτ

a

)ε�ε

)(daτ)ε

= 1

aε�

(�a

), (3.103)


Property 17 (Conjugate for the local fractional Fourier transform operator).Suppose that

g (τ ) , G (�) ∈ L1,ε [R]

and

� [g (τ )] = G (�) .

Then

�[g (−τ)

]= � [g (τ )]. (3.104)

Proof. The definition of the generalized local fractional Fourier transform operatorimplies that

�[g (−τ)

]= 1√

� (1 + ε)

∫ ∞

−∞g (−τ)Eε


= 1√� (1 + ε)

∫ ∞

−∞θ (−τ)Eε ((2π i)ε τ ε�ε) (dτ)ε

= 1√� (1 + ε)

∫ ∞

−∞g (−τ)Eε (− (2π i)ε (−τ)ε �ε) (dτ)ε

= 1√� (1 + ε)

∫ ∞

−∞g (τ )Eε (− (2π i)ε τ ε�ε) (dτ)ε

= � [g (τ )], (3.105)

which proves the asserted result.


Property 18 (Translation for the local fractional Fourier transform operator).Suppose that

g (τ ) , G (�) ∈ L1,ε [R] ,

� [g (τ )] = G (�) ,


� [Eε

((2π i)ε aετ ε

)g (τ )

] = G (� − a) . (3.106)

Proof. From the definition of the generalized local fractional Fourier transformoperator, we get

� [Eε

(− (2π i)ε aετ ε)

g (τ )] = 1√

� (1 + ε)

∫ ∞

−∞[g (τ )Eε

((2π i)ε aετ ε

)]Eε


= 1√� (1 + ε)

∫ ∞

−∞g (τ )Eε

(− (2π i)ε

(� − a)ε τ ε)(dτ)ε

= G (� − a) , (3.107)

which completes the proof.

Property 19 (Duality for local fractional Fourier transform operator). Supposethat

g (τ ) , G (�) ∈ L1,ε [R]

and

� [g (τ )] = G (�) .

Then,

� [g (−τ)] = � [G (τ )] . (3.108)


g (τ ) = 1√� (1 + ε)

∫ ∞

−∞G (�)Eε

((2π i)ε τ ε�ε

)(d�)ε = �−1

[G (�)] .

(3.109)

Now, upon interchanging τ and � , from (3.109), we arrive at

g (�) = 1√� (1 + ε)

∫ ∞

−∞G (τ )Eε


)(dτ)ε. (3.110)


Replacing � by −� , we obtain

g (−�) = 1√� (1 + ε)

∫ ∞

−∞G (τ )Eε

(− (2π i)ε τ ε�ε)(dτ)ε, (3.111)

which evidently proves the result (3.108).

Property 20 (Composition for the local fractional Fourier transform operator).Suppose that

g1 (τ ) , g2 (τ ) , G1 (�) , G2 (�) ∈ L1,ε [R] ,

� [g1 (τ )] = G1 (�) ,

and

� [g2 (τ )] = G2 (�) .

Then,∫ ∞

−∞g1 (�)G2 (�)Eε


)(d�)ε =

∫ ∞

−∞g2 (η)G1 (η − τ) (dη)ε.

(3.112)

Proof. Let us write the left-hand side of (3.112) as follows:∫ ∞

−∞g1 (�)G2 (�)Eε


)(d�)ε

=∫ ∞

−∞g1 (�)

[1√

� (1 + ε)

∫ ∞

−∞g2 (η)Eε

(− (2π i)ε ηε�ε)(dη)ε

]Eε


)(d�)ε

=∫ ∞

−∞

[1√

� (1 + ε)

∫ ∞

−∞g1 (�)Eε

(− (2π i)ε (η − τ)ε �ε)(d�)ε

]g2 (η) (dη)

ε

=∫ ∞

−∞G1 (η − τ) g2 (η) (dη)

ε, (3.113)

which obviously establishes the result (3.112).


g (τ ) , G (�) ∈ L1,ε [R] ,

� [g (τ )] = G (�) ,

and

lim|τ |→∞ g (τ ) = 0.


Then,

�[g(ε) (τ )

]= (2π i�)ε � [g (τ )] . (3.114)

Proof. From the definition of the generalized local fractional Fourier transformoperator, we find that

�[g(ε) (τ )

]= 1√

� (1 + ε)

∫ ∞

−∞g(ε) (τ )Eε

(− (2π i)ε τ ε�ε)(dτ)ε. (3.115)

Now, by integration by part for the local fractional integral operator in (3.115), weobtain

�[g(ε) (τ )

]= 1√

� (1 + ε)

∫ ∞

−∞g(ε) (τ )Eε


=[√� (1 + ε)g (τ )Eε

(− (2π i)ε τ ε�ε)]∞

−∞

+ (2π i)ε �ε

√� (1 + ε)

∫ ∞

−∞g (τ )Eε

(− (2π i)ε τ ε�ε)(dτ)ε, (3.116)

which, by using

lim|τ |→∞ g (τ ) = 0,

leads us to

�[g(ε) (τ )

]= (2π i)ε �ε

[1√

� (1 + ε)

∫ ∞

−∞g (τ )Eε


]= (2π i�)ε � [g (τ )] . (3.117)

Thus, we get the asserted result (3.114).

In similar manner, by repeating this process, we get

�[g(kε) (τ )

]= (2π i�)kε � [g (τ )] (k ∈ N), (3.118)

where

g((k−1)ε) (0) = · · · = g(ε) (0) = g (0) = 0. (3.119)


g (τ ) , G (�) ∈ L1,ε [R] ,

� [g (τ )] = G (�) ,

and

limτ→∞ −∞I(ε)τ g (τ ) = 0.


Then,

�[−∞I(ε)τ g (τ )

]= 1

(2π i�)ε� [g (τ )] . (3.120)

Proof. In view of the definition of the generalized local fractional Fourier transformoperator, we conclude that

�[−∞I(ε)τ g (τ )

]= 1√

� (1+ε)∫ ∞

−∞

(−∞I(ε)τ g (τ )

)Eε

(− (2π i)ε τ ε�ε)(dτ)ε,

(3.121)

which, by integrating by part for the local fractional integral operator, yields

�[−∞I(ε)τ g (τ )

]=

[√� (1 + ε)−∞I(ε)τ g (τ )Eε

(− (2π i)ε τ ε�ε)]∣∣∣ ∞−∞

+[

1

(2π� i)ε√� (1+ε)

∫ ∞

−∞g (τ)Eε


].

(3.122)

Now, by taking

limτ→∞ −∞I(ε)τ g (τ ) = 0

into account in (3.122), we have

�[−∞I(ε)τ g (τ )

]= 1

(2π� i)ε√� (1 + ε)

∫ ∞

−∞g (τ )Eε


= 1

(2π� i)ε� [g (τ )] , (3.123)

which completes the proof.


�[−∞I(kε)τ g (τ )

]= 1

(2π� i)kε� [g (τ )] , (3.124)

where

limτ→∞ −∞I(kε)τ g (τ ) = 0. (3.125)


g (τ ) , G (�) ∈ L1,ε [R]

and

� [g (τ )] = G (�) .


Then,

� [τ εg (τ )

] =(

i

2π

)εG(ε) (�) . (3.126)

Proof. From the definition of the inverse generalized local fractional Fouriertransform operator, we observe that

�−1[(

i

2π

)εG(ε) (�)

]= 1√

� (1 + ε)

∫ ∞

−∞

[(i

2π

)εG(ε) (�)

]Eε


)(d�)ε , (3.127)

which, after integrating by part for local fractional integral operator, yields

�−1[(

i

2π

)εG(ε) (�)

]=

[(i

2π

)εG (�)Eε


)]∞

−∞

+ τ ε[

1√� (1+ε)

∫ ∞

−∞G (�)Eε


)(d�)ε

].

(3.128)

Now, by taking

lim|ω|→∞�(ω) = 0

into account in (3.128), we have

�−1[(

i

2π

)εG(ε) (�)

]= τ ε

[1√

� (1 + ε)

∫ ∞

−∞G (�)Eε


)(d�)ε

]= τ εg (τ ) , (3.129)

which proves the claimed result.

In a similar way, we obtain

�[τ kεg (τ )

]=

(i

2π

)kε

G(kε) (�) . (3.130)

Definition 3.7. The local fractional convolution of two functions g1 (τ ) and g2 (τ )

via local fractional integral operator, denoted by

(g1 ∗ g2) (τ ) = g1 (τ ) ∗ g2 (τ )

is defined as follows:

(g1 ∗ g2) (τ ) = g1 (τ ) ∗ g2 (τ )

= √� (1 + ε)−∞I(ε)∞ [g1 (t) g2 (τ − t)]

= 1√� (1 + ε)

∫ ∞

−∞g1 (t) g2 (τ − t) (dt)ε. (3.131)


By using the above definition of the local fractional convolution of two functions,we have the following properties:

(a) g1 (τ ) ∗ g2 (τ ) = g2 (τ ) ∗ g1 (τ ) and(b) g1 (τ ) ∗ (g2 (τ )+ g3 (τ )) = g1 (τ ) ∗ g2 (τ )+ g1 (τ ) ∗ g3 (τ ).


g1 (τ ) , g2 (τ ) , G1 (�) , G2 (�) ∈ L1,ε [R] ,

� [g1 (τ )] = G1 (�) ,

and

� [g2 (τ )] = G2 (�) .

Then,

� [g1 (τ ) ∗ g2 (τ )] = G1 (�)G2 (�) (3.132)

or

g1 (τ ) ∗ g2 (τ ) = �−1[G1 (�)G2 (�)] (3.133)

or, equivalently,

1√� (1 + ε)

∫ ∞

−∞g1 (t) g2 (τ − t) (dt)ε = 1√

� (1 + ε)

∫ ∞

−∞G1 (�)G2 (�) (d�)

ε.

(3.134)


� [g1 (τ ) ∗ g2 (τ )] = 1√� (1 + ε)

∫ ∞

−∞Eε


×[

1√� (1 + ε)

∫ ∞

−∞g1 (η) g2 (τ − η) (dη)ε

]= 1√

� (1 + ε)

∫ ∞

−∞g1 (η)Eε

(− (2π i)ε ηε�ε)

×[∫ ∞

−∞ Eε (− (2π i)ε (τ − η)ε �ε) θ2 (τ − η) (dτ)ε√� (1 + ε)

](dη)ε ,

(3.135)

where

G2 (�) = 1√� (1 + ε)

∫ ∞

−∞Eε

(− (2π i)ε (τ − η)ε �ε)θ2 (τ − η) (dτ)ε

(3.136)


and

� [g1 (τ ) ∗ g2 (τ )] = 1√� (1 + ε)

∫ ∞

−∞g1 (η)Eε

(− (2π i)ε ηε�ε)

G2 (�) (dη)ε

= G2 (�)1√

� (1 + ε)

∫ ∞

−∞g1 (η)Eε

(− (2π i)ε ηε�ε)(dη)ε

= �1 (�)�2 (�) . (3.137)

This completes the proof.


g1 (τ ) , g2 (τ ) , G1 (�) , G2 (�) ∈ L1,ε [R] ,

� [g1 (τ )] = G1 (�) ,

and

� [g2 (τ )] = G2 (�) .

� [g1 (τ ) g2 (τ )] = G1 (�) ∗ G2 (�) (3.138)

or

g1 (τ ) g2 (τ ) = �−1[G1 (�) ∗ G2 (�)] (3.139)

or, equivalently,

1√� (1 + ε)

∫ ∞

−∞g1 (τ ) g2 (τ ) (dτ)

ε = 1√� (1 + ε)

∫ ∞

−∞G1 (t)G2 (� − t) (dt)ε.

(3.140)

Proof. From the definition of the generalized local fractional Fourier transformoperator, we have

�−1[G1 (�) ∗ G2 (�)] = 1√

� (1 + ε)

∫ ∞

−∞

×[

1√� (1 + ε)

∫ ∞

−∞G1 (�)G2 (ω −�) (d�)ε

]Eε

((2π i)ε τ εωε

)(dω)ε

= 1√� (1 + ε)

∫ ∞

−∞Eε


)G1 (�)

×[∫ ∞

−∞ G2 (ω−�)Eε ((2π i)ε τ ε (ω−�)ε) (dω)ε√� (1+ε)

](d�)ε,

(3.141)


where

g2 (τ ) = 1√� (1 + ε)

∫ ∞

−∞G2 (ω −�)Eε

((2π i)ε τ ε (ω −�)ε

)(dω)ε

(3.142)

and

�−1[G1 (�) ∗ G2 (�)] = 1√

� (1 + ε)

∫ ∞

−∞Eε


)G1 (�) g2 (τ ) (d�)

ε

=[

1√� (1+ε)

∫ ∞

−∞Eε


)�1 (�) (d�)

ε

]g2 (τ )

= g1 (τ ) g2 (τ ) . (3.143)


Theorem 3.15 (Convolution theorem for generalized local fractional Fouriertransform operator). Suppose that

g1 (τ ) , g2 (τ ) , G1 (�) , G2 (�) ∈ L1,ε [R] ,

� [g1 (τ )] = G1 (�) ,

and

� [g2 (τ )] = G2 (�) .

Then,

1√� (1 + ε)

∫ ∞

−∞g1 (τ ) g2 (τ ) (dτ)

ε = 1√� (1 + ε)

∫ ∞

−∞G1 (�)G2 (�) (d�)

ε.

(3.144)

Proof. We consider

g2 (τ ) = 1√� (1 + ε)

∫ ∞

−∞G2 (�)Eε ((2π i)ε τ ε�ε) (d�)ε

= 1√� (1 + ε)

∫ ∞

−∞G2 (�)Eε

(− (2π i)ε τ ε�ε)(d�)ε, (3.145)

which, in view of (3.145), yields

1√� (1 + ε)

∫ ∞

−∞g1 (τ ) g2 (τ ) (dτ)

ε

= 1√� (1 + ε)

∫ ∞

−∞g1 (τ )

(1√

� (1 + ε)

∫ ∞

−∞G2 (�)Eε

(− (2π i)ε τ ε�ε)(d�)ε

)(dτ )ε

= 1√� (1 + ε)

∫ ∞

−∞G2 (�)

(1√

� (1 + ε)

∫ ∞

−∞g1 (τ )Eε


)(d�)ε ,

(3.146)


where

G1 (�) = 1√� (1 + ε)

∫ ∞

−∞g1 (τ )Eε

(− (2π i)ε τ ε�ε)(dτ)ε (3.147)

and

1√� (1 + ε)

∫ ∞

−∞g1 (τ ) g2 (τ ) (dτ)

ε = 1√� (1 + ε)

∫ ∞

−∞G1 (�)G2 (�) (d�)

ε.

(3.148)


Theorem 3.16 (Parseval’s theorem for generalized local fractional Fourier trans-form operator). Suppose that

g (τ ) ∈ L1,ε [R]

and

� [g (τ )] = G (�) .

Then,

1√� (1 + ε)

∫ ∞

−∞|g (τ )|2 (dτ)ε = 1√

� (1 + ε)

∫ ∞

−∞|G (�)|2 (d�)ε. (3.149)

Proof. By setting

g1 (τ ) = g2 (τ ) = g (τ )

in (3.144), we conclude that

g1 (τ ) g2 (τ ) = g (τ ) g (τ ) = |g (τ )|2 (3.150)

and

G1 (�)G2 (�) = G (�)G (�) = |G (�)|2 . (3.151)

We thus obtain the desired result (3.149).


3.3.1 The analogous distributions defined on Cantor sets

In light of Definition 3.5, we define the function δε (τ ), which is called the analogousDirac (Dirac-like function) distribution via the local fractional integral operator (alsocalled the local fractional Dirac function).


Definition 3.8. The analogous Dirac distribution via the local fractional integraloperator is defined by

ψ (0) = 1

� (1 + ε)

∫ ∞

−∞δε (τ ) ψ (τ) (dτ)

ε

= (δε ∗ ψ) (τ)= δε (τ ) ∗ ψ (τ) , (3.152)

and it has the following properties:

(a) δε (τ ) ≥ 0, for τ ∈ R;(b) δε (τ ) = 0, for τ �= 0; and(c) 1

�(1+ε)∫ ∞−∞ δε (τ ) (dτ)ε = 1.

It follows that

1

� (1 + ε)

∫ ∞

−∞δε (τ − t) ψ (τ) (dτ)ε = ψ (t) , (3.153)

1

� (1 + ε)

∫ ∞

−∞δε (τ ) ψ

(kε) (τ ) (dτ)ε = ψ(kε) (0) , (3.154)

and

1

� (1 + ε)

∫ ∞

−∞δε (τ − t) ψ(kε) (τ ) (dτ)ε = ψ(kε) (t) , (3.155)

where k ∈ N.According to the earlier works [1–27], for μ ∈ R, we have

1

� (1 + ε)

∫ ∞

−∞1

(4πμ)ε/2

�(1+ε)Eε

(− τ 2ε

(4μ)ε

)(dτ)ε = 1, (3.156)

from which we obtain the analogous Dirac distribution as follows:

δε (τ ) = limμ→0

1(4πμ)ε/2

�(1+ε)Eε

(− τ 2ε

(4μ)ε

). (3.157)

For μ ∈ R, we also have

�ε (μ, τ) = 1(4πμ)ε/2

�(1+ε)Eε

(− τ 2ε

(4μ)ε

) (μ = 1

4π

), (3.158)

which yields

�ε

(1

4π, τ

)= � (1 + ε)Eε

(−πετ 2ε

). (3.159)


–0.4 –0.2 0 0.2 0.4 0.60.2

0.4

0.6

0.8

1

t

–2 –1.5 –1 –0.5 0 0.5 1 1.5 20.1

0.2

0.3

0.4

t

(a)

(b)

Figure 3.1 The plots of a family of good kernels: (a) the plot of �ε(

14π , τ

)with fractal

dimension ε = ln 2/ ln 3 and (b) the plot of �ε (1, τ) with fractal dimension ε = ln 2/ ln 3.

In this case, we get

1

� (1 + ε)

∫ ∞

−∞� (1 + ε)Eε

(−πετ 2ε

)(dτ)ε = 1. (3.160)

We also call �ε(

14π , τ

)a good kernel and �ε (μ, τ) a family of good kernels as

μ → 0. The graphs of the functions�ε(

14π , τ

)and�ε (1, τ) are shown in Figure 3.1.

Theorem 3.17. If

�ε (τ) = � (1 + ε)Eε(−πετ 2ε

), (3.161)

then,

�ε (�) = �ε (�). (3.162)

Proof. We define

X (�) = �ε (�) = 1√� (1 + ε)

∫ ∞

−∞�ε (τ)Eε


(3.163)


and find that

X (0) = �ε (0) = √� (1 + ε).

We now have

�(ε)ε (τ ) = − (2π)ε τ ε�ε (τ ) , (3.164)

so that

X(ε) (�) =[

1√� (1 + ε)

∫ ∞

−∞�ε (τ)

(− (2π i)ε τ ε)


](dτ)ε

]= iε

[1√

� (1 + ε)

∫ ∞

−∞�(ε)ε (τ )Eε


]. (3.165)

Using (3.165) together with (3.118), we conclude that

X(ε) (�) = iε (2π i�)ε �ε (�) = − (2π�)ε �ε (�) , (3.166)

which, for any 2π� , yields

�ε (�) = �ε (�) . (3.167)

Now, for any good kernel �ε (μ, τ), we can write

�ε (μ, τ)∗ψ (τ)−ψ (τ) = 1

� (1 + ε)

∫ ∞

−∞δε (τ ) [ψ (t − τ)− ψ (t)] (dτ)ε. (3.168)

In this case, we have

θ (τ ) = 1

(2π)ε

∫ ∞

−∞Eε

(iετ ερε

) [1

� (1 + ε)

∫ ∞

−∞θ (t)Eε

(−iεtερε)(dt)ε

](dρ)ε

= 1

� (1 + ε)

∫ ∞

−∞θ (t)

[1

(2π)ε

∫ ∞

−∞Eε

(iε (τ − t)ε ρε

)(dρ)ε

](dt)ε

= 1

� (1 + ε)

∫ ∞

−∞θ (t) δε (τ − t) (dt)ε, (3.169)

where

δε (τ − t) = 1

(2π)ε

∫ ∞

−∞Eε

(iε (τ − t)ε ρε

)(dρ)ε (3.170)

and

g (τ ) = 1√� (1 + ε)

∫ ∞

−∞

[∫ ∞−∞ g (t)Eε

[− (2π i)ε tευε](dt)ε√

� (1 + ε)

]Eε[

(2π i)ε τ ευε](dυ)ε

= 1√� (1 + ε)

∫ ∞

−∞g (t)

{∫ ∞−∞ Eε

[(2π i)ε (τ − t)ε υε

](dυ)ε√

� (1 + ε)

}(dt)ε

= 1√� (1 + ε)

∫ ∞

−∞g (t)

(δε (τ − t)√� (1 + ε)

)(dt)ε. (3.171)


Here,

δε (τ − t) =∫ ∞

−∞Eε

[(2π i)ε (τ − t)ε υε

](dυ)ε. (3.172)

In fact, by virtue of (3.169) and (3.171), we find from (3.170) and (3.172) that

δε (τ ) = 1

(2π)ε

∫ ∞

−∞Eε

(iετ ερε

)(dρ)ε (3.173)

and

δε (τ ) =∫ ∞

−∞Eε

[(2π i)ε τ ευε

](dυ)ε, (3.174)

where t = 0.In a similar way, we have

�(ω) = 1

� (1 + ε)

∫ ∞

−∞Eε

(−iερεωε) [

1

(2π)ε

∫ ∞

−∞�(t)Eε

(iερεtε

)(dt)ε

](dρ)ε

= 1

(2π)ε

∫ ∞

−∞�(t)

[1

� (1 + ε)

∫ ∞

−∞Eε

(−iερε (ω − t)ε)(dρ)ε

](dt)ε

= 1

(2π)ε

∫ ∞

−∞�(t)

[(2π)ε

� (1 + ε)δε (ω − t)

](dt)ε, (3.175)

where

(2π)ε

� (1 + ε)δε (ω − t) = 1

� (1 + ε)

∫ ∞

−∞Eε

(−iερε (ω − t)ε)(dρ)ε (3.176)

and

G (�)= 1√� (1 + ε)

∫ ∞

−∞

{∫ ∞−∞ G (ρ)Eε

[(2π i)ε tερε

](dρ)ε√

� (1+ε)

}Eε

[− (2π i)ε tε�ε](dt)ε

= 1√� (1 + ε)

∫ ∞

−∞G (ρ)

{∫ ∞−∞ Eε

[− (2π i)ε tε (� − ρ)ε](dt)ε√

� (1 + ε)

}(dρ)ε

= 1√� (1 + ε)

∫ ∞

−∞G (ρ)

{1√

� (1 + ε)δε (� − ρ)

}(dρ)ε , (3.177)

so that

δε (� − ρ) =∫ ∞

−∞Eε

[− (2π i)ε tε (� − ρ)ε](dt)ε. (3.178)

In fact, in light of (3.175) and (3.177), we find by using (3.176) and (3.178) that

(2π)ε

� (1 + ε)δε (ω) = 1

� (1 + ε)

∫ ∞

−∞Eε

(−iερεωε)(dρ)ε (3.179)


and

δε (�) =∫ ∞

−∞Eε

[− (2π i)ε tε�ε](dt)ε, (3.180)

so that t = 0 and ρ = 0. Therefore, we show the following results:

1

� (1 + ε)

∫ ∞

−∞Eε

(−iεωετ ε)(dτ)ε = (2π)ε

� (1 + ε)δε (ω) , (3.181)

1

(2π)ε

∫ ∞

−∞Eε

(iετ εωε

)(dω)ε = δε (τ ) , (3.182)

1√� (1 + ε)

∫ ∞

−∞Eε

[− (2π i)ε �ετ ε](dτ)ε = δε (�)√

� (1 + ε), (3.183)

and

1√� (1 + ε)

∫ ∞

−∞Eε


](d�)ε = δε (τ )√

� (1 + ε). (3.184)

Definition 3.9. Let Hε (τ ) be the Heaviside function defined on Cantor sets asfollows:

Hε (τ ) ={

0, if x < 0,1, if x ≥ 0,

(3.185)

with a distribution given by the formula:

1

� (1 + ε)

∫ ∞

−∞Hε (τ ) ψ (τ) (dτ)

ε = 1

� (1 + ε)

∫ ∞

0ψ (τ) (dτ)ε. (3.186)

Definition 3.10. The local fractional derivative of the analogous Dirac distribution,denoted by uε (τ ), is defined by

uε (τ ) = dεδε (τ )

dτ ε(3.187)

with the following properties:

1

� (1 + ε)

∫ ∞

−∞uε (τ ) ψ (τ) (dτ)

ε = −ψ(ε) (0) . (3.188)

Definition 3.11. The analogous rectangular pulse defined on Cantor sets, denotedby rectε (τ ), is defined by

rectε (τ ) ={

1 if |τ | ≤ 12 ,

0 if |τ | > 12 .

(3.189)

For finding the local fractional Fourier transform of the analogous rectangularpulse, it is observed that


� [rectε (τ )] = (� rectε) (ω)

= 1

� (1 + ε)

∫ ∞

−∞rectε (τ )Eε


= 1

� (1 + ε)

∫ 1/2

−1/2Eε


= Eε(iε

(ω2

)ε) − Eε(−iε

(ω2

)ε)iω

= 2 sinε(ω2

)εωε

. (3.190)

We thus find that

limω→0

2 sinε(ω2

)εωε

= limω→0

∂ε

∂ωε2 sinε

(ω2

)ε∂εωε

∂ωε

= 21+ε

� (1 + ε)(3.191)

and

limω→∞

2 sinε(ω2

)εωε

= 0. (3.192)

Hence, we get

(� rectε) (0) = 21+ε

� (1 + ε). (3.193)

In Figure 3.2, rectε (τ ) and (� rectε) (ω) are sketched.From (3.191), we have

limω→0

sinε(ω2

)ε( ω2 )

ε

�(1+ε)= 1 (3.194)

or

limω→0

sinε(ω2

)ε(ω2

)ε = 1

� (1 + ε). (3.195)

When ω = 1, we obtain

limω→0

sin ω2

ω2

= 1. (3.196)

Definition 3.12. The analogous triangle function defined on Cantor sets, denotedby trigε (τ ) is defined as

trigε (τ ) ={

(1−|τ |ε)�(1+ε) if |τ | ≤ 1,

0 if |τ | > 1.(3.197)


–0.5 –0.4 –0.3 –0.2 –0.1 0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

t

rec

t e(t

)

–2 –1.5 –1 –0.5 0 0.5 1 1.5 25

5.5

6

6.5

w

(Á r

ect e

)(w

)

(a)

(b)

Figure 3.2 The graphs of analogous rectangular pulse and its local fractional Fouriertransform: (a) the graph of rectε (τ ) and (b) the graph of (� rectε) (ω).

Find the local fractional Fourier transform of analogous triangle function, namely,

� [triangε (τ )

] = (� triangε)(ω)

= 1

� (1 + ε)

∫ ∞

−∞triangε (τ )Eε


= 1

� (1 + ε)

∫ 1

−1

(1 − |τ |ε)� (1 + ε)

Eε(−iετ εωε

)(dτ)ε

= 1

� (1 + ε)

∫ 1

−1

(1 − |τ |ε)� (1 + ε)

[cosε

(τ εωε

) − iε sinε(τ εωε

)](dτ)ε

= 1

� (1 + ε)

∫ 1

−1

(1 − |τ |ε)� (1 + ε)

cosε(τ εωε

)(dτ)ε

= 2

� (1 + ε)

∫ 1

0

(1 − τ ε)

� (1 + ε)cosε

(τ εωε

)(dτ)ε

= 2

ωε

[(1−τ ε)� (1+ε) sinε

(τ εωε

)]1

0+ 2

ωε

[1

� (1 + ε)

∫ 1

0sinε

(τ εωε

)(dτ)ε

]= 2

ωε

[1

� (1 + ε)

∫ 1

0sinε

(τ εωε

)(dτ)ε

]= 2

ω2ε

[1 − cosε

(ωε

)]= 4 sin2

ε

(ω2

)εω2ε

. (3.198)


–1 –0.5 0 0.5 10

0.5

1

1.5

t

trige(t)

–2 –1.5 –1 –0.5 0 0.5 1 1.5 24

5

6

7

8

w

(Á tr

ige)

(w)

(a)

(b)

Figure 3.3 The graphs of the analogous triangle function and its local fractional Fouriertransform: (a) the plot of triangε (τ ) and (b) the plot of

(� triangε)(ω).

Using (3.128), we find that(� triangε)(0) = lim

ω→0

4 sin2ε

(ω2

)εω2ε = lim

ω→0

2

� (1 + 2ε)(3.199)

and (� triangε)(∞) = lim

ω→∞4 sin2

ε

(ω2

)εω2ε = 0. (3.200)

In Figure 3.3, the graphs of triangε (τ ) and(� triangε

)(ω) are drawn.

Definition 3.13. The two-sided Mittag–Leffler distribution defined on Cantor setswith the positive parameter a, denoted by Ma.ε (τ ), is defined by

Ma.ε (τ ) ={

Eε (− (aτ)ε) , τ > 0,−Eε ((aτ)ε) , τ < 0.

(3.201)

We see that

� [Ma.ε (τ )] = (�Ma.ε) (ω)

= 1

� (1 + ε)

∫ ∞

−∞Ma.ε (τ )Eε



= 1

� (1 + ε)

∫ ∞

0Eε

(−τ ε (iεωε + aε

))(dτ)ε

− 1

� (1 + ε)

∫ 0

−∞Eε

(τ ε

(aε − iεωε

))(dτ)ε

= 1

iεωε + aε− 1

aε − iεωε

= 1

iεωε + aε− 1

aε − iεωε

= −2iεωε

a2ε + ω2ε , (3.202)

which leads us to

(�Ma.ε) (0) = 0 (3.203)

and

(�Ma.ε) (∞) = 0. (3.204)

Definition 3.14. The complex distribution via the Mittag–Leffler function definedon Cantor sets with the positive parameter a, denoted by �a.ε (τ ), is defined by

�a.ε (τ ) = Eε(−aε |τ |ε) . (3.205)

We thus find that

� [�a.ε (τ )] = (��a.ε) (ω)

= 1

� (1 + ε)

∫ ∞

−∞�a.ε (τ )Eε


= 1

� (1 + ε)

∫ ∞

0Eε

(−τ ε (iεωε − aε

))(dτ)ε

+ 1

� (1 + ε)

∫ 0

−∞Eε

(τ ε

(−ωεiε − aε))(dτ)ε

= 1

iεωε − aε− 1

ωεiε + aε

= 2aε

a2ε + ω2ε , (3.206)

which leads us to

(��a.ε) (0) = 2aε

a2ε (3.207)

and

(��a.ε) (∞) = 0. (3.208)


Definition 3.15. The signum distribution defined on Cantor sets, denoted bysgnε (τ ), is defined as

sgnε (τ ) ={

1, τ ≥ 0,−1, τ < 0.

(3.209)

Thus, we have

� [sgnε (τ )

] = (� sgnε)(ω)

= 1

� (1 + ε)

∫ ∞

−∞sgnε (τ )Eε


= 1

� (1 + ε)

∫ ∞

0Eε


− 1

� (1 + ε)

∫ 0

−∞Eε

(iετ ε (−ω)ε) (dτ)ε

= 2

iεωε, (3.210)

which leads to(� sgnε)(0) = ∞ (3.211)

and

(��a.ε) (∞) = 0. (3.212)

Definition 3.16. The Mittag–Leffler distribution defined on Cantor sets with thesquared variable and positive parameter a, denoted by �ε (τ), is defined as

�ε (τ) = Eε(−aτ 2ε

). (3.213)

As a result, by the definition of the local fractional Fourier transform operator,we have

� [�ε (τ)] = (��ε) (ω)= 1

� (1 + ε)

∫ ∞

−∞�εEε


= 1

� (1 + ε)

∫ ∞

−∞Eε

(−aτ 2ε − iετ εωε

)(dτ)ε

= 2

� (1 + ε)

∫ ∞

0Eε

(−aτ 2ε

)cosε

(τ εωε

)(dτ)ε. (3.214)


In addition, we find that

�(ε) [�ε (τ)] = −2

� (1 + ε)

∫ ∞

0Eε

(−aτ 2ε

)τ ε sinε

(τ εωε

)(dτ)ε

=[

21−ε

aEε

(−aτ 2ε

)sinε

(τ εωε

)]∞

0

− 21−εωε

a

[1

� (1 + ε)

∫ ∞

0Eε

(−aτ 2ε

)cosε

(τ εωε

)(dτ)ε

],

(3.215)

which leads us to[21−ε

aEε

(−aτ 2ε

)sinε

(τ εωε

)]∞

0= 0 (3.216)

and

�(ε) [�ε (τ)] = −21−εωε

a

[1

� (1 + ε)

∫ ∞

0Eε

(−aτ 2ε

)cosε

(τ εωε

)(dτ)ε

]= −2−εωε

a

1

� (1 + ε)

∫ ∞

−∞Eε

(−aτ 2ε − iετ εωε

)(dτ)ε

= −2−εωε

a� [�ε (τ)] . (3.217)

In this case, by making use of (3.217), we have

�(ε) [�ε (τ)] + 2−ε

aωε� [�ε (τ)] = 0, (3.218)

which yields to the nondifferentiable solution in the form

� [�ε (τ)] = (��ε) (ω) = M0Eε

(−2−2ε

aω2ε

), (3.219)

where M0 can be confirmed by

M0 = (��ε) (0) . (3.220)

By using (3.156), we rewrite (3.220) as follows:

M0 = (��ε) (0)= 1

� (1 + ε)

∫ ∞

∞Eε

(−aτ 2ε

)(dτ)ε

=πε/2

√1a

� (1 + ε). (3.221)


Thus, from (3.219) and (3.221), we obtain

�[Eε

(−aτ 2ε

)]=

πε/2√

1a

� (1 + ε)Eε

[−1

a

(ω2

)2ε]

. (3.222)

In this case, for ε = 1, we get∫ ∞

−∞exp

(−aτ 2 − iτω

)dτ =

√π

aexp

(− 1

4aω2

). (3.223)

For a = 1, we rewrite (3.222) as follows:

�[Eε

(−τ 2ε

)]= πε/2

� (1 + ε)Eε

[−

(ω2

)2ε]

. (3.224)

For a = πε, we obtain

�[Eε

(−πετ 2ε

)]= 1

� (1 + ε)Eε

[− 1

πε

(ω2

)2ε]

. (3.225)

In fact, we rewrite Hε (τ ) as follows:

Hε (τ ) = 1 + sgnε (τ )

2. (3.226)

In this case, we find that

� [Hε (τ )] = (�Hε) (ω)

= 1

� (1 + ε)

∫ ∞

−∞Hε (τ )Eε


= 1

� (1 + ε)

∫ ∞

−∞

[1 + sgnε (τ )

2

]Eε


= 1

� (1 + ε)

∫ ∞

−∞1

2Eε


+ 1

� (1 + ε)

∫ ∞

−∞sgnε (τ )

2Eε

(−iετ εωε)(dτ)ε, (3.227)

which, by using (3.181) and (3.210), becomes

� [Hε (τ )] = 1

2

(2π)ε

� (1 + ε)δε (ω)+ 1

iεωε. (3.228)

3.3.2 Applications of signal analysis on Cantor sets

Let us consider the signal given by

ψ (τ) = δε (τ − τ0) . (3.229)


Also, let its local fractional Fourier transform be read as follows:

� [ψ (τ)] = (�ψ) (ω)= 1

� (1 + ε)

∫ ∞

−∞δε (τ − τ0)Eε


= Eε(−iετ ε0ω

ε)

. (3.230)

We now find the local fractional Fourier transform of the signal given by

ψ (τ) = δε (τ − τ0) θ (τ ) .

Indeed, by using the local fractional Fourier transform operator, we have

� [ψ (τ)] = (�ψ) (ω)= 1

� (1 + ε)

∫ ∞

−∞δε (τ − τ0) θ (τ )Eε


= θ (τ0)Eε(−iετ ε0ω

ε)

. (3.231)

The next step is to find the local fractional Fourier transform of the signal given by

ψ (τ) = Eε(iετ εωε0

).

Taking into account the local fractional Fourier transform operator, we have

� [ψ (τ)] = (�ψ) (ω)= 1

� (1 + ε)

∫ ∞

−∞Eε

(iετ εωε0

)Eε


= 1

� (1 + ε)

∫ ∞

−∞Eε

(−iετ ε (ω − ω0)ε)(dτ)ε

= (2π)ε

� (1 + ε)δε (ω − ω0) . (3.232)

Now, it is natural to find the local fractional Fourier transform of the signal givenby

ψ (τ) = sinε(aετ ε

).

Indeed we have

� [ψ (τ)] = (�ψ) (ω)= 1

� (1 + ε)

∫ ∞

−∞sinε

(aετ ε

)Eε


= 1

� (1 + ε)

∫ ∞

−∞Eε (iεaετ ε)− Eε (−iεaετ ε)

2iεEε


= (2π)ε

� (1 + ε)

[δε (ω + a)− δε (ω − a)]

2iε. (3.233)


In order to find the local fractional Fourier transform of the signal given by

ψ (τ) = cosε(aετ ε

),

we observe that

� [ψ (τ)] = (�ψ) (ω)= 1

� (1 + ε)

∫ ∞

−∞cosε

(aετ ε

)Eε


= 1

� (1 + ε)

∫ ∞

−∞Eε (iεaετ ε)+ Eε (−iεaετ ε)

2Eε


= (2π)ε

� (1 + ε)

[δε (ω + a)+ δε (ω − a)]

2. (3.234)

We now find the local fractional Fourier transform of the signal given by

ψ (τ) = δ(ε)ε (τ ) .

In fact, from the definition of the local fractional Fourier transform operator,we have

� [ψ (τ)] = (�ψ) (ω)= 1

� (1 + ε)

∫ ∞

−∞δ(ε)ε (τ )Eε


= iεωε1

� (1 + ε)

∫ ∞

−∞δε (τ )Eε


= iεωε. (3.235)

In order to find the local fractional Fourier transform of the constant signalgiven by

ψ (τ) = C,

we observe from the definition of the local fractional Fourier transform operatorthat

� [ψ (τ)] = (�ψ) (ω)= 1

� (1 + ε)

∫ ∞

−∞CEε


= C1

� (1 + ε)

∫ ∞

−∞Eε


= C (2π)ε

� (1 + ε)δε (ω) . (3.236)

Find the local fractional Fourier transform of the signal given by

ψ (τ) = τ ε.


From the definition of the local fractional Fourier transform operator, we get

� [ψ (τ)] = (�ψ) (ω)= 1

� (1 + ε)

∫ ∞

−∞τ εEε


= iε�(ε) [1]

= (2π i)ε

� (1 + ε)δ(ε)ε (ω) . (3.237)

Find the local fractional Fourier transform of the signal given by

ψ (τ) = Hε (τ )Eε(−τ εaε) .

By using the definition of the local fractional Fourier transform operator, we have

� [ψ (τ)] = (�ψ) (ω)= 1

� (1 + ε)

∫ ∞

−∞[Hε (τ )Eε

(−τ εaε)] Eε(−iετ εωε

)(dτ)ε

= 1

� (1 + ε)

∫ ∞

0Eε

(−τ εaε) Eε(−iετ εωε

)(dτ)ε

= 1

� (1 + ε)

∫ ∞

0Eε

(−τ ε (ωεiε + aε

))(dτ)ε

= 1

ωεiε + aε. (3.238)



We consider the fractal relaxation equation governed by the local fractional ordinarydifferential equation in the form

∂ε� (μ)

∂με+ p�(μ) = δε (μ) (3.239)

subject to the initial condition:

�(0) = 1. (3.240)

Taking the local fractional Fourier transform of (3.239), we first obtain

iεωε (��) (ω)+ p (��) (ω) = 1, (3.241)

which implies that

(��) (ω) = 1

iεωε + p. (3.242)


0 0.2 0.4 0.6 0.8 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

m

Φ(m

)

P = 1P = 2P = 3

Figure 3.4 The graphs of �(μ) when ε = ln 2/ ln 3, p = 1, p = 2, and p = 3.

Now, taking the inverse fractional Fourier transform in (3.242), we have

�(μ) = Hε (τ )Eε(−pτ ε

).

We have drawn the graphs when p = 1, p = 2, and p = 3 in Figure 3.4.We write the local fractional ordinary differential equation in the form

∂ε� (μ)

∂με+ 2�(μ) = Hε (τ )Eε (−μ) (3.243)


�(0) = 0. (3.244)

By using the local fractional Fourier transform in (3.243), we have

iεωε (��) (ω)+ 2 (��) (ω) = 1

iεωε + 1. (3.245)

Considering (3.245), we obtain

(��) (ω) = 1

iεωε + 1− 1

2 + iεωε(3.246)


0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14× 1029

m

Φ(m

)

Figure 3.5 The graph of �(μ) when ε = ln 2/ ln 3.

such that

�(μ) = Hε (τ ) [Eε (−μ)− Eε (−2μ)] . (3.247)

The graph of �(μ) when ε = ln 2/ ln 3 is shown in Figure 3.5.Let us now consider the local fractional ordinary differential equation with the

positive parameter p in the form

∂2ε� (μ)

∂μ2ε + p�(μ) = δε (τ ) (3.248)

subject to initial condition:

�(0) = 0. (3.249)

Taking the local fractional Fourier transform in (3.248), we get

i2εω2ε (��) (ω)+ p (��) (ω) = 1, (3.250)

which yields

(��) (ω) = 1

p − ω2ε . (3.251)


0 0.2 0.4 0.6 0.8 1–3.5

–3

–2.5

–2

–1.5

–1

–0.5× 1030

m

Φ(m

)

P = 1P = 2P = 3

Figure 3.6 The graph of �(μ) when ε = ln 2/ ln 3.

Hence, we have

(��) (ω) =√

p

2

[1√

p − ωε+ 1√

p + ωε

](3.252)

such that

�(μ) = −√

p

2Hε (τ )

[Eε

(√pμ

) − Eε(−√

pμ)]

. (3.253)

The graph for the parameters p = 1, p = 2, and p = 3 is shown in Figure 3.6.


We consider here the local fractional diffusion equation in 1 + 1 fractal dimensionalspace as follows:

∂ε� (μ, τ)

∂τ ε= k2 ∂

2ε� (μ, τ)

∂μ2ε , τ > 0, −∞ < μ < ∞ (3.254)


�(μ, 0) = δε (μ) . (3.255)


Taking the local fractional Fourier transform with fractal space in (3.254) and(3.255), we get

∂ε� (ω, τ)

∂τ ε= −kεω2ε� (ω, τ) (3.256)

and�(μ, 0) = 1. (3.257)

Making use of (3.256) and (3.257), we have

�(ω, τ) = Eε(−kεω2ετ ε

). (3.258)

Taking the inverse local fractional Fourier transform in (3.258), we obtain

�(μ, τ) = �−1 [�(ω, τ)]

= 1

(2π)ε

∫ ∞

−∞�(ω, τ)Eε

(iεμεωε

)(dω)ε

= 1

(2π)ε

∫ ∞

−∞Eε

(−kετ εω2ε

)Eε

(iεμεωε

)(dω)ε, (3.259)

which leads us to

�−1{

Eε[−uω2ε

]}= 1

πε/2√

4εu�(1+ε)

Eε

(−1

u

(τ2

)2ε)

(3.260)

and

�(μ, τ) = � (1 + ε)

(4πkτ)ε/2Eε

(− μ2ε

(4kτ)ε

). (3.261)

In this case, we obtain the same result as with the local fractional similarity solution[23]. The corresponding solution was discussed in Chapter 1.

Let us consider the local fractional Laplace equation in 1 + 1 fractal dimensionalspace as follows:

∂2ε� (μ, η)

∂μ2ε + ∂2ε� (μ, η)

∂η2ε = 0, η > 0, −∞ < μ < ∞ (3.262)

subject to the initial conditions:

�(μ, 0) = δε (μ) (3.263)

and

limη→∞�(μ, η) = 0. (3.264)

Let us take the local fractional Fourier transform with the variable μ in (3.262),(3.263), and (3.264). We then obtain

∂2ε� (ω, η)

∂η2ε − ω2ε� (ω, η) = 0, (3.265)


�(ω, 0) = 1, (3.266)

and

limη→∞�(ω, η) = 0. (3.267)

From (3.265), we have the general solution in the form

�(ω, η) = P (ω)Eε(−ωεηε) + Q (ω)Eε

(ωεηε

), (3.268)

which, in view of (3.267), is rewritten as follows:

�(ω, η) = P (ω)Eε(− |ω|ε ηε) (3.269)

and

Q (ω) = 0. (3.270)

From (3.266) and (3.269), we have

P (ω) = 1 (3.271)

such that

�(ω, η) = Eε(− |ω|ε ηε) . (3.272)

Taking the inverse local fractional Fourier transform in (3.272) yields

�(μ, η) = �−1 [�(ω, η)]

= 1

(2π)ε

∫ ∞

−∞�(ω, η)Eε

(iεμεωε

)(dω)ε

= 1

(2π)ε

∫ ∞

−∞Eε

(− |ω|ε ηε) Eε(iεμεωε

)(dω)ε

= 1

(2π)ε

∫ ∞

0Eε

(−ωεηε) Eε(iεμεωε

)(dω)ε

+ 1

(2π)ε

∫ 0

−∞Eε

(ωεηε

)Eε

(iεμεωε

)(dω)ε

= 1

(2π)ε

∫ ∞

0Eε

[−ωε (ηε − iεμε

)](dω)ε

+ 1

(2π)ε

∫ 0

−∞Eε

[ωε

(ηε + iεμε

)](dω)ε

= � (1 + ε)

(2π)ε

(1

ηε − iεμε+ 1

ηε + iεμε

)= � (1 + ε)

(2π)ε2ηε

μ2ε + η2ε. (3.273)


We thus conclude that

�(ω, τ) = � (1 + ε)

(2π)ε2ηε

(μ− τ)2ε + η2ε, (3.274)

so that

�[� (1 + ε)

(2π)ε2ηε

(μ− τ)2ε + η2ε

]= 1

(2π)ε

∫ ∞

−∞Eε

(− |ω|ε ηε) (2π)ε

� (1 + ε)δε (ω) (dω)

ε

= 1, (3.275)

which leads us to

δε (τ ) = limτ→0

�(ω, τ) = limτ→0

� (1 + ε)

(2π)ε2ηε

(μ− τ)2ε + η2ε(3.276)

and

1

� (1 + ε)

∫ ∞

−∞� (1 + ε)

(2π)ε2ηε

(μ− τ)2ε + η2ε(dτ)ε = 1, (3.277)

respectively.

4Local fractional Laplace transformand applications

4.1 Introduction

A complex number has the form z = x + iy, where x, y ∈ R and i is the imaginary unitthat satisfies i2 = −1. Recently, the number in the fractal dimension space, namely,xε+ iεyε, where x, y ∈ R and iε is the fractal imaginary unit fulfilling i2ε = −1, whichhas the relation zε = xε + iεyε ∈ �. Thus, we can derive both the local fractionalderivative and integral operators based on the numbers in the fractal dimension space.

The local fractional derivative operator of ψ (z) of order ε (0 < ε ≤ 1) at the pointz0 is defined as [16, 21, 108]

z0Dεzψ (z0) = �ε (ψ (z)− ψ (z0))

(z − z0)ε , (4.1)

where �ε (ψ (z)− ψ (z0)) ∼= � (1 + ε) (ψ (z)− ψ (z0)).The local fractional integral operator of�(z) of order ε (0 < ε ≤ 1) from the point

zp to the point zq is defined as given below [16, 21, 108]:

zpIεzqψ (z) = 1

� (1 + ε)lim�zi→0

k∑i=1

�(zi) (�zi)ε

= 1

� (1 + ε)

∫ zq

zp

�(z) (dz)ε, (4.2)

where �zi = zi − zi−1, z0 = zp, and zn = zq.We recall that the function

ψ (z) = ν (μ, η)+ iευ (μ, η) (4.3)

is a local fractional analytic in the region � if

∂εν (μ, η)

∂με− ∂ευ (μ, η)

∂ηε= 0 (4.4)

and

∂ευ (μ, η)

∂με+ ∂εν (μ, η)

∂ηε= 0. (4.5)

Also, there is

1

(2π)ε iε· 1

� (1 + ε)

∮C

ψ (z)

(z − z0)ε (dz)ε = ψ (z0) , (4.6)


http://dx.doi.org/10.1016/B978-0-12-804002-7.00004-8


whereψ (z) denotes the local fractional analytic within and on a simple closed contourC and z0 is any point interior to C.

Generally, we have [16–21]1

(2π)ε iε· 1

� (1 + ε)

∮C

f (z)

(z − z0)(n+1)ε

(dz)ε = f (nε) (z0) . (4.7)

When ψ (z) = 1, from (4.6), we conclude

1

� (1 + ε)

∮C

(dz)ε

(z − z0)ε = (2π)ε iε, (4.8)

where z0 is any point interior to C.For C: |z − z0| ≤ R, a local fractional Laurent series of ψ (z) has the form [16–21]

ψ (z) =∞∑

k=−∞γ (k, z0) (z − z0)

kα , (4.9)

where

γ (k, z0) = 1

(2π)ε iε· 1

� (1 + ε)

∮C

ψ (z)

(z − z0)(k+1)ε

(dz)ε. (4.10)

We observe that from (4.10), a generalized residue of f (z) at the point z = z0 via thelocal fractional integral operator, denoted by Res

z=z0ψ (z), is [16, 21]

1

(2π)ε iε� (1 + ε)

∮Cψ (z) (dz)ε = Res

z=z0ψ (z) , (4.11)

where C: |z − z0| ≤ R.As a result, we obtain a new transform (Laplace-like transform) based upon the

numbers in the fractal dimension space. There is a local fractional Fourier transformof �(ω) as

�(ω) = 1

� (1 + ε)

∫ ∞

−∞φ (τ)Eε

(−iετ εωε)(dτ)ε, (4.12)

which leads to a new transform in the fractal dimension space, namely,

�(s) = 1

� (1 + ε)

∫ ∞

−∞φ (τ)Eε

(−τ εsε) (dτ)ε, (4.13)

where s = iω and ω → ∞.We generalized (4.13) as

�(s) = 1

� (1 + ε)

∫ ∞

−∞φ (τ)Eε

(−τ εsε) (dτ)ε, (4.14)

where s = β + iω, sε = βε + iεωε and ω → ∞.We observe that when ε = 1, we get the Laplace transform as [109]

�(s) =∫ ∞

−∞φ (τ) exp (−τ s) dτ , (4.15)

where s = β + iω and ω → ∞.

Local fractional Laplace transform and applications 149

Thus, we have

φ (τ) = 1

(2π)ε

∫ ∞

−∞�(ω)Eε

(iετ εωε

)(dω)ε (4.16)

such that the inverse formula takes the form

φ (τ) = 1

(2π)ε

∫ iω

−iω� (s)Eε

(τ εsε

)(ds)ε, (4.17)

where s = iω and ω → ∞.In a similar manner, (4.17) is generalized in the following form:

φ (τ) = 1

(2π)ε

∫ β+iω

β−iω� (s)Eε

(τ εsε

)(ds)ε, (4.18)

where s = β + iω, sε = βε + iεωε and ω → ∞.When ε = 1, by using (4.18), we conclude that the inverse Laplace transform has

the form [109]

φ (τ) = 1

2π

∫ β+iω

β−iω� (s) exp (τ s) ds, (4.19)

where s = β + iω and ω → ∞.Based on the relation (4.11), we compute (4.18).In this case, we have

φ (τ) = 1

(2π)ε

∫ β+iω

β−iω� (s)Eε

(τ εsε

)(ds)ε

= 1

(2π)ε

∫ β+iω

β−iω

[1

� (1 + ε)

∫ ∞

−∞φ (τ)Eε

(−τ εsε) (dτ)ε]Eε(τ εsε

)(ds)ε,

(4.20)

where s = β + iω, sε = βε + iεωε and ω → ∞.In the particular case, we have

φ (τ) = 1

(2π)ε

∫ β+iω

β−iω� (s)Eε

(τ εsε

)(ds)ε

= 1

(2π)ε

∫ β+iω

β−iω

[1

� (1 + ε)

∫ ∞

0φ (τ)Eε

(−τ εsε) (dτ)ε]Eε(τ εsε

)(ds)ε,

(4.21)

such that

�(s) = 1

� (1 + ε)

∫ ∞

0φ (τ)Eε

(−τ εsε) (dτ)ε, (4.22)

φ (τ) = 1

(2π)ε

∫ β+iω

β−iω� (s)Eε

(τ εsε

)(ds)ε, (4.23)

where s = β + iω, sε = βε + iεωε and ω → ∞.


The structure of this chapter is as follows. In Section 4.2, we present the definitionsand properties of the local fractional Laplace transform operators. In Section 4.3, wediscuss the application of the local fractional Laplace transform operator to signalanalysis. In Section 4.4, the local fractional Laplace transform operator was utilizedto solve some local fractional differential equation (ODEs and PDEs).


Below, we introduce the basic definitions of the local fractional Laplace operators andits properties [16, 21, 110–115].

4.2.1 The basic definitions of the local fractional Laplacetransform operators

Definition 4.1. Setting θ ∈ L1,ε [R+] and ‖θ‖1,ε < ∞, the local fractional Laplacetransform operator, denoted by M [θ (τ )] = �(s), is defined as

M [θ (τ )] = �(s) = 1

� (1 + ε)

∫ ∞

0θ (τ )Eε

(−τ εsε) (dτ)ε , (4.24)

where M is called the local fractional Laplace transform operator.

Definition 4.2. The inverse local fractional Laplace transform operator, denoted byM−1 [�(s)] = θ (τ ), is defined as

M−1 [�(s)] = θ (τ ) = 1

(2π)ε

∫ β+i∞

β−i∞�(s)Eε

(τ εsε

)(ds)ε, (4.25)

where M−1 is called the inverse local fractional Laplace transform operator.

A sufficient condition for convergence is presented as follows:

1

� (1 + ε)

∫ ∞

0|ψ (τ)| (dτ)ε < K < ∞. (4.26)

Definition 4.3. Setting θ ∈ L1,ε [R] and ‖θ‖1,ε < ∞, the two-sided local fractionalLaplace transform operator, denoted by A [θ (τ )] = �(s), is defined as

A [θ (τ )] = �(s) = 1

� (1 + ε)

∫ ∞

∞θ (τ )Eε

(−τ εsε) (dτ)ε , (4.27)

where A is called the two-sided local fractional Laplace transform operator.

Definition 4.4. The inverse two-sided local fractional Laplace transform operator,denoted by A−1 [�(s)] = θ (τ ), is defined as


A−1 [�(s)] = θ (τ ) = 1

(2π)ε

∫ β+i∞

β−i∞�(s)Eε

(τ εsε

)(ds)ε, (4.28)

where A−1 is called the inverse two-sided local fractional Laplace transform operator.

A sufficient condition for convergence is presented as

1

� (1 + ε)

∫ ∞

−∞|ψ (τ)| (dτ)ε < K < ∞. (4.29)

4.2.2 The properties and theorems for the local fractionalLaplace transform operator

Property 21 (Linearity for local fractional Laplace transform operator). Suppose thatθ1 (τ ) , θ2 (τ ) ∈ L1,ε [R+], M [θ1 (τ )] = �1 (s) and M [θ2 (τ )] = �2 (s), then there is

M [aθ1 (τ )± bθ2 (τ )] = a�1 (s)± b�2 (s) , (4.30)


Proof. We have, by definition of the local fractional Laplace transform operator,

M [aθ1 (τ )± bθ2 (τ )] = 1

� (1 + ε)

∫ ∞

0[aθ1 (τ )± bθ2 (τ )] Eε

(−τ εsε) (dτ)ε= a

1

� (1 + ε)

∫ ∞

0

[θ1 (τ )Eε

(−τ εsε)] (dτ)ε± b

1

� (1 + ε)

∫ ∞

0

[θ2 (τ )Eε

(−τ εsε)] (dτ)ε= a�1 (s)± b�2 (s) . (4.31)

Thus, the proof is finished.

Property 22 (Shifting time for local fractional Laplace transform operator).Suppose that θ (τ ) ∈ L1,ε [R+], M [θ (τ )] = �(s) and a is a constant, then there is

M [θ (τ − a)] = Eε(−aεsε

)M [θ (τ )] . (4.32)

Proof. By using the definition of the local fractional Laplace transform operator,we obtain

M [θ (τ − a)] = 1

� (1 + ε)

∫ ∞

0θ (τ − a)Eε

(−τ εsε) (dτ)ε= Eε

(−aεsε) 1

� (1 + ε)

∫ ∞

0θ (τ − a)Eε

(− (τ − a)ε sε)(dτ)ε


= Eε(−aεsε

)�(s)

= Eε(−aεsε

)M [θ (τ )] . (4.33)

Therefore, the proof is finished.

Property 23 (Scaling time for local fractional Laplace transform operator).Suppose that θ (τ ) ∈ L1,ε [ R+], M [θ (τ )] = �(s) and a (a > 0) is a constant,then there is

M [θ (aτ)] = 1

aε�

( s

a

). (4.34)

Proof. With the help of the definition of the local fractional Laplace transformoperator, we obtain

M [θ (aτ)] = 1

� (1 + ε)

∫ ∞

0θ (aτ)Eε

(−τ εsε) (dτ)ε= 1

aε1

� (1 + ε)

∫ ∞

0θ (aτ)Eε

(−(aτ

a

)εsε)(daτ)ε

= 1

aε�

( s

a

). (4.35)

Thus, we completed this proof.

Property 24 (Translation for local fractional Laplace transform operator). Sup-pose that θ (τ ) ∈ L1,ε [ R+], M [θ (τ )] = �(s) and a is a constant, then there is

M[Eε

(aετ ε

)θ (τ )

] = �(s − a) . (4.36)

Proof. By utilizing the definition of the local fractional Laplace transform operator,we conclude that

M[Eε

(−aετ ε)θ (τ )

] = 1

� (1 + ε)

∫ ∞

0

[θ (τ )Eε

(aετ ε

)]Eε


� (1 + ε)

∫ ∞

0θ (τ )Eε

(− (s − a)ε τ ε)(dτ)ε

= �(s − a) . (4.37)

Therefore, we proved our claim.

Theorem 4.1. Suppose that θ (τ ) ∈ L1,ε [ R+], M [θ (τ )] = �(s) andlimτ→∞ θ (τ ) = 0, then there is

M[θ(ε) (τ )

]= sεM [θ (τ )] − θ (0) . (4.38)


Proof. Once more, by using the definition of the local fractional Laplace transformoperator, we obtain

M[θ(ε) (τ )

]= 1

� (1 + ε)

∫ ∞

0θ(ε) (τ )Eε

(−τ εsε) (dτ)ε. (4.39)

Taking the integration by part for local fractional integral operator and using (4.36),we conclude

M[θ(ε) (τ )

]= 1

� (1 + ε)

∫ ∞

0θ(ε) (τ )Eε

(−τ εsε) (dτ)ε= [

θ (τ )Eε(−τ εsε)]∞0 + sε

� (1 + ε)

∫ ∞

0θ (τ )Eε

(−τ εsε) (dτ)ε,(4.40)

which, using limτ→∞ θ (τ ) = 0, leads to

M[θ(ε) (τ )

]= sε

[1

� (1 + ε)

∫ ∞

0θ (τ )Eε

(−τ εsε) (dτ)ε] − θ (0)

= sεM [θ (τ )] − θ (0) . (4.41)

Thus, we completed the proof.

We remark that there is

M[θ(kε) (τ )

]= skεM [θ (τ )]−s(k−1)εθ (0)−s(k−2)εθ (ε) (0)−·· ·−θ((k−1)ε) (0) ,

(4.42)

where k ∈ N.

Theorem 4.2. Suppose that θ (τ ) ∈ L1,ε [ R+], M [θ (τ )] = �(s) and

limτ→∞ 0I(ε)τ θ (τ ) = 0, then there is

M[0I(ε)τ θ (τ )

]= 1

sεM [θ (τ )] . (4.43)

Proof. In accordance with the definition of the local fractional Laplace transformoperator, we have

M[0I(ε)τ θ (τ )

]= 1

� (1 + ε)

∫ ∞

0

(0I(ε)τ θ (τ )

)Eε

(−τ εsε) (dτ)ε, (4.44)

which is, using the integration by part for local fractional integral operator,

M[0I(ε)τ θ (τ )

]=

[0I(ε)τ θ (τ )Eε

(−τ εsε)]∣∣∣∞0

+[

sε

� (1 + ε)

∫ ∞

0θ (τ )Eε

(−τ εsε) (dτ)ε] . (4.45)


Taking limτ→∞ 0I(ε)τ θ (τ ) = 0 into account in (4.45), we have the following result:

M[0I(ε)τ θ (τ )

]=

[sε

� (1 + ε)

∫ ∞

0θ (τ )Eε

(−τ εsε) (dτ)ε]= sεM [θ (τ )] . (4.46)

In this way, we finish the proof.


M[0I(kε)τ θ (τ )

]= 1

skεM [θ (τ )] , (4.47)

where

limτ→∞ 0I(kε)τ θ (τ ) = 0. (4.48)

Theorem 4.3. Suppose θ (τ ) ∈ L1,ε [R+] and M [θ (τ )] = �(s), then there is

M[τ εθ (τ )

] = �(ε) (s) . (4.49)

Proof. Again, by using the definition of the inverse local fractional Laplacetransform operator, we conclude that

M−1[�(ε) (s)

]= 1

(2π)ε

∫ β+i∞

β−i∞

[�(ε) (s)

]Eε

(τ εsε

)(ds)ε, (4.50)

which becomes, by using the related integrating by part,

M−1[�(ε) (s)

]= � (1 + ε)

(2π)ε[�(s)Eε

(τ εsε

)]β+i∞β−i∞

+ τ ε

[1

(2π)ε

∫ β+i∞

β−i∞�(s)Eε

(τ εsε

)(ds)ε

]. (4.51)

Taking lim|s|→∞�(s) = 0 into account in (4.51), the final result can be seen as

M−1[�(ε) (s)

]= τ ε

[1

(2π)ε

∫ β+i∞

β−i∞�(s)Eε

(τ εsε

)(ds)ε

]

= τ εθ (τ ) . (4.52)

Therefore, we finished the proof.

Using the sane way of thinking, we have

M[τ kεθ (τ )

]= (−1)kε �(kε) (s) , (4.53)

where

lim|s|→∞�(kε) (s) = 0, k ∈ N. (4.54)


Definition 4.5. The local fractional convolution of two functions θ1 (τ ) and θ2 (τ )

via local fractional integral operator, denoted by (θ1 ∗ θ2) (τ ) = θ1 (τ ) ∗ θ2 (τ ), isdefined as

(θ1 ∗ θ2) (τ ) = θ1 (τ ) ∗ θ2 (τ )

= 0I(ε)∞ [θ1 (t) θ2 (τ − t)]

= 1

� (1 + ε)

∫ ∞

0θ1 (t) θ2 (τ − t) (dt)ε. (4.55)

From the definition of local fractional convolution, we obtain the followingproperties, namely:

(a) θ1 (τ ) ∗ θ2 (τ ) = θ2 (τ ) ∗ θ1 (τ );(b) θ1 (τ ) ∗ (θ2 (τ )+ θ3 (τ )) = θ1 (τ ) ∗ θ2 (τ )+ θ1 (τ ) ∗ θ3 (τ ).

Theorem 4.4. Suppose that θ1 (τ ) , θ2 (τ ) ∈ L1,ε [R+], M [θ1 (τ )] = �1 (s) andM [θ2 (τ )] = �2 (s), then there is

M [θ1 (τ ) ∗ θ2 (τ )] = �1 (s)�2 (s) (4.56)

or

θ1 (τ ) ∗ θ2 (τ ) = M−1 [�1 (s)�2 (s)] (4.57)

or, equivalently,

1

� (1 + ε)

∫ ∞

0θ1 (t) θ2 (τ − t) (dt)ε = 1

(2π)ε

∫ β+iω

β−iω�1 (s)�2 (s)Eε

(τ εsε

)(ds)ε.

(4.58)

Proof. From the definition of the local fractional Laplace transform operator, weconclude that

M [θ1 (τ ) ∗ θ2 (τ )] = 1

� (1 + ε)

∫ ∞

0Eε

(−τ εsε) (dτ)ε×

[1

� (1 + ε)

∫ ∞

0θ1 (η) θ2 (τ − η) (dη)ε

]

= 1

� (1 + ε)

∫ ∞

0θ1 (η)Eε

(−ηεsε)×

[1

� (1 + ε)

∫ ∞

0Eε

(− (τ − η)ε sε)θ2 (τ − η) (dτ)ε

](dη)ε .

(4.59)

From

�2 (s) = 1

� (1 + ε)

∫ ∞

0Eε

(− (τ − η)ε sε)θ2 (τ − η) (dτ)ε, (4.60)


we conclude that

M [θ1 (τ ) ∗ θ2 (τ )] = 1

� (1 + ε)

∫ ∞

0θ1 (η)Eε

(−ηεsε)�2 (s) (dη)ε

= �2 (s)1

� (1 + ε)

∫ ∞

0θ1 (η)Eε

(−ηεsε) (dη)ε= �1 (s)�2 (s) . (4.61)

Thus, the desired result is proved.

Theorem 4.5. Suppose that θ1 (τ ) , θ2 (τ ) ∈ L1,ε [R+], M [θ1 (τ )] = �1 (s) andM [θ2 (τ )] = �2 (s), then there is

M [θ1 (τ ) θ2 (τ )] = �1 (s) ∗�2 (s) (4.62)

or

θ1 (τ ) θ2 (τ ) = M−1 [�1 (s) ∗�2 (s)] (4.63)

or, equivalently,

1

� (1 + ε)

∫ ∞

0θ1 (τ ) θ2 (τ )Eε


(2π)ε

∫ β−iω

β−iω�1 (s)�2 (s − s) (ds)ε. (4.64)

Proof. In addition, we conclude that

M−1 [�1 (s) ∗�2 (s)] = 1

(2π)ε

∫ β+i∞

β−i∞

[1

(2π)ε

∫ β+i∞

β−i∞�1 (s)�2 (s − s) (ds)ε

]

× Eε(τ εsε

)(ds)ε

= 1

(2π)ε

∫ β+i∞

β−i∞Eε

(τ ε sε

)�1 (s)

×[

1

(2π)ε

∫ β+i∞

β−i∞�2 (s − s)Eε

(τ ε (s − s)ε

)(ds)ε

](ds)ε .

(4.65)

Thus, we have

θ2 (τ ) = 1

(2π)ε

∫ β+i∞

β−i∞�2 (s − s)Eε

(τ ε (s − s)ε

)(ds)ε (4.66)

such that (4.65) is expressed as


M−1 [�1 (s) ∗�2 (s)] = 1

(2π)ε

∫ β+i∞

β−i∞Eε

(τ ε sε

)�1 (s) θ2 (τ ) (ds)ε

=[

1

(2π)ε

∫ β+i∞

β−i∞Eε

(τ ε sε

)�1 (s) (ds)ε

]θ2 (τ )

= θ1 (τ ) θ2 (τ ) . (4.67)

Therefore, we completed the proof.

Theorem 4.6 (Convolution theorem for local fractional Laplace transform oper-ator). Suppose θ1 (τ ) , θ2 (τ ) ∈ L1,ε [R+], M [θ1 (τ )] = �1 (s), and M [θ2 (τ )] =�2 (s), then there is

1

� (1 + ε)

∫ ∞

0θ1 (τ ) θ2 (τ ) (dτ)

ε = 1

(2π)ε

∫ β+i∞

β−i∞�1 (s)�2 (s) (ds)ε. (4.68)

Proof. We consider that

θ2 (τ ) = 1

(2π)ε

∫ β+i∞

β−i∞�2 (s)Eε (τ εsε) (ds)ε

= 1

(2π)ε

∫ β+i∞

β−i∞�2 (s)Eε

(−τ εsε) (ds)ε. (4.69)

In this case, from (4.68), we write

1

� (1 + ε)

∫ ∞

0θ1 (τ ) θ2 (τ ) (dτ)

ε

= 1

� (1 + ε)

∫ ∞

0θ1 (τ )

[1

(2π)ε

∫ β+i∞

β−i∞�2 (s)Eε

(−τ εsε) (ds)ε](dτ)ε

=[

1

(2π)ε

∫ β+i∞

β−i∞�2 (s)

[1

� (1 + ε)

∫ ∞

0θ1 (τ )Eε

(−τ εsε) (dτ)ε] (ds)ε]

.

(4.70)

Therefore, observing

�1 (s) = 1

� (1 + ε)

∫ ∞

0θ1 (τ )Eε

(−τ εsε) (dτ)ε, (4.71)

we clearly say that

1

� (1 + ε)

∫ ∞

0θ1 (τ ) θ2 (τ ) (dτ)

ε = 1

(2π)ε

∫ β+i∞

β−i∞�1 (s)�2 (s) (ds)ε. (4.72)

Thus, the proof is completed.


Theorem 4.7 (The initial value theorem for local fractional Laplace transformoperator). Suppose that M [θ (τ )] = �(s) and M

[θ(ε) (τ )

] = sεM [θ (τ )] − θ (0−),then there is

limτ→0+

θ(τ ) = θ(0+) = lims→∞ sεM [θ (τ )] . (4.73)

Proof. Due to

lims→∞ Eε

(−sετ ε) = 0, (4.74)

it follows that

lims→∞

1

� (1 + ε)

∫ ∞

0+

dεθ (τ )

dτ εEε

(−sετ ε)(dτ)ε

= 1

� (1 + ε)

∫ ∞

0+

dεθ (τ )

dτ ε

(lim

s→∞ Eε(−sετ ε

))(dτ)ε

= 0. (4.75)

Thus, we have

M[θ(ε) (τ )

]= sεM [θ (τ )] − θ (0−)

= lims→∞

1

� (1 + ε)

∫ ∞

0−

dεθ (τ )

dτ εEε

(−sετ ε)(dτ)ε. (4.76)

In this case, we conclude

1

� (1 + ε)

∫ 0+

0−

dεθ (τ )

dτ εEε

(−sετ ε)(dτ)ε = θ (0+)− θ (0−) , (4.77)

which leads to

1

� (1 + ε)

∫ ∞

0−dεθ (τ )

dτ εEε


= 1

� (1 + ε)

∫ 0+

0−dεθ (τ )

dτ εEε


+ 1

� (1 + ε)

∫ ∞

0+dεθ (τ )

dτ εEε


= θ (0+)− θ (0−)+ 1

� (1 + ε)

∫ ∞

0+dεθ (τ )

dτ εEε


= θ (0+)− θ (0−) . (4.78)

Thus, we conclude

lims→∞ M

[θ(ε) (τ )

]= lim

s→∞[sεM [θ (τ )] − θ (0−)

](4.79)


such that

lims→∞ sεM [θ (τ )] = lim

s→∞{

M[θ(ε) (τ )

]+ θ (0−)

}= lim

s→∞

[1

� (1 + ε)

∫ ∞

0−

dεθ (τ )

dτ εEε

(−sετ ε)(dτ)ε + θ (0−)

]

= lims→∞

[θ (0+)+ 1

� (1 + ε)

∫ ∞

0+

dεθ (τ )

dτ εEε


]= θ (0+)= limτ→0+

θ (τ ) . (4.80)

Thus, we obtained the result.

Theorem 4.8 (The final value theorem for local fractional Laplace transformoperator). Suppose that M [θ (τ )] = �(s) and M

[θ(ε) (τ )

] = sεM [θ (τ )] − θ (0−),then there is

limτ→∞ θ(τ ) = θ(+∞) = lim

s→0sεM [θ (τ )] . (4.81)

Proof. We consider

lims→0

M[θ(ε) (τ )

]= lim

s→0

[sεM [θ (τ )] − θ (0−)

], (4.82)

which leads to

lims→0

sεM [θ (τ )] = lims→0

[M

[θ(ε) (τ )

]+ θ (0−)

]. (4.83)

In this case, we may easily put it in the form

lims→0

M[θ(ε) (τ )

]= lim

s→0

[1

� (1 + ε)

∫ ∞

0−

dεθ (τ )

dτ εEε


]

= 1

� (1 + ε)

∫ ∞

0−

dεθ (τ )

dτ ε

[lims→0

Eε(−sετ ε

)](dτ)ε

= 1

� (1 + ε)

∫ ∞

0−

dεθ (τ )

dτ ε(dτ)ε

= θ (∞)− θ (0−) . (4.84)

Thus, making use of (4.83), we have the following result:

lims→0

sεM [θ (τ )] = lims→0

[M

[θ(ε) (τ )

]+ θ (0−)

]= lim

s→0θ (∞)

= θ (∞) . (4.85)

Therefore, the proof of this theorem is reported.


For the tables of the local fractional Laplace transform operators, the reader cansee Appendix F.


We consider the signal defined on Cantor sets [16–21].Let us take the local fractional Laplace transform of the signal defined as

θ (τ ) = 1, τ > 0. (4.86)

We observe that, by using the definition of the local fractional Laplace transformoperator, we have

M [1] = 1

� (1 + ε)

∫ ∞

0Eε


sε. (4.87)

We determine the signal defined on Cantor sets, which is given as

θ (τ ) = Eε(aετ ε

), τ > 0. (4.88)

With the help of the local fractional Laplace transform operator, we report the result:

M[Eε

(aετ ε

)] = 1

� (1 + ε)

∫ ∞

0Eε

(aετ ε

)Eε


� (1 + ε)

∫ ∞

0Eε

(−τ ε (s − a)ε)(dτ)ε

= 1

(s − a)ε

= 1

sε − aε. (4.89)

Taking aε = με + iεηε in (4.88), where μ and η are constants, it gives

M[Eε

((με + iεηε

)τ ε

)] = 1

sε − (με + iεηε). (4.90)

Taking aε = με − iεηε in (4.88), we may create

M[Eε

((με − iεηε

)τ ε

)] = 1

sε − (με − iεηε). (4.91)

Taking μ = 0 in (4.90), the final result reads as

M[Eε

(iεηετ ε

)] = 1

sε − iεηε. (4.92)

Taking μ = 0 in (4.91), this may be added in the form

M[Eε

(−iεηετ ε)] = 1

sε + iεηε. (4.93)


We report the local fractional Laplace transform of the signal on Cantor denoted by

θ (τ ) = cosε(ηετ ε

), τ > 0. (4.94)

From (4.92) and (4.93), we conclude that

M[cosε

(ηετ ε

)] = 1

� (1 + ε)

∫ ∞

0cosε

(ηετ ε

)Eε


� (1 + ε)

∫ ∞

0

[Eε (iεηετ ε)+ Eε (−iεηετ ε)

2

]Eε

(−τ εsε) (dτ)ε

=1

sε−iεηε + 1sε+iεηε

2

= sε

s2ε + η2ε . (4.95)

We determine the local fractional Laplace transform of the signal on Cantor as

θ (τ ) = sinε(ηετ ε

), τ > 0. (4.96)

Utilizing both (4.92) and (4.93), we see that

M[sinε

(ηετ ε

)] = 1

� (1 + ε)

∫ ∞

0sinε

(ηετ ε

)Eε


� (1 + ε)

∫ ∞

0

[Eε (iεηετ ε)− Eε (−iεηετ ε)

2iε

]Eε


=1

sε−iεηε − 1sε+iεηε

2iε

= ηε

s2ε + η2ε . (4.97)

We find the local fractional Laplace transform of the signal on Cantor, namely,

θ (τ ) = coshε(ηετ ε

), τ > 0. (4.98)

By using the formula (4.89), we found that

M[coshε

(ηετ ε

)] = 1

� (1 + ε)

∫ ∞

0coshε

(ηετ ε

)Eε


� (1 + ε)

∫ ∞

0

[Eε (ηετ ε)+ Eε (−ηετ ε)

2

]Eε

(−τ εsε) (dτ)ε=

1sε−ηε + 1

sε+ηε2

= sε

s2ε − η2ε . (4.99)

We show the local fractional Laplace transform of the signal on Cantor

θ (τ ) = sinhε(ηετ ε

), τ > 0. (4.100)


With the help of formula (4.89), we present the final result as

M[sinhε

(ηετ ε

)] = 1

� (1 + ε)

∫ ∞

0sinhε

(ηετ ε

)Eε


� (1 + ε)

∫ ∞

0

[Eε (ηετ ε)− Eε (−ηετ ε)

2

]Eε


1sε−ηε − 1

sε+ηε2

= ηε

s2ε − η2ε . (4.101)

We compute the local fractional Laplace transform of the signal on Cantor as

θ (τ ) = τ kε

� (1 + kε), τ > 0, k ∈ N. (4.102)

With the help of the definition of the local fractional Laplace transform, we have

M

[τ kε

� (1 + kε)

]= 1

� (1 + ε)

∫ ∞

0

τ kε

� (1 + kε)Eε


sε1

� (1 + ε)

∫ ∞

0

τ (k−1)ε

� (1 + (k − 1) ε)Eε


s2ε

1

� (1 + ε)

∫ ∞

0

τ (k−2)ε

� (1 + (k − 2) ε)Eε


sε(k+1). (4.103)

We give the signal defined on Cantor sets by the following expression:

θ (τ ) = τ kε

� (1 + kε)Eε

(aετ ε

), τ > 0, k ∈ N. (4.104)

Its graphs with different parameters a and k are depicted in Figure 4.1.With the help of (4.36) and (4.103), we conclude that

M

[τ kε

� (1 + kε)Eε

(aετ ε

)] = 1

� (1 + ε)

∫ ∞

0

[τ kε

� (1 + kε)Eε

(aετ ε

)]Eε


(s − a)(k+1)ε. (4.105)

We write the local fractional Laplace transform of the signal on Cantor sets as


)cosε

(ηετ ε

), τ > 0. (4.106)

The corresponding graphs, with different parameters a and η, are shown inFigure 4.2.


0 0.2 0.4 0.6 0.8 11.5

2

2.5

3

3.5

4

4.5

5

t

q(t)

k= 2, aε = 1

Figure 4.1 The graph of θ (τ ) when ε = ln 2/ ln 3.

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

t

q(t

)

ae = 1, he = 0.2



0 0.2 0.4 0.6 0.8 1–0.45

–0.4

–0.35

–0.3

–0.25

–0.2

–0.15

–0.1

–0.05

0

t

q(t

)

ae = 1, he = 0.2


From formulas (4.90) and (4.91), we conclude that

M[Eε

(aετ ε

)cosε

(ηετ ε

)]= 1

� (1 + ε)

∫ ∞

0

[Eε

(aετ ε

)cosε

(ηετ ε

)]Eε


� (1 + ε)

∫ ∞

0Eε

(aετ ε

) [Eε (iεηετ ε)+ Eε (−iεηετ ε)

2

]Eε


=1(

(s−a)ε−iεηε) + 1(

(s−a)ε+iεηε)

2= (s − a)2ε

(s − a)2ε + η2ε. (4.107)

We consider the local fractional Laplace transform of the signal on Cantor sets as


)sinε

(ηετ ε

), τ > 0. (4.108)

Its graphs with different parameters a and η are presented in Figure 4.3.Using (4.92) and (4.93), it implies that

M[Eε

(aετ ε

)sinε

(ηετ ε

)]= 1

� (1 + ε)

∫ ∞

0

[Eε

(aετ ε

)sinε

(ηετ ε

)]Eε


� (1 + ε)

∫ ∞

0Eε

(aετ ε

)


0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

4

4.5

5

t

q(t

)

ae = 1, he = 0.2


×[

Eε (iεηετ ε)− Eε (−iεηετ ε)

2iε

]Eε


=1

(s−a)ε−iεηε − 1(s−a)ε+iεηε

2iε

= ηε

(s − a)2ε + η.2ε. (4.109)

We treat the local fractional Laplace transform of the signal on Cantor sets as


)coshε

(ηετ ε

), τ > 0. (4.110)

As we did before, the corresponding graphs for different parameters of a and η areplotted in Figure 4.4.

In view of (4.36) and (4.100), the final result is given below:

M[Eε

(aετ ε

)coshε

(ηετ ε

)] = 1

� (1 + ε)

∫ ∞

0

[Eε

(aετ ε

)coshε

(ηετ ε

)]Eε


� (1 + ε)

∫ ∞

0Eε

(aετ ε

) [Eε (ηετ ε)+ Eε (−ηετ ε)2

]Eε


1(s−a)ε−ηε + 1

(s−a)ε+ηε2

= (s − a)ε

(s − a)2ε − η2ε. (4.111)


0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t

q(t

)

ae = 1, he = 0.2


Let us compute the local fractional Laplace transform of the signal on Cantor setsdefined by


)sinhε

(ηετ ε

), τ > 0. (4.112)

In Figure 4.5, its graphs with different parameters a and η are illustrated.Adopting (4.36) and (4.101), we conclude that

M[Eε

(aετ ε

)sinhε

(ηετε

)] = 1

� (1 + ε)

∫ ∞0

[Eε

(aετ ε

)sinhε

(ηετε

)]Eε

(−τεsε)(dτ)ε

= 1

� (1 + ε)

∫ ∞0

Eε(aετ ε

) [Eε(ηετε

) − Eε(−ηετε)

2

]

Eε(−τεsε

)(dτ)ε

=1

(s − a)ε − ηε− 1

(s − a)ε + ηε

2= ηε

(s − a)2ε − η.2ε. (4.113)

We present the local fractional Laplace transform of the signal on Cantor sets as

θ (τ ) = Eε(−aτ ε

) − Eε(−bτ ε

), τ > 0. (4.114)

Thus, from the definition of the local fractional Laplace transform operator, weconclude


M[Eε

(−aτ ε) − Eε

(−bτ ε)] = 1

� (1 + ε)

∫ ∞

0

[Eε

(−aτ ε) − Eε

(−bτ ε)]

Eε(−τ εsε) (dτ)ε

=1

sε + a− 1

sε + b2

= b − a

(sε + a) (sε + b). (4.115)

We take the local fractional Laplace transform of the signal on Cantor setsdefined by

θ (τ ) = Eε(−aτ ε

) + aτ ε

� (1 + ε)− 1, τ > 0. (4.116)

From the definition of the local fractional Laplace transform operator, it followsthat

M

[Eε

(−aτ ε) + aτ ε

� (1 + ε)− 1

]

= 1

� (1 + ε)

∫ ∞

0

[Eε

(−aτ ε) + aτ ε

� (1 + ε)− 1

]Eε


sε + a+ a

s2ε − 1

sε

= a2

(sε + a) s2ε . (4.117)

We consider the local fractional Laplace transform of the signal on Cantor setsdefined by

θ (τ ) = 1

η2

τ ε

� (1 + ε)− 1

η3 sinε(ητε

), τ > 0. (4.118)

Thus, we obtain from the definition of the local fractional Laplace transformoperator

M

[1

η2

τ ε

� (1 + ε)− 1

η3 sinε(ητε

)]

= 1

� (1 + ε)

∫ ∞

0

[1

η2

τ ε

� (1 + ε)− 1

η3 sinε(ητε

)]Eε


η2

(1

s2ε − 1

s2ε + η2

),

= 1

s2ε(s2ε + η2

) . (4.119)


Now, let us consider the local fractional Laplace transform of the signal on Cantorsets as

θ (τ ) = 1

η2 − μ2

(sinε (μτε)

μ− sinε (ητ ε)

η

), τ > 0. (4.120)

By using the definition of the local fractional Laplace transform operator, the finalresult is written as

M

[1

η2 − μ2

(sinε (μτε)


η

)]

= 1

� (1 + ε)

∫ ∞

0

[1

η2 − μ2

(sinε (μτε)


η

)]Eε


η2 − μ2

[1

s2ε + μ2 − 1

s2ε + η2

]

= 1(s2ε + μ2

) (s2ε + η2

) . (4.121)

We find the local fractional Laplace transform of the signal on Cantor sets by thefollowing expression:

θ (τ ) = μ− (μ− η)Eε (−ητε)η

, τ > 0. (4.122)

We have

M

[μ− μEε (−ητε)

η

]= 1

� (1 + ε)

∫ ∞

0

[μ− μEε (−ητε)

η

]Eε

(−τ εsε) (dτ)ε= μ

η

1

sε− μ

η

1

sε + η

= μη

sε (sε + η). (4.123)

We handle the local fractional Laplace transform of the signal on Cantor sets as

θ (τ ) =(

1 + ηετ ε

� (1 + ε)

)Eε

(ηετ ε

), τ > 0. (4.124)

Taking the local fractional Laplace transform of (4.124), we conclude that

M

[(1 + ηετ ε

� (1 + ε)

)Eε

(ηετ ε

)]

= 1

� (1 + ε)

∫ ∞

0

[(1 + ηετ ε

� (1 + ε)

)Eε

(ηετ ε

)]Eε


sε − ηε+ ηε

(s − η)2ε

= sε

(s − η)2ε. (4.125)


0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

90

100

t

q(t

)


We give the local fractional Laplace transform of the signal, which is a localfractional Dirac function:

θ (τ ) = δε (τ ) , τ > 0. (4.126)

Similarly, we have

M [δε (τ )] = 1

� (1 + ε)

∫ ∞

0δε (τ )Eε

(−τ εsε) (dτ)ε= 1. (4.127)

We consider the local fractional Laplace transform of the signal, which is given as

θ (τ ) = τ−ε/2, τ > 0, (4.128)

and we depict its graph in Figure 4.6.It easily follows that

M[τ−ε/2] = 1

� (1 + ε)

∫ ∞

0τ−ε/2Eε

(−τ εsε) (dτ)ε= � (1 + 2ε)

�2 (1 + ε)

[s−ε/2

(1

� (1 + ε)

∫ ∞

0Eε

(−τ 2ε

)(dτ)ε

)]

= � (1 + 2ε)

2�3 (1 + ε)

(πs

)ε/2. (4.129)


Find the local fractional Laplace transform of the analogous rectangular pulse,defined by

rectε (τ , τ0, τ1) =⎧⎨⎩

0, τ0 ≤ τ ,1, τ0 < τ ≤ τ1,0, τ > τ1,

(4.130)

that is to say,

M [rectε (τ , τ0, τ1)] = 1

� (1 + ε)

∫ ∞

0rectε (τ , τ0, τ1)Eε


� (1 + ε)

∫ τ1

τ0

Eε(−τ εsε) (dτ)ε

= 1

sε[Eε

(−τ ε0 sε) − Eε

(−τ ε1 sε)]

, (4.131)

where 0 < τ0 < τ1.For some illustrative examples, the reader can see Appendix F.


We now present the local fractional Laplace transform operators to obtain thenondifferentiable solution of local ordinary and partial differential equations.


The potential applications of the local fractional Laplace transform operators areparticularly effective for linear local fractional ODEs with constant coefficients. Now,we illustrate the methods with the following examples:

Let us consider the local fractional ordinary differential equation

dεθ (μ)

dμε+ aθ (μ) = b sinε

(cμε

), (4.132)

subject to initial value condition

θ (0) = 1, (4.133)

where a, b, and c are constants.Taking the local fractional Laplace transform in (4.131), it gives

sε� (s)− 1 + a�(s) = bc

s2ε + c2. (4.134)


In this case, we can rearrange (4.133) to obtain

�(s) = bc

(sε + a)(s2ε + c2

) + 1

sε + a

=(

bc

a2 + c2 + 1

)1

sε + a− bc(

a2 + c2) ( sε

s2ε + c2

)+ ab(

a2 + c2) ( c

s2ε + c2

),

(4.135)

which leads to

θ (μ) =(

bc

a2 + c2 + 1

)Eε

(−aμε)− bc(

a2 + c2) cosε

(cμε

)+ ab(a2 + c2

) sinε(cμε

).

(4.136)

We consider the local fractional ordinary differential equation

dεθ (μ)

dμε+ aθ (μ) = b cosε

(cμε

), (4.137)

with the initial value condition

θ (0) = 1, (4.138)

such that a, b, and c are constants.Similarly, making the local fractional Laplace transform in (4.136), we conclude

sε� (s)− 1 + a�(s) = bsε

s2ε + c2 . (4.139)

We may reformulate (4.138) as

�(s) = bsε

(sε + a)(s2ε + c2

) + 1

sε + a

=(

1 − ab

c2 + a2

)1

sε + a+ ab

c2 + a2

(sε

s2ε + c2

)

+⎛⎝b − a2b

c2+a2

c

⎞⎠(

c

s2ε + c2

). (4.140)

Thus, it follows that

θ (μ) =(

1 − ab

c2 + a2

)Eε

(−aμε) + ab

c2 + a2 cosε(cμε

)

+⎛⎝b − a2b

c2+a2

c

⎞⎠ sinε

(cμε

). (4.141)


The local fractional ordinary differential equation takes the form

dεθ (μ)

dμε+ aθ (μ) = bEε

(cμε

), (4.142)

subject to the initial value condition

θ (0) = 1, (4.143)

where a, b, and c are constants.Let us take the local fractional Laplace transform in (4.141). That is to say,

sε� (s)− 1 + a�(s) = b

sε − c. (4.144)

Thus, we have

�(s) = b

(sε − c) (sε + a)+ 1

sε + a. (4.145)

Therefore, we have the following result:

θ (μ) = Eε(−aμε

) + b

a + c

[Eε

(aμε

) + Eε(−cμε

)]. (4.146)

We consider that the local fractional ordinary differential equation

d2εθ (μ)

dμ2ε + a2θ (μ) = b sinε(cμε

), (4.147)

subject to the initial value conditions

θ (0) = 0, (4.148)

dεθ (0)

dμε= 0, (4.149)

where a, b, and c are constants.Taking the local fractional Laplace transform in (4.146), we conclude that

s2ε� (s)+ a2�(s) = b

s2ε + c2 , (4.150)

which leads to

�(s) = b(s2ε + c2

) (s2ε + a2

) . (4.151)

Thus, from (4.121), this may be added in the form

θ (μ) = b

a2 − c2

(sinε (cμε)

c− sinε (aμε)

a

), (4.152)

where a = c.


For a = c, we found that

�(s) = b(s2ε + c2

)2 = b

2c2

[1

s2ε + c2 − s2ε − c2(s2ε + c2

)2

], (4.153)

where

M

[με

� (1 + ε)cosε

(ζμε

)] = s2ε − ζ 2(s2ε + ζ 2ε

)2 , (4.154)

such that

θ (μ) = b

2c2

[sinε (cμε)

c− με

� (1 + α)cosε

(cμε

)]. (4.155)

We investigate the local fractional ordinary differential equation

d2εθ (μ)

dμ2ε + a2θ (μ) = b cosε(cμε

), (4.156)

in the presence of the initial value conditions

θ (0) = 0, (4.157)

dεθ (0)

dμε= 0, (4.158)

where a, b, and c are constants.Similarly, we present

s2ε� (s)+ a2�(s) = bsε

s2ε + c2 , (4.159)

which becomes

�(s) = bsε(s2ε + c2

) (s2ε + a2

) . (4.160)

For a = c, we have

M

[1

a2 − b2

(cosε

(bμε

) − cosε(aμε

))] = sε(s2ε + a2

) (s2ε + b2

) , (4.161)

such that

θ (μ) = b

a2 − c2

(cosε

(cμε

) − cosε(aμε

)). (4.162)

In case a = c, we have

�(s) = bsε(s2ε + a2

)2 = b

a

asε(s2ε + a2

)2 (4.163)


in such a way that

θ (μ) = b

a

με

� (1 + ε)sinε

(aμε

), (4.164)

where

M

[με

� (1 + ε)sinε

(ζμε

)] = ζ sε(s2ε + ζ 2ε

)2 . (4.165)

Now, let us discuss the local fractional ordinary differential equation

d2εθ (μ)

dμ2ε − a2θ (μ) = b sinε(cμε

)(4.166)

equipped with the initial value conditions

θ (0) = 0, (4.167)

dεθ (0)

dμε= 0, (4.168)

such that a, b, and c are constants.After calculating the local fractional Laplace transform in (4.166), we conclude

s2ε� (s)− a2�(s) = b

s2ε + c2 , (4.169)

which is rearranged in the form

�(s) = b(s2ε + c2

) (s2ε − a2

) = b

c2 + a2

[1

s2ε − a2 − 1

s2ε + c2

]. (4.170)

Thus, from (4.170), it results that

θ (μ) = b

c2 + a2

[sinhε

(aμε

) − sinε(cμε

)]. (4.171)

Let us consider the local fractional ordinary differential equation

d2εθ (μ)

dμ2ε − a2θ (μ) = b cosε(cμε

)(4.172)

subjected to the initial value conditions

θ (0) = 0, (4.173)

dεθ (0)

dμε= 0, (4.174)

with a, b, and c being constants.In this case, we have

s2ε� (s)− a2�(s) = bsε

s2ε + c2 , (4.175)


which becomes

�(s) = bsε(s2ε + c2

) (s2ε − a2

) = b

c2 + a2

[sε

s2ε − a2 − sε

s2ε + c2

]. (4.176)

Thus, we write that

θ (μ) = b

c2 + a2

[coshε

(aμε

) − cosε(cμε

)]. (4.177)

The next step is to find the solution of the local fractional ordinary differential equation

d2εθ (μ)

dμ2ε + a2θ (μ) = 0 (4.178)

in the presence of the initial value conditions

θ (0) = b, (4.179)

dεθ (0)

dμε= c, (4.180)

where a, b, and c are constants. Using (4.42), we note that

s2ε� (s)− sεb − c + a2�(s) = 0, (4.181)

which leads us to

�(s) = sεb + c

s2ε + a2 . (4.182)

Thus, we get

θ (μ) = b cosε(aμε

) + c

asinε

(aμε

). (4.183)

Below, we present the solution of the local fractional ordinary differential equation

d2εθ (μ)

dμ2ε − a2θ (μ) = 0 (4.184)

subjected to the initial value conditions

θ (0) = b, (4.185)

dεθ (0)

dμε= c, (4.186)

where a, b, and c are constants.Adopting (4.42), we observe that

s2ε� (s)− sεb − c + a2�(s) = 0, (4.187)

which can be written as

�(s) = sεb + c

s2ε − a2 . (4.188)


Thus, the final result becomes

θ (μ) = b coshε(aμε

) + c

asinhε

(aμε

). (4.189)


We start with the local fractional partial differential equations

∂εθ (μ, τ)

∂με+ ∂εθ (μ, τ)

∂τ ε+ θ (μ, τ) = cosε

(τ ε

), (4.190)

subject to the initial-boundary conditions

θ (μ, 0) = 1, (4.191)

θ (0, τ) = 0. (4.192)

Taking the local fractional Laplace transform in (4.190), we find that

∂εθ (μ, s)

∂με+ (

sε + 1)θ (μ, s)− 1 = sε

s2ε + 1(4.193)

and

θ (0, s) = 0. (4.194)

In this case, (4.173) gives the following result:

θ (μ, s) =sε

s2ε+1+ 1

sε + 1

(1 − Eε

(− (sε + 1

)με

))=

{1

2sε

(1

s2ε − 1+ 1

s2ε + 1

)− 1

2

(1

s2ε − 1− 1

s2ε + 1

)}(1 − Eε

(− (sε + 1

)με

))=

{1

2sε

(1

s2ε − 1+ 1

s2ε + 1

)− 1

2

(1

s2ε − 1− 1

s2ε + 1

)}(1 − Eε

(−sεμε)

Eε(−με)) , (4.195)

which reduces to

θ (μ, τ) = 1

2

{coshε

(τ ε

) + cosε(τ ε

) − sinhε(τ ε

) + sinε(τ ε

)− Eε

(−με)Hε (τ − μ)[coshε

((τ − μ)ε

) + cosε((τ − μ)ε

)−Eε

(−με)Hε (τ − μ)[sinε

((τ − μ)ε

) − sinhε((τ − μ)ε

)]}.

(4.196)


The local fractional wave equation for the fractal vibrating string has the form

∂2εθ (μ, τ)

∂τ 2ε = a2ε ∂2εθ (μ, τ)

∂μ2ε , (4.197)

with the initial-boundary conditions

θ (μ, 0) = 0, (4.198)

∂εθ (μ, 0)

∂τ ε= 0, (4.199)

θ (0, τ) = sinε(τ ε

), (4.200)

|θ (μ, τ)| < �, (4.201)

where the constant aε denotes the speed of fractal wave travels.Taking the local fractional Laplace transform in (4.197), it gives

a2ε ∂2εθ (μ, s)

∂μ2ε − s2εθ (μ, s) = 0, (4.202)

which is rewritten in the form

∂2εθ (μ, s)

∂μ2ε − s2ε

a2ε θ (μ, s) = 0, (4.203)

where

θ (0, s) = 1

s2ε + 1. (4.204)

We observe that the general solution of (4.203) is

θ (μ, s) = ζ1 (s)Eε(sεμε/aε

) + ζ2 (s)Eε(−sεμε/aε

), (4.205)

which, from (4.201), leads to

ζ1 (s) = 0 (4.206)

or

θ (μ, s) = ζ2 (s)Eε(−sεμε/aε

). (4.207)

From (4.204), we see that (4.207) can be rewritten as

θ (μ, s) = 1

s2ε + 1Eε

(−sεμε/aε)

. (4.208)

Thus, the inverse local fractional Laplace transform of (4.208) reduces to

θ (μ, τ) = Hε (τ − μ/a) sinε[(τ − μ/a)ε

]. (4.209)


Let us consider now the local fractional diffusion equation in 1 + 1 fractal dimensionalspace, namely,

∂εθ (μ, τ)

∂τ ε− k2ε ∂

2εθ (μ, τ)

∂μ2ε = 0, (4.210)

subject to the initial-boundary conditions

θ (0, τ) = δε (τ ) , (4.211)

∂εθ (μ, 0)

∂με= 0, (4.212)

|θ (μ, τ)| < �. (4.213)

Taking the local fractional Laplace transform in (4.210), we have

sεθ (μ, s)− k2ε ∂2εθ (μ, s)

∂μ2ε = 0. (4.214)

From (4.211), we write

θ (μ, s) = ζ1 (s)Eε(

sε/2με/kε)

+ ζ2 (s)Eε(−sε/2με/kε

). (4.215)

From (4.213), it follows that

θ (μ, s) = ζ2 (s)Eε(−sε/2με/kε

), (4.216)

ζ1 (s) = 0. (4.217)

From (4.211), θ (0, s) = 1, we conclude that

ζ2 (s) = 1. (4.218)

We now observe from (4.216) and (4.218) that

θ (μ, s) = Eε(−sε/2με/kε

). (4.219)

Thus, we have the final result:

θ (μ, τ) = � (1 + ε)√4επεkε

Eε(−μ2ε/4επεkε

). (4.220)

5Coupling the local fractionalLaplace transform with analyticmethods

5.1 Introduction

Many challenging problems, such as vibrating strings, traffic flow, mass, and heattransfer in fractal dimensional time-space, have opened new frontiers in physics,mathematics, and engineering applications. The local fractional partial differentialequations were used to investigate some anomalous, still unsolved nondifferentialphenomena in nature. The local fractional Fourier and Laplace transform operatorshandle these types of equations (e.g., Chapters 3 and 4). Also, there are analytic andnumerical methods to deal with the local fractional partial differential equations. Thischapter presents the variational iteration, decomposition methods, and the couplingmethods of the Laplace transform with them within the local fractional operators.

In order to clearly illustrate the analytic methods, we consider the local fractionalpartial differential equation in a local fractional operator form, which is given by

L(n)ε � + Rε� = 0, (5.1)

where L(n)ε is linear local fractional operators of nε order and Rε is the linear local

fractional operators of order less than L(n)ε .

The structure of the chapter is given below. In Section 5.2, the variational iterationmethod of local fractional operator is presented. Section 5.3 gives the decompositionmethod of local fractional operator. In Section 5.4, the coupling Laplace transformwith variational iteration method of local fractional operator is given. Section 5.5is devoted to the coupling Laplace transform with decomposition method of localfractional operator.

5.2 Variational iteration method of the local fractionaloperator

In this section, the idea of the variational iteration method of local fractional operatoris briefly introduced. The variational iteration method, proposed by He, was used tofind the approximate solutions for the linear partial differential equations [116, 117].The variational iteration method of the local fractional operator was employed to solvethe local fractional partial differential equations [75, 104, 118–125].

Local Fractional Integral Transforms and Their Applications. http://dx.doi.org/10.1016/B978-0-12-804002-7.00005-XCopyright © 2016 Xiao-Jun Yang, Dumitru Baleanu and Hari M. Srivastava. Published by Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/B978-0-12-804002-7.00005-X


As a starting point, we consider the local fractional variational iteration algorithm,namely,

�n+1 (τ ) = �n (τ ) + 1

� (1 + ε)

∫ τ

0λ

{L(n)

ε �n (ξ) + Rε�n (ξ)}

(dξ)ε. (5.2)

The local fractional correction functional is written as

�n+1 (τ ) = �n (τ ) + 1

� (1 + ε)

∫ τ

0λ

{{L(n)

ε �n (ξ) + Rε�n (ξ)}}

(dξ)ε, (5.3)

where �n is considered as a restricted local fractional variation, that is, δε�n = 0 (formore details, the reader can read the references [75, 104, 118–125] and the referencestherein).

After identifying the multiplier, namely,

λ = (−1)n (ξ − τ)(n−1)ε

� (1 + (n − 1) ε), (5.4)

we have

�n+1 (τ ) = �n (τ ) + 1

� (1 + ε)

∫ τ

0

(−1)n (ξ − τ)(n−1)ε

� (1 + (n − 1) ε){{L(n)

ε �n (ξ) + Rε�n (ξ)}}

(dξ)ε

= �n (τ ) + 0I(ε)τ

{(−1)n (ξ − τ)(n−1)ε

� (1 + (n − 1) ε)

[L(n)

ε �n (ξ) + Rε�n (ξ)]}

.

(5.5)

Finally, the nondifferentiable solution of (5.1) admits

�(τ) = limn→∞ �n (τ ) . (5.6)

Let us consider now the local fractional partial differential equation

∂2ε�(μ, τ)

∂τ 2ε− ∂2ε�(μ, τ)

∂μ2ε− ∂3ε�(μ, τ)

∂με∂τ 2ε= 0, (5.7)

subjected to the initial-boundary condition

�(μ, 0) = μ2ε

� (1 + 2ε), 0 ≤ μ ≤ 1, (5.8)

∂ε�(μ, 0)

∂τ ε= 0, 0 ≤ μ ≤ 1, (5.9)

�(l, τ) = �(0, τ) = 0, τ > 0, (5.10)

∂ε�(l, τ)

∂με= ∂ε�(0, τ)

∂με= 0, τ > 0. (5.11)

We structure the local fractional variational iteration algorithm as

Coupling the local fractional Laplace transform with analytic methods 181

�n+1 (μ, τ) = �n (μ, τ) + 0I(ε)τ

{(ξ − τ)ε

� (1 + ε)

[∂2ε�n(μ, τ)

∂τ 2ε

−∂2ε�n(μ, τ)

∂μ2ε− ∂3ε�n(μ, τ)

∂με∂τ 2ε

]}, (5.12)

where

�0 (μ, τ) = μ2ε

� (1 + 2ε). (5.13)

With the help of (5.12) and (5.13), we obtain the following approximations:

�1 (μ, τ) = �0 (μ, τ) + 0I(ε)τ

{(ξ − τ)ε

� (1 + ε)

[∂2ε�0(μ, τ)

∂τ 2ε

−∂2ε�0(μ, τ)

∂μ2ε− ∂3ε�0(μ, τ)

∂με∂τ 2ε

]}= μ2ε

� (1 + 2ε)+ 0I(ε)

τ

{(ξ − τ)ε

� (1 + ε)

}= μ2ε

� (1 + 2ε)+ τ 2ε

� (1 + 2ε), (5.14)

�2 (μ, τ) = �1 (μ, τ) + 0I(ε)τ

{(ξ − τ)ε

� (1 + ε)

[∂2ε�1(μ, τ)

∂τ 2ε

−∂2ε�1(μ, τ)

∂μ2ε− ∂3ε�1(μ, τ)

∂με∂τ 2ε

]}= μ2ε

� (1 + 2ε)+ τ 2ε

� (1 + 2ε), (5.15)

�3 (μ, τ) = �2 (μ, τ) + 0I(ε)τ

{(ξ − τ)ε

� (1 + ε)

[∂2ε�2(μ, τ)

∂τ 2ε

−∂2ε�2(μ, τ)

∂μ2ε− ∂3ε�2(μ, τ)

∂με∂τ 2ε

]}= μ2ε

� (1 + 2ε)+ τ 2ε

� (1 + 2ε), (5.16)

...

�n (μ, τ) = μ2ε

� (1 + 2ε)+ τ 2ε

� (1 + 2ε). (5.17)

Thus, we obtain

�(μ, τ) = limn→∞ �n (μ, τ) = μ2ε

� (1 + 2ε)+ τ 2ε

� (1 + 2ε), (5.18)


00.2

0.40.6

0.81

0

0.5

10

50

100

150

200

mt

Φ(m

,t)

Figure 5.1 The plot of � (μ, τ) in fractal dimension ε = ln 2/ ln 3.

and the related graph is shown in Figure 5.1.We consider the following local fractional partial differential equation:

∂3ε� (η, μ)

∂μ3ε− ∂ε� (η, μ)

∂με= ∂4ε� (η, μ)

∂η4ε, (5.19)

equipped with the initial value conditions

∂2ε� (0, μ)

∂η2ε= Eε(μ

ε), (5.20)

∂ε� (0, μ)

∂ηε= 0, (5.21)

�(0, μ) = 0. (5.22)

We structure the local fractional variational iteration algorithm as

�n+1 (η, μ) = �n (η, μ) − 0I(ε)τ

{(ξ − τ)2ε

� (1 + 2ε)

[∂3ε�n (η, μ)

∂μ3ε

−∂ε�n (η, μ)

∂με− ∂4ε�n (η, μ)

∂η4ε

] }, (5.23)

where

�0 (η, μ) = η2ε

� (1 + 2ε)Eε(μ

ε). (5.24)


Thus, the approximations read as

�1 (η, μ) = �0 (η, μ) − 0I(ε)τ

{(ξ − τ)2ε

� (1 + 2ε)

[∂3ε�0 (η, μ)

∂μ3ε

−∂ε�0 (η, μ)

∂με− ∂4ε�0 (η, μ)

∂η4ε

]}

= η2ε

� (1 + 2ε)Eε(μ

ε) − 0I(ε)τ

{(ξ − τ)2ε

� (1 + 2ε)

[η2ε

� (1 + 2ε)Eε(μ

ε)

− η2ε

� (1 + 2ε)Eε(μ

ε) − 0

]}

= η2ε

� (1 + 2ε)Eε(μ

ε), (5.25)

�2 (η, μ) = �1 (η, μ) − 0I(ε)τ

{(ξ − τ)2ε

� (1 + 2ε)

[∂3ε�1 (η, μ)

∂μ3ε

−∂ε�1 (η, μ)

∂με− ∂4ε�1 (η, μ)

∂η4ε

] }

= η2ε

� (1 + 2ε)Eε(μ

ε) − 0I(ε)τ

{(ξ − τ)2ε

� (1 + 2ε)

[η2ε

� (1 + 2ε)Eε(μ

ε)

− η2ε

� (1 + 2ε)Eε(μ

ε) − 0

] }

= η2ε

� (1 + 2ε)Eε(μ

ε), (5.26)

�3 (η, μ) = �2 (η, μ) − 0I(ε)τ

{(ξ − τ)2ε

� (1 + 2ε)

[∂3ε�2 (η, μ)

∂μ3ε

−∂ε�2 (η, μ)

∂με− ∂4ε�2 (η, μ)

∂η4ε

] }

= η2ε

� (1 + 2ε)Eε(μ

ε) − 0I(ε)τ

{(ξ − τ)2ε

� (1 + 2ε)

[η2ε

� (1 + 2ε)Eε(μ

ε)

− η2ε

� (1 + 2ε)Eε(μ

ε) − 0

]}


00.2

0.40.6

0.81

0

0.5

10

1

2

3

4

mh

Φ(h

,m)

Figure 5.2 The plot of � (η, μ) in fractal dimension ε = ln 2/ ln 3.

= η2ε

� (1 + 2ε)Eε(μ

ε), (5.27)

...

�n (η, μ) = η2ε

� (1 + 2ε)Eε(μ

ε). (5.28)

As a result, we report

�(η, μ) = limn→∞ �n (η, μ) = η2ε

� (1 + 2ε)Eε(μ

ε), (5.29)

and its graph is shown in Figure 5.2.

5.3 Decomposition method of the local fractionaloperator

In this section, the idea of the decomposition method of local fractional operatoris considered. The decomposition method, proposed by Adomian, was used tofind the approximate solutions for linear partial differential equations [126, 127].Also, the decomposition method of local fractional operator was considered to findthe nondifferentiable solutions of the local fractional partial differential equations[125, 128, 129].


When L(n)ε in (5.1) is a local fractional differential operator of order nε, we denote

it as

L(n)ε = L(nε)

μ = ∂nε

∂μnε(5.30)

and

Rε� (μ) =∑n=0

an∂(n−1)ε

∂μ(n−1)ε. (5.31)

Defining the n-fold local fractional integral operator

L(−n)ε � (μ) = 0I(nε)

μ � (μ) , (5.32)

we conclude

L(−n)ε L(n)

ε � = L(−n)ε Rε�. (5.33)

Therefore, (5.33) implies

�(μ) = I (μ) + L(−n)ε Rε� (μ) , (5.34)

where I (μ) is to be determined from the fractal initial conditions.Hence, the iterative formula is expressed as

�n (μ) = L(−n)ε Rε�n (μ) , (5.35)

where �0 (μ) = I (μ).Thus, for n ≥ 0, the following recurrence formula reads{

�n (μ) = L(−n)ε Rε�n (μ) ,

�0 (μ) = I (μ) .(5.36)

Finally, we obtain the following result:

�(μ) = limn→∞

∞∑n=0

�n (μ). (5.37)

The next step is to consider the local fractional partial differential equation:

∂2ε�(μ, η)

∂η2ε= 2

∂2ε�(μ, η)

∂μ2ε− 1, (5.38)

subject to the following initial-boundary conditions

�(μ, 0) = Eε

(με

), 0 ≤ μ ≤ 1, (5.39)

∂ε�(μ, 0)

∂με= 0, 0 ≤ μ ≤ 1, (5.40)

�(l, η) = �(0, η) = 0, η > 0, (5.41)


∂ε�(l, η)

∂με= ∂ε�(0, η)

∂με= 0, η > 0. (5.42)

Next, we consider the iterative formula, namely,

�n+1 (μ, η) = L(−2ε)η

(2∂2ε�n (μ, η)

∂μ2α

), (5.43)

together with the initial value

�0 (μ, η) = Eε

(με

) + L(−2ε)η 1 = Eε

(με

) + η2ε

� (1 + 2ε). (5.44)

Using (5.43) and (5.44), the approximations read as

�1 (μ, η) = L(−2ε)η

(2∂2ε�0 (μ, η)

∂μ2α

)= 2η2ε

� (1 + 2ε)Eε

(με

), (5.45)

�2 (μ, η) = L(−2ε)η

(2∂2ε�1 (μ, η)

∂μ2α

)= 4η4ε

� (1 + 4ε)Eε

(με

), (5.46)

�3 (μ, η) = L(−2ε)η

(2∂2ε�2 (μ, η)

∂μ2α

)= 8η6ε

� (1 + 6ε)Eε

(με

), (5.47)

�4 (μ, η) = L(−2ε)η

(2∂2ε�3 (μ, η)

∂μ2α

)= 16η8ε

� (1 + 8ε)Eε

(με

), (5.48)

...

�n (μ, η) = 2nη2nε

� (1 + 2nε)Eε

(με

). (5.49)

Finally, the solution containing the nondifferentiable terms is given by

�(μ, η) = limn→∞

n∑n=0

�n (μ, η) = Eε

(με

) n∑n=0

2nη2nε

� (1 + 2nε)+ η2ε

� (1 + 2ε). (5.50)

Below, we consider the following local fractional partial differential equation:

∂2ε� (η, μ)

∂μ2ε+ ∂ε� (η, μ)

∂με+ �(η, μ) = ∂5ε� (η, μ)

∂η5ε, (5.51)

and its initial values read as

�(0, μ) = Eε(με), (5.52)

∂ε� (0, μ)

∂ηε= 0. (5.53)


The corresponding local fractional iteration algorithms become

�k+1 (η, μ) = L(−5ε)η

(∂2ε�k (η, μ)

∂μ2ε+ ∂ε�k (η, μ)

∂με+ �k (η, μ)

), k ≥ 0,

(5.54)

where

�0 (η, μ) = Eε(με). (5.55)

The components of the algorithm are given below:

�0 (η, μ) = Eε(με), (5.56)

�1 (η, μ) = L(−5ε)η

(∂2ε�0 (η, μ)

∂μ2ε+ ∂ε�0 (η, μ)

∂με+ �0 (η, μ)

)= L(−5ε)

η

(3Eε(μ

ε))

= 3η5ε

�(1 + 5ε)Eε(μ

ε), (5.57)

�2 (η, μ) = L(−5ε)η

(∂2ε�1 (η, μ)

∂μ2ε+ ∂ε�1 (η, μ)

∂με+ �1 (η, μ)

)= L(−5ε)

η

(6η5ε

�(1 + 5ε)Eε(μ

ε)

)= 6η10ε

�(1 + 10ε)Eε(μ

ε), (5.58)

�3 (η, μ) = L(−5ε)η

(∂2ε�2 (η, μ)

∂μ2ε+ ∂ε�2 (η, μ)

∂με+ �2 (η, μ)

)= L(−5ε)

η

(9η10ε

�(1 + 10ε)Eε(μ

ε)

)= 9η15ε

�(1 + 15ε)Eε(μ

ε), (5.59)

...

�n (η, μ) = 3nη3nε

�(1 + 3nε)Eε(μ

ε), (5.60)

and so on.Thus, the corresponding solution is given by

�(μ, η) = limn→∞

n∑n=0

�n (μ, η) = Eε

(με

) n∑n=0

3nη3nε

�(1 + 3nε). (5.61)


5.4 Coupling the Laplace transform with variationaliteration method of the local fractional operator

In this section, we consider the idea of the local fractional Laplace variational iterationmethod [113, 114], which is coupled by the variational iteration method and Laplacetransform of the local fractional operator.

Using the local fractional Laplace transform, we present the new iteration algo-rithm in the following form:

M {�n+1 (μ)} = M {�n (μ)} + (−1)k 1

skεM {Lε�n (μ) + Rε�n (μ)} , (5.62)

where the initial value condition is indicated as

s(k−1)ε� (0) + s(k−2)ε�(ε) (0) + · · · + �((k−1)ε) (0)

skε= 0. (5.63)

Therefore, we report that

M {�} = limn→∞ M {�n} (5.64)

such that

�n+1 (μ) = M−1 [M {�n+1 (μ)}]= M−1 [M {�n (μ)}] + (−1)k M−1

[1

skεM

{(L(n)

ε �n (μ) + Rε�n (μ))}]

.

(5.65)

Thus, we finally conclude

�(μ) = limn→∞ M−1 {M {�n (μ)}} . (5.66)

Next, we analyze the local fractional partial differential equation:

∂3ε� (η, μ)

∂μ3ε+ �(η, μ) = ∂2ε� (η, μ)

∂η2ε, (5.67)

subjected to the following initial value:

∂ε� (0, μ)

∂ηε= 0, (5.68)

�(0, μ) = Eε(−με). (5.69)

Below, we show the local fractional Laplace variational iteration algorithm, namely,

�n+1 (η, μ) = M−1 [M {�n+1 (η, μ)}]

= M−1 [M {�n (η, μ)}] + M−1[

1

s2εM

{∂2ε�n (η, μ)

∂η2ε

−�n (η, μ) − ∂3ε�n (η, μ)

∂μ3ε

}], (5.70)


where

�0 (η, μ) = Eε(με). (5.71)

The corresponding approximations are given by

�1 (η, μ) = M−1 [M {�1 (η, μ)}]

= M−1 [M {�0 (η, μ)}] + M−1[

1

s2εM

{∂2ε�0 (η, μ)

∂η2ε

−�0 (η, μ) − ∂3ε�0 (η, μ)

∂μ3ε

}]

= Eε(−με) + M−1[

1

s2εM

{∂3εEε(−με)

∂μ3ε+ Eε(−με) − ∂2εEε(−με)

∂η2ε

}]= Eε(−με) + M−1

[1

s2εM

{−Eε(−με) + Eε(−με) − 0}]

= Eε(−με), (5.72)

�2 (η, μ) = M−1 [M {�2 (η, μ)}]

= M−1 [M {�1 (η, μ)}] + M−1[

1

s2εM

{∂3ε�1 (η, μ)

∂μ3ε

+�1 (η, μ) − ∂2ε�1 (η, μ)

∂η2ε

}]= Eε(−με) + M−1

[1

s2εM

{−Eε(−με) + Eε(−με) − 0}]

= Eε(−με), (5.73)

�3 (η, μ) = M−1 [M {�3 (η, μ)}]

= M−1 [M {�2 (η, μ)}] + M−1[

1

s2εM

{∂3ε�2 (η, μ)

∂μ3ε

+�2 (η, μ) − ∂2ε�2 (η, μ)

∂η2ε

}]= Eε(−με) + M−1

[1

s2εM

{−Eε(−με) + Eε(−με) − 0}]

= Eε(−με), (5.74)

...

�n (η, μ) = M−1 [M {�n (η, μ)}]= Eε(−με). (5.75)


00.2

0.40.6

0.81

0

0.5

10.4

0.5

0.6

0.7

0.8

0.9

1

mh

Φ(h

,m)


As a result, we conclude that

�(η, μ) = limn→∞ M−1 {M {�n (η, μ)}} = Eε(−με). (5.76)

In Figure 5.3, we show the corresponding graph.Let us analyze now the local fractional partial differential equation:

∂3ε� (η, μ)

∂μ3ε− ∂2ε� (η, μ)

∂μ2ε= ∂3ε� (η, μ)

∂η3ε, (5.77)

equipped with the initial value

∂2ε� (0, μ)

∂η2ε= 0, (5.78)

∂ε� (0, μ)

∂ηε= Eε(μ

ε), (5.79)

�(0, μ) = 0. (5.80)

In this case, we calculate the local fractional Laplace variational iteration algorithm,namely,

�n+1 (η, μ) = M−1 [M {�n+1 (η, μ)}]

= M−1 [M {�n (η, μ)}] + M−1[

1

s3εM

{∂3ε�n (η, μ)

∂μ3ε

−∂2ε�n (η, μ)

∂μ2ε− ∂3ε�n (η, μ)

∂η3ε

}], (5.81)


where

�0 (η, μ) = ηε

� (1 + ε)Eε(μ

ε). (5.82)

Therefore, we obtain the following approximations:

�1 (η, μ) = M−1 [M {�1 (η, μ)}]

= M−1 [M {�0 (η, μ)}] + M−1[

1

s3εM

{∂3ε�0 (η, μ)

∂μ3ε

−∂2ε�0 (η, μ)

∂μ2ε− ∂3ε�0 (η, μ)

∂η3ε

}]= ηε

� (1 + ε)Eε(μ

ε), (5.83)

�2 (η, μ) = M−1 [M {�2 (η, μ)}]

= M−1 [M {�1 (η, μ)}] + M−1[

1

s3εM

{∂3ε�1 (η, μ)

∂μ3ε

−∂2ε�1 (η, μ)

∂μ2ε− ∂3ε�1 (η, μ)

∂η3ε

}]= ηε

� (1 + ε)Eε(μ

ε), (5.84)

�3 (η, μ) = M−1 [M {�3 (η, μ)}]

= M−1 [M {�2 (η, μ)}] + M−1[

1

s3εM

{∂3ε�2 (η, μ)

∂μ3ε

−∂2ε�2 (η, μ)

∂μ2ε− ∂3ε�2 (η, μ)

∂η3ε

}]= ηε

� (1 + ε)Eε(μ

ε), (5.85)

...

�n (η, μ) = M−1 [M {�n (η, μ)}]= ηε

� (1 + ε)Eε(μ

ε). (5.86)

At this point, we conclude that

�(η, μ) = limn→∞ M−1 {M {�n (η, μ)}} = ηε

� (1 + ε)Eε(μ

ε), (5.87)

and we depict the corresponding graph in Figure 5.4.


00.2

0.40.6

0.81

0

0.5

10

1

2

3

4

5

mh

Φ(h

,m)


5.5 Coupling the Laplace transform with decompositionmethod of the local fractional operator

In this section, we show the core of the local fractional Laplace decomposition method[88], which is coupled with the decomposition method and the Laplace transform ofthe local fractional operator.

Taking the local fractional Laplace transform in (5.1), we have

M{

L(n)ε �(μ, η)

}+ M {Rε�(μ, η)} = 0, (5.88)

such that

M {�(μ, η)} =n∑

k=1

1

skε�((k−1)ε)(0, η) − M {Rε�(μ, η)} − 1

skεM {�(μ, η)} .

(5.89)

Taking the inverse of the local fractional Laplace transform in (5.89), it gives

�(μ, η) = M−1

[n∑

k=1

1

skε�((k−1)ε)(0, η) − M {Rε�(μ, η)} − 1

skεM {�(μ, η)}

],

(5.90)


which leads to the local fractional recursive relation as

�n+1(μ, η) = M−1{− 1

skεM {�(μ, η)}

}, (5.91)

where

�0(μ, η) = M−1

[n∑

k=1

1

skε�((k−1)ε)(0, η) − M {Rε�(μ, η)}

]. (5.92)

Below, we investigate the local fractional partial differential equation

∂2ε�(η, μ)

∂μ2ε+ ∂ε�(η, μ)

∂με= ∂3ε�(η, μ)

∂η3ε(5.93)

subjected to the initial values

�(0, μ) = 0,∂ε�(0, μ)

∂ηε= Eε(μ

ε). (5.94)

In our case, the local fractional iteration algorithms are constructed as

�k+1 (η, μ) = M−1(

1

s3εM

{∂2ε�k (η, μ)

∂μ2ε+ ∂ε�k (η, μ)

∂με

}), k ≥ 0, (5.95)

where

�0 (η, μ) = ηε

�(1 + ε)Eε(μ

ε). (5.96)

Therefore, from (5.96), we present the components as

�1 (η, μ) = M−1(

1

s3εM

{∂2ε�0 (η, μ)

∂μ2ε+ ∂ε�0 (η, μ)

∂με

})= M−1

(2

s5εEε

(με

))= 2η4ε

�(1 + 4ε)Eε

(με

), (5.97)

�2 (η, μ) = M−1(

1

s3εM

{∂2ε�1 (η, μ)

∂μ2ε+ ∂ε�1 (η, μ)

∂με

})= M−1

(4

s8εEε

(με

))= 4η7ε

�(1 + 7ε)Eε

(με

), (5.98)


�3 (η, μ) = M−1(

1

s3εM

{∂2ε�2 (η, μ)

∂μ2ε+ ∂ε�2 (η, μ)

∂με

})= M−1

(8

s11εEε(μ

ε)

)= 8η10ε

�(1 + 10ε)Eε(μ

ε), (5.99)

and so on.As a result, there is

�(η, μ) = limn→∞

n∑n=0

M−1 {M {�n (η, μ)}}

=∞∑

n=0

2nη(1+3n)ε

�(1 + (1 + 3n) ε)Eε

(με

). (5.100)

The next step is to examine the local fractional partial differential equation

∂ε�(η, μ)

∂με+ �(η, μ) = ∂4ε�(η, μ)

∂η4ε, (5.101)

in the presence of the following initial values:

�(0, μ) = Eε(με). (5.102)

For this specific case, the local fractional iteration algorithms are written as

�k+1 (η, μ) = M−1(

1

s4εM

{∂ε�k (η, μ)

∂με+ �k (η, μ)

}), k ≥ 0, (5.103)

where

�0 (η, μ) = Eε(με). (5.104)

Hence, we evaluate the components, namely,

�1 (η, μ) = M−1(

1

s4εM

{∂ε�0 (η, μ)

∂με+ �0 (η, μ)

})= M−1

(2

s5εEε(μ

ε)

)= 2η4ε

�(1 + 4ε)Eε(μ

ε), (5.105)

�2 (η, μ) = M−1(

1

s4εM

{∂ε�1 (η, μ)

∂με+ �1 (η, μ)

})= M−1

(4

s9εEε(μ

ε)

)= 4η8ε

�(1 + 8ε)Eε(μ

ε), (5.106)


�3 (η, μ) = M−1(

1

s4εM

{∂ε�2 (η, μ)

∂με+ �2 (η, μ)

})= M−1

(8

s13εEε(μ

ε)

)= 8η12ε

�(1 + 12ε)Eε(μ

ε), (5.107)

and so on.Therefore, we report

�(η, μ) = limn→∞

n∑n=0

M−1 {M {�n (η, μ)}}

=∞∑

n=0

2nη(1+4n)ε

�(1 + (1 + 4n) ε)Eε

(με

). (5.108)

Now, we concentrate on solving the following local fractional partial differentialequation:

∂3ε� (η, μ)

∂μ3ε+ ∂2ε� (η, μ)

∂μ2ε= ∂2ε� (η, μ)

∂η2ε, (5.109)

subjected to the initial values

�(0, μ) = 0, (5.110)

∂ε� (0, μ)

∂ηε= 0, (5.111)

∂2ε� (0, μ)

∂η2ε= Eε(μ

ε). (5.112)

The local fractional iteration algorithms have the forms

�k+1 (η, μ) = M−1(

1

s2εM

{∂3ε�k (η, μ)

∂μ3ε+ ∂2ε�k (η, μ)

∂μ2ε

}), k ≥ 0,

(5.113)

where

�0 (η, μ) = η2ε

�(1 + 2ε)Eε(μ

ε). (5.114)

The components containing nondifferentiable terms are written as


�1 (η, μ) = M−1(

1

s2εM

{∂3ε�0 (η, μ)

∂μ3ε+ ∂2ε�0 (η, μ)

∂μ2ε

})= M−1

(2

s5εEε(μ

ε)

)= 2η4ε

�(1 + 4ε)Eε(μ

ε), (5.115)

�2 (η, μ) = M−1(

1

s2εM

{∂3ε�1 (η, μ)

∂μ3ε+ ∂2ε�1 (η, μ)

∂μ2ε

})= M−1

(4

s7εEε(μ

ε)

)= 4η6ε

�(1 + 6ε)Eε(μ

ε), (5.116)

�3 (η, μ) = M−1(

1

s2εM

{∂3ε�2 (η, μ)

∂μ3ε+ ∂2ε�2 (η, μ)

∂μ2ε

})= M−1

(6

s9εEε(μ

ε)

)= 6η8ε

�(1 + 8ε)Eε(μ

ε), (5.117)

and so on.Thus, we finally conclude

�(η, μ) = limn→∞

n∑n=0

M−1 {M {�n (η, μ)}}

=∞∑

n=0

2nη(2+2n)ε

�(1 + (2 + 2n) ε)Eε

(με

). (5.118)

Appendix AThe analogues of trigonometricfunctions defined on Cantor sets

In order to understand the first chapter, we present the analogues of the classicaltrigonometric functions now defined on Cantor sets. Here, we provide the proofs ofthe analogous trigonometric functions [1, 16, 21], namely,

sin2ε

(με

) = 1 − cosε (2μ)ε

2.

Proof.

sin2ε

(με

) =(

Eε (iεμε) − Eε (−iεμε)

2iε

)2

= (Eε (iεμε) − Eε (−iεμε))2

4i2ε

= E2ε (iεμε) + E2

ε (−iεμε) − 2

4i2ε

= Eε (iε (2μ)ε) + Eε (−iε (2μ)ε) − 2

4i2ε

= 1 − cosε (2μ)ε

2.

Thus, the proof is completed.

cos2ε

(με

) = 1 + cosε (2μ)ε

2.

198 Appendix A The analogues of trigonometric functions defined on Cantor sets

Proof.

cos2ε

(με

) =(

Eε (iεμε) + Eε (−iεμε)

2

)22

= (Eε (iεμε) + Eε (−iεμε))2

4i2ε

= E2ε (iεμε) + E2

ε (−iεμε) + 2

4

= Eε (iε (2μ)ε) + Eε (−iε (2μ)ε) + 2

4

= 1 + cosε (2μ)ε

2.

This completes the proofs.

cosε (2μ)ε = cos2ε

(με

) − sin2ε

(με

).

Proof.

cos2ε

(με

) − sin2ε

(με

) = 1 + cosε (2μ)ε

2− 1 − cosε (2μ)ε

2= cosε (2μ)ε .

Hence, the proof is completed.

cos2ε

(με

) + sin2ε

(με

) = 1.

Proof.

cos2ε

(με

) + sin2ε

(με

) = 1 + cosε (2μ)ε

2+ 1 − cosε (2μ)ε

2= 1.

The proof is completed.

cosε

(με

)cosε

(ηε

) = cosε (με + ηε) + cosε (με − ηε)

2.

Appendix A The analogues of trigonometric functions defined on Cantor sets 199

Proof.

cosε

(με

)cosε

(ηε

) =(


2

) (Eε (iεηε) + Eε (−iεηε)

2

)

= Eε (iεμε + iεηε) + Eε (iεμε − iεηε) + Eε (iεηε − iεμε) + Eε (−iεμε − iεηε)

4

= cosε (με + ηε) + iε sinε (με + ηε)

4+ cosε (με − ηε) + iε sinε (με − ηε)

4

+ cosε (με − ηε) − iε sinε (με − ηε)

4+ cosε (με + ηε) − iε sinε (με + ηε)

4

= cosε (με + ηε) + cosε (με − ηε)

2

that completes the proof.

sinε

(με

)sinε

(ηε

) = −cosε (με + ηε) − cosε (με − ηε)

2.

Proof.

sinε

(με

)sinε

(ηε

) =(


2iε

) (Eε (iεηε) − Eε (−iεηε)

2iε

)

= − Eε (iεμε + iεηε)− Eε (iεηε − iεμε)− Eε (iεμε− iεηε) + Eε (−iεμε− iεηε)

4

= − cosε (με + ηε) + iε sinε (με + ηε)

4+ cosε (ηε − με) + iε sinε (ηε − με)

4

+ cosε (με − ηε) + iε sinε (με − ηε)

4− cosε (με + ηε) − iε sinε (με + ηε)

4

= − cosε (με + ηε) + iε sinε (με + ηε)

4+ cosε (με − ηε) − iε sinε (με − ηε)

4

+ cosε (με − ηε) + iε sinε (με − ηε)

4− cosε (με + ηε) − iε sinε (με + ηε)

4

= − cosε (με + ηε) − cosε (με − ηε)

2.


sinε

(με

)cosε

(ηε

) = sinε (με + ηε) + sinε (με − ηε)

2.


Proof.

sinε

(με

)cosε

(ηε

) =(


2iε

)(Eε (iεηε) + Eε (−iεηε)

2

)

= Eε (iεμε + iεηε) + Eε (iεμε− iεηε) − Eε (−iεμε+ iεηε) − Eε (−iεμε− iεηε)

4iε


4iε+ cosε (με − ηε) + iε sinε (με − ηε)

4iε

− cosε (με − ηε) − iε sinε (με − ηε)

4iε− cosε (με + ηε) − iε sinε (με + ηε)

4iε

= sinε (με + ηε) + sinε (με − ηε)

2.

Hence, it completes the proof.

cosε

(με

)sinε

(ηε

) = sinε (με + ηε) − sinε (με − ηε)

2.

Proof.

cosε

(με

)sinε

(ηε

) =(


2

)(Eε (iεηε) − Eε (−iεηε)

2iε

)

= Eε (iεμε + iεηε) + Eε (iεηε − iεμε) − Eε (iεμε − iεηε) − Eε (−iεμε − iεηε)

4iε


4iε+ cosε (με − ηε) − iε sinε (με − ηε)

4iε

− cosε (με − ηε) + iε sinε (με − ηε)

4iε− cosε (με + ηε) − iε sinε (με + ηε)

4iε

= sinε (με + ηε) − sinε (με − ηε)

2.


sinε

(με + ηε

) = sinε

(με

)cosε

(ηε

) + cosε

(με

)sinε

(ηε

).

Proof.

sinε

(με + ηε

) = sinε (με + ηε) − sinε (με − ηε)

2

+ sinε (με + ηε) + sinε (με − ηε)

2= sinε

(με

)cosε

(ηε

) + cosε

(με

)sinε

(ηε

).


sinε

(με − ηε

) = sinε

(με

)cosε

(ηε

) + cosε

(με

)sinε

(ηε

).


Proof.

sinε

(με − ηε

) = sinε (με + ηε) + sinε (με − ηε)

2

− sinε (με + ηε) − sinε (με − ηε)

2= sinε

(με

)cosε

(ηε

) − cosε

(με

)sinε

(ηε

).

Hence, completing the proof, we get

cosε

(με + ηε

) = cosε

(με

)cosε

(ηε

) − sinε

(με

)sinε

(ηε

).

Proof.

cosε

(με + ηε


2

+ cosε (με + ηε) − cosε (με − ηε)

2= cosε

(με

)cosε

(ηε

) − sinε

(με

)sinε

(ηε

)and this proof is completed.

cosε

(με − ηε

) = cosε

(με

)cosε

(ηε

) + sinε

(με

)sinε

(ηε

).

Proof.

cosε

(με − ηε


2

− cosε (με + ηε) − cosε (με − ηε)

2= cosε

(με

)cosε

(ηε

) + sinε

(με

)sinε

(ηε

).

This proof is completed.

sinε

(με

) + sinε

(με

) = 2 sinε

((μ + η

2

)ε)cosε

((μ − η

2

)ε)

Proof.

2sinε

[(μ + η

2

)ε]cosε

[(μ − η

2

)ε]= sinε

[(μ + η

2

)ε

+(

μ − η

2

)ε]

+ sinε

[(μ + η

2

)ε

−(

μ − η

2

)ε]= sinε

(με

) + sinε

(ηε

)This proof is completed.


sinε

(με

) − sinε

(με

) = 2 cosε

[(μ + η

2

)ε]sinε

[(μ − η

2

)ε].

Proof.

2 cosε

[(μ + η

2

)ε]sinε

[(μ − η

2

)ε]= sinε

[(μ + η

2

)ε

+(

μ − η

2

)ε]

− sinε

[(μ + η

2

)ε

−(

μ − η

2

)ε]= sinε

(με

) + sinε

(ηε

).


cosε

(με

) + cosε

(με

) = 2 cosε

((μ + η

2

)ε)cosε

((μ − η

2

)ε).

Proof.

2 cosε

((μ + η

2

)ε)cosε

((μ − η

2

)ε)= cosε

[(μ + η

2

)ε

+(

μ − η

2

)ε]

+ cosε

[(μ + η

2

)ε

−(

μ − η

2

)ε]= cosε

(με

) + cosε

(ηε

).


cosε

(με

) − cosε

(με

) = −2 sinε

[(μ + η

2

)ε]sinε

[(μ − η

2

)ε].

Proof.

−2 sinε

[(μ + η

2

)ε]sinε

[(μ − η

2

)ε]= cosε (με + ηε) − cosε (με − ηε)

2

= cosε

[(μ + η

2

)ε

+(

μ − η

2

)ε]

− cosε

[(μ + η

2

)ε

−(

μ − η

2

)ε]= cosε

(με

) − cosε

(με

).


[Eε

(iεμε

)]k = cosε

[(kμ)ε

] + iε sinε

[(kμ)ε

].


Proof. Let k = 0; it is right.Let k = 1; it is right.When k = n, we suppose that[

Eε

(iεμε

)]k = (cosε

[(kμ)ε

] + iε sinε

[(kμ)ε

]).

When k = n + 1, we present that[Eε

(iεμε

)]k+1 = (cosε

[(kμ)ε

] + iε sinε

[(kμ)ε

]) (cosε

(με

) + iε sinε

(με

))= cosε

[(kμ)ε

]cosε

(με

) + iε cosε

[(kμ)ε

]sinε

(με

)+ iε sinε

[(kμ)ε

]cosε

(με

) − sinε

[(kμ)ε

]sinε

(με

)= cosε

[(kμ)ε + με

] + cosε

[(kμ)ε − με

]2

+ iεsinε

[(kμ)ε + με

] − sinε

[(kμ)ε − με

]2

+ iεsinε

[(kμ)ε + με

] + sinε

[(kμ)ε − με

]2

+ cosε

[(kμ)ε + με

] − cosε

[(kμ)ε − με

]2

= cosε

[(kμ)ε + με

] + iε sinε

[(kμ)ε + με

]= cosε

[(k + 1)ε με

] + iε sinε

[(k + 1)ε με

].

Hence, we completed the proof.

In this case, we use the following formula:[Eε

(iεμε

)]k = (cosε

[(kμ)ε

] + iε sinε

[(kμ)ε

]) = Eε

[iε (kμ)ε

].

Let us define

θε =k∑

n=1

cosε (nμ)ε

and

ϑε =k∑

n=1

sinε (nμ)ε.

Then, we may structure a function

� = θε + iεϑε

=k∑

n=1

Eε

[iε (nμ)ε

].


This allows to obtain

Eε

(iεμε

)� = Eε

(iεμε

) (θε + iεϑε

)= Eε

(iεμε

) {k∑

n=1

Eε

[iε (nμ)ε

]}

=k+1∑n=2

Eε

[iε (nμ)ε

].

Therefore, we have

�(Eε

(iεμε

) − 1) = Eε

[iε ((k + 1) μ)ε

] − Eε

(iεμε

),

which leads to

� = θε + iεϑε

= Eε

[iε ((k + 1) μ)ε

] − Eε (iεμε)

(Eε (iεμε) − 1)

= Eε

(−iε(

μ2

)ε)Eε

[iε ((k + 1) μ)ε

] − Eε (iεμε)

Eε

(−iε(

μ2

)ε)(Eε (iεμε) − 1)

=Eε

[iε

((k + 1

2

)μ

)ε] − Eε

(iε

(μ2

)ε)Eε

[iε

(μ2

)ε] − Eε

[−iε(

μ2

)ε]

= Eε

[iε

((k + 1

2

)μ

)ε] ⎧⎪⎨⎪⎩

Eε

[iε

(kμ2

)ε] − Eε

(−iε

(kμ2

)ε)Eε

[iε

(μ2

)ε] − Eε

[−iε(

μ2

)ε]⎫⎪⎬⎪⎭

= Eε

[iε

((k + 1

2

)μ

)ε] ⎧⎪⎨⎪⎩

Eε

[iε

(kμ2

)ε]−Eε

(−iε

(kμ2

)ε)2iε

Eε[iε( μ2 )

ε]−Eε[−iε( μ2 )

ε]2iε

⎫⎪⎬⎪⎭

= Eε

[iε

((k + 1

2

)μ

)ε] ⎧⎪⎨⎪⎩

sinε

[(kμ2

)ε]sinε

[(μ2

)ε]⎫⎪⎬⎪⎭

=

⎧⎪⎨⎪⎩

sinε

[(kμ2

)ε]sinε

[(μ2

)ε]⎫⎪⎬⎪⎭

{cosε

[((k + 1

2

)μ

)ε]+ iε sinε

[((k + 1

2

)μ

)ε]}.

Consequently, we obtain

θε =k∑

n=1

cosε (nμ)ε =

⎧⎪⎨⎪⎩

sinε

[(kμ2

)ε]sinε

[(μ2

)ε]⎫⎪⎬⎪⎭ cosε

[((k + 1

2

)μ

)ε]


and

ϑε =k∑

n=1

sinε (nμ)ε =

⎧⎪⎨⎪⎩

sinε

[(kμ2

)ε]sinε

[(μ2

)ε]⎫⎪⎬⎪⎭ sinε

[((k + 1

2

)μ

)ε],

where

sinε

[(μ

2

)ε] �= 0.

Now, we can present the following formula as

1

2+

k∑n=1

cosε (nμ)ε = 1

2+

⎧⎪⎨⎪⎩

sinε

[(kμ2

)ε]sinε

[(μ2

)ε]⎫⎪⎬⎪⎭ cosε

[((k + 1

2

)μ

)ε]

=sinε

[(μ2

)ε] + 2 sinε

[(kμ2

)ε]cosε

[((k+1

2

)μ

)ε]2 sinε

[(μ2

)ε]

=sinε

[(μ2

)ε] + sinε

[(kμ2

)ε]cosε

[((k+1

2

)μ

)ε]2 sinε

[(μ2

)ε]

=sinε

[((k + 1

2

)μ

)ε]2 sinε

[(μ2

)ε]

Dk,ε (t) = 1

2+

k∑n=1

cosε (nμ)ε =sinε

[((k + 1

2

)μ

)ε]2 sinε

[(μ2

)ε] .

Let us define the tangent function defined on Cantor sets, namely,

tanε

(με

) = sinε (με)

cosε (με).

Similarly, the cotangent function defined on Cantor sets is

cotε(με

) = cosε (με)

sinε (με),

where μ ∈ R and 0 < ε ≤ 1

tanε

(με

) = sinε

[(2μ)ε

]1 + cosε

[(2μ)ε

] = 1 − cosε

[(2μ)ε

]sinε

[(2μ)ε

] .


Proof.

tanε

(με

) = 2 sinε (με) cosε (με)

2 cosε (με) cosε (με)

= sinε

[(2μ)ε

]1 + cosε

[(2μ)ε

]and consequently, one obtains

tanε

(με

) = 2 sinε (με) sinε (με)

2 sinε (με) cosε (με)

= 1 − cosε

[(2μ)ε

]sinε

[(2μ)ε

] .

Appendix BLocal fractional derivativesof elementary functions

Consider the function

Eε

(Cμε

) =∞∑

k=0

Ckμkε

� (1 + kε).

Then, we have

dε

dμεEε

(Cμε

) = dε

dμε

( ∞∑k=0

Ckμkε

� (1 + kε)

)

=∞∑

k=1

Ckμ(k−1)ε

� (1 + (k − 1) ε)

= C∞∑

k=1

C(k−1)μ(k−1)ε

� (1 + (k − 1) ε).

Hence, we get

dε

dμεEε

(Cμε

) = CEε

(Cμε

).

Further, when C = −1, we have

dε

dμεEε

(−με) = −Eε

(−με)

.

Using the chain rule, one obtains

dε

dμε

[Eε

(μ2ε

)]= dεEε

(μ2ε

)d(μ2)ε

[dμ2

dμ

]ε

= (2μ)ε Eε

(μ2ε

).

In a similar manner, we have

dε

dμεEε

(Cμ2ε

)= (2μ)ε CEε

(Cμ2ε

)

208 Appendix B Local fractional derivatives of elementary functions

and

dε

dμεEε

(−μ2ε

)= − (2μ)ε Eε

(−μ2ε

)

dε

dμεsinε

(με) = dε

dμε

[Eε (iεμε) − Eε (−iεμε)

2iε

]

=[

iεEε (iεμε) + iεEε (−iεμε)

2iε

]

= Eε (iεμε) + Eε (−iεμε)

2= cosε

(με)

.

Therefore, applying the same style of calculations, we get the following formulas:

dε

dμεsinε

(Cμε

) = C cosε

(με)

;

dε

dμεcosε

(με) = − sinε

(με)

;

dε

dμεcosε

(Cμε

) = −C sinε

(Cμε

);

dε

dμεsinhε

(με) = dε

dμε

[Eε (με) − Eε (−με)

2

]

= Eε (με) + Eε (−με)

2= coshε

(με)

;

dε

dμεcoshε

(με) = dε

dμε

[Eε (με) + Eε (−με)

2

]

= Eε (με) − Eε (−με)

2= sinhε

(με)

.

Similarly, we may obtain that

dε

dμεsinhε

(Cμε

) = dε

dμε

[Eε (Cμε) − Eε (−Cμε)

2

]

= CEε (Cμε) + CEε (−Cμε)

2= C coshε

(Cμε

)

Appendix B Local fractional derivatives of elementary functions 209

dε

dμεcoshε

(Cμε

) = dε

dμε

[Eε (Cμε) + Eε (−Cμε)

2

]

= Eε (Cμε) − CEε (−Cμε)

2= C sinhε

(Cμε

).

Appendix CLocal fractional Maclaurin’s seriesof elementary functions

In this appendix, we will present the local fractional Maclaurin’s series ofnondifferentiable elementary functions.

For a given nondifferentiable function ϕ (μ), the local fractional Maclaurin’s seriestakes the form

ϕ (μ) =∞∑

k=0

D(kε)ϕ (0)

� (1 + kε)μkε.

Moreover, the local fractional Maclaurin polynomial can be presented as

Tn [ϕ (μ)] =n∑

k=0

D(kε)ϕ (0)

� (1 + kε)μkε.

Further, using the relation D(kε)ϕ (0) = 1, where ϕ (μ) = Eε (με), one obtains

Eε

(με

) =∞∑

k=0

μkε

� (1 + kε).

Similarly, we may obtain that

D(kε)ϕ (0) = (−1)k ,

with ϕ (μ) = Eε (−με) and k ∈ R0, such that

Eε

(−με) =

∞∑k=0

(−1)k μkε

� (1 + kε).

Next, we have

D(kε)ϕ (0) ={

(−1)k , 2k + 10, 2k,

where ϕ (μ) = sinε (με) and k ∈ R0, such that

sinε

(με

) =∞∑

k=0

(−1)k μ(2k+1)ε

� (1 + (2k + 1) ε).

212 Appendix C Local fractional Maclaurin’s series of elementary functions

In the same way, we have

D(kε)ϕ (0) ={

(−1)k , 2k0, 2k + 1,

where ϕ (μ) = cosε (με) and k ∈ R0, such that

cosε

(με

) =∞∑

k=0

(−1)k μ2kε

� (1 + 2kε).

In addition,

D(kε)ϕ (0) ={

1, 2k + 10, 2k,

where ϕ (μ) = sinhε (με) and k ∈ R0, such that

sinhε

(με

) =∞∑

k=0

μ(2k+1)ε

� (1 + (2k + 1) ε).

Similarly,

D(kε)ϕ (0) ={

1, 2k0, 2k + 1,

where ϕ (μ) = coshε (με) and k ∈ R0, such that

coshε

(με

) =∞∑

k=0

μ2kε

� (1 + 2kε).

In the above formulas, we notice that R0 = R ∪ 0.

Appendix DCoordinate systems ofCantor-type cylindrical andCantor-type spherical coordinates

Let us consider the coordinate system of the Cantor-type cylindrical coordinates

r = Rε cosε(θε

)eε1 + Rε sinε

(θε

)eε2 + σεeε3

= rReεR + rθeεθ + rσ eεσ ,

where⎧⎨⎩με = Rε cosε(θε),ηε = Rε sinε(θε),σε = σε,

with R ∈ (0, +∞), z ∈ (−∞, +∞), θ ∈ (0,π ], and μ2ε + η2ε = R2ε.We have⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

HεR = 1

� (1 + ε)

∂εr∂Rε

= cosε (θε) eε1 + sinε (θε) eε2,

Hεθ = 1

� (1 + ε)

∂εr∂θε

= − Rε�(1+ε) sinε (θε) eε1 + Rε

�(1+ε) cosε (θε) eε2,

Hε3 = 1

� (1 + ε)

∂εr∂σ ε

= eε3,

such that⎧⎨⎩

eεR = cosε (θε) eε1 + sinε (θε) eε2,eεθ = − sinε (θε) eε1 + cosε (θε) eε2,eεσ = eε3,

where⎧⎨⎩

HεR = eεR,Hεθ = Rε

�(1+α)eεθ ,

Hε3 = eεσ .

We have⎛⎝ eεR

eεθeεε

⎞⎠ =

⎛⎝ cosε (θε) sinε (θε) 0

− sinε (θε) cosε (θε) 00 0 1

⎞⎠

⎛⎝ eε1

eε2eε3

⎞⎠

214 Appendix D Coordinate systems of Cantor-type cylindrical and Cantor-type spherical coordinates

or

Wεi = GεijW

εj ,

where

Wεi =

⎛⎝ eεR

eεθeεσ

⎞⎠ ,

Gεij =⎛⎝ cosε (θε) sinε (θε) 0

− sinε (θε) cosε (θε) 00 0 1

⎞⎠ ,

Wεj =

⎛⎝ eε1

eε2eε3.

⎞⎠ .

In this case, we have that

∂ε

∂Rε=

(∂μ

∂R

)ε∂ε

∂με+

(∂η

∂R

)ε∂ε

∂ηε+

(∂σ

∂R

)ε∂ε

∂σ ε= eεR · ∇ε = ∇ε

R,

∂ε

∂θε=

(∂μ

∂θ

)ε∂ε

∂με+

(∂η

∂θ

)ε∂ε

∂ηε+

(∂σ

∂θ

)ε∂ε

∂σ ε= Rεeεθ · ∇ε = Rε∇ε

θ ,

∂ε

∂σ ε=

(∂μ

∂σ

)ε∂ε

∂με+

(∂η

∂σ

)ε∂ε

∂ηε+

(∂σ

∂σ

)ε∂ε

∂σ ε= eεσ · ∇ε = ∇ε

σ ,

where⎧⎨⎩

∇εR = eεR · ∇ε = ∂ε

∂Rε ,∇εθ = Rεeεθ · ∇ε = 1

Rε∂ε

∂θε,

∇εσ = eεσ · ∇ε = ∂ε

∂σ ε.

The local fractional gradient operator in the Cantor-type cylindrical coordinatesreads as

∇ε = eεR∇εR + eεθ∇ε

θ + eεσ∇εσ = eεR

∂ε

∂Rε+ eεθ

1

Rε∂ε

∂θε+ eεσ

∂ε

∂σ ε.

There is

∇ε · r = eεR · ∂εr∂Rε

+ eεθ · 1

Rε∂εr∂θε

+ eεσ · ∂εr∂σ ε

,

where

eεR · ∂εr∂Rε

= eεR ·(∂εrR

∂RεeεR + ∂εrθ

∂Rεeεθ + ∂εrσ

∂Rεeεσ + rR

∂εeεR∂Rε

+ rθ∂εeεθ∂Rε

+ rσ∂εeεσ∂Rε

)

= eεR ·(∂εrR



∂Rεeεσ + 0 + 0 + 0

)

Appendix D Coordinate systems of Cantor-type cylindrical and Cantor-type spherical coordinates 215

= eεR ·(∂εrR



∂Rεeεσ

)

= ∂εrR

∂Rε,

eεθ · 1

Rε∂εr∂θε

= eεθ · 1

Rε

(∂εrR

∂θεeεR + ∂εrθ

∂θεeεθ + ∂εrσ

∂θεeεσ + rR

∂εeεR∂θε

+ rθ∂εeεθ∂θε

+ rσ∂εeεσ∂θε

)

= eεθ · 1

Rε

(∂εrR



∂θεeεσ + rReεθ − rθ eεR + 0

)

= eεθ · 1

Rε

(∂εrR



∂θεeεσ + rReεθ − rθ eεR

)

= 1

Rε∂εrθ∂θε

+ rR

Rε,

eεσ · ∂εr∂σ ε

= eεσ ·(∂εrR

∂σ εeεR + ∂εrθ

∂σ εeεθ + ∂εrz

∂σ εeεσ + rR

∂εeεR∂σ ε

+ ∂εeεθ∂σ ε

rθ + ∂εeεσ∂σ ε

rσ

)

= eεσ ·(∂εrR



∂σ εeεσ + 0 + 0 + 0

)

= eεσ ·(∂εrR



∂σ εeεσ

)

= ∂εrz

∂σ ε.

Hence, we get the local fractional divergence operator in the Cantor-type cylindri-cal coordinates

∇ε · r = ∂εrR

∂Rε+ 1

Rε∂εrθ∂θε

+ rR

Rε+ ∂εrz

∂σ ε.

The local fractional curl operator in the Cantor-type cylindrical coordinates ispresented as follows:

∇ε × r =(

eεR∂ε

∂Rε+ eεθ

1

Rε∂ε

∂θε+ eεσ

∂ε

∂σ ε

)× (

eεRrR + eεθrθ + eεσ rσ)

= eεR × ∂εr∂Rε

+ eεθ × 1

Rε∂εr∂θε

+ eεσ × ∂εr∂σ ε

,

where

eεR × ∂εr∂Rε

= eεR ×(∂εrR



∂Rεeεσ + rR

∂εeεR∂Rε


+ rσ∂εeεσ∂Rε

)

= eεR ×(∂εrR



∂Rεeεσ + 0 + 0 + 0

)


= eεR ×(∂εrR



∂Rεeεσ

)

= ∂εrθ∂Rε

eεσ − ∂εrσ∂Rε

eεθ ,

eεθ × 1

Rε∂εr∂θε

= eεθ × 1

Rε

(∂εrR



∂θεeεσ + rR

∂εeεR∂θε


+ rσ∂εeεσ∂θε

)

= eεθ × 1

Rε

(∂εrR



∂θεeεσ + rReεθ − rθ eεR + 0

)

= −eεσ1

Rε∂εrR

∂θε+ eεR

1

Rε∂εrσ∂θε

+ eεσrθRε

,

eεσ × ∂εr∂σ ε

= eεσ ×(∂εrR


∂σ εeεθ + ∂εrσ

∂σ εeεσ + rR

∂εeεR∂σ ε

+ rθ∂εeεθ∂σ ε

+ rσ∂εeεσ∂σ ε

)

= eεσ ×(∂εrR


∂σ εeεθ + ∂εrσ

∂σ εeεσ + 0 + 0 + 0

)

= eεθ∂εrR

∂σ ε− eεR

∂εrθ∂σ ε

.

Thus, we obtain the local fractional curl operator in the Cantor-type cylindricalcoordinates:

∇ε × r =(

1

Rε∂εrσ∂θε

− ∂εrθ∂σ ε

)eεR +

(∂εrR

∂σ ε− ∂εrσ∂Rε

)eεθ +

(∂εrθ∂Rε

+ rθRε

− 1

Rε∂εrR

∂θε

)eεσ .

We get the local fractional Laplace operator in the Cantor-type cylindricalcoordinates:

∇2εψ (R, θ , σ) =(

eεR∂ε

∂Rε+ eεθ

1

Rε∂ε

∂θε+ eεσ

∂ε

∂σ ε

)·(

eεR∂εψ

∂Rε+ eεθ

1

Rε∂εψ

∂θε+ eεσ

∂εψ

∂σ ε

)

= eεR · ∂ε

∂Rε

(eεR∂εψ

∂Rε+ eεθ

1

Rε∂εψ

∂θε+ eεσ

∂εψ

∂σ ε

)

+ eεθ · 1

Rε∂ε

∂θε

(eεR∂εψ

∂Rε+ eεθ

1

Rε∂εψ

∂θε+ eεσ

∂εψ

∂σ ε

)

+ eεσ · ∂ε

∂σ ε

(eεR∂εψ

∂Rε+ eεθ

1

Rε∂εψ

∂θε+ eεσ

∂εψ

∂σ ε

),

where

eεR · ∂ε

∂Rε

(eεR∂εψ

∂Rε+ eεθ

1

Rε∂εψ

∂θε+ eεσ

∂εψ

∂σ ε

)

= eεR ·(

eεR∂2εψ

∂R2ε + eεθ∂εφ

∂Rε

(1

Rε∂εψ

∂θε

)+ eεσ

∂2εψ

∂Rε∂σ ε

)

= ∂2εψ

∂R2ε ,


eεθ · 1

Rε∂ε

∂θε

(eεR∂εψ

∂Rε+ eεθ

1

Rε∂εψ

∂θε+ eεσ

∂εψ

∂σ ε

)= 1

Rε∂ε

∂θε

(1

Rε∂εψ

∂θε

)+ 1

Rε∂εψ

∂Rε

= 1

R2ε

∂2εψ

∂θ2ε + 1

Rε∂εψ

∂Rε,

eεσ · ∂ε

∂σ ε

(eεR∂εψ

∂Rε+ eεθ

1

Rε∂εψ

∂θε+ eεσ

∂εψ

∂σ ε

)

= eεσ ·(

eεR∂2εψ

∂σ ε∂Rε+ eεθ

∂ε

∂σ ε

(1

Rε∂εψ

∂θε

)+ eεσ

∂2εψ

∂σ 2ε

)

= ∂2εψ

∂σ 2ε.

Hence, the local fractional Laplace operator in the Cantor-type cylindrical coordi-nates can be written in the form

∇2εψ (R, θ , σ) = ∂2εψ

∂R2ε + 1

R2ε

∂2εψ

∂θ2ε + 1

Rε∂εψ

∂Rε+ ∂2εψ

∂σ 2ε .

We consider the coordinate system of the Cantor-type spherical coordinates

r = Rε cosε(θε

)sinε

(ϑε

)eε1 + Rε sinε

(θε

)sinε

(ϑε

)eε2 + Rε cosε

(ϑε

)eε3

= rReεR + rϑeεϑ + rθeεθ ,

where⎧⎨⎩με = Rε cosε (θε) sinε (ϑε) ,ηε = Rε sinε (θε) sinε (ϑε) ,σε = Rε cosε (ϑε) ,

with R ∈ (0, +∞), ϑ ∈ (0,π), θ ∈ (0, 2π), and μ2ε + η2ε + σ 2ε = R2ε.We have⎧⎨

⎩eεR = sinε (ϑε) cosε (θε) eε1 + sinε (ϑε) sinε (θε) eε2 + cosε (ϑε) eε3,eεϑ = cosε (ϑε) cosε (θε) eε1 + cosε (ϑε) sinε (θε) eε2 − sinε (ϑε) eε3,eεθ = − sinε (θε) eε1 + cosε (θε) eε2.

We rewrite (5.5) as⎛⎝ eεR

eεϑeεθ

⎞⎠ =

⎛⎝ sinε (ϑε) cosε (θε) sinε (ϑε) sinε (θε) cosε (ϑε)

cosε (ϑε) cosε (θε) cosε (ϑε) sinε (θε) − sinε (ϑε)− sinε (θε) cosε (θε) 0

⎞⎠

⎛⎝ eε1

eε2eε3

⎞⎠

or

Sεi = DεijSεj ,

where


Sεi =⎛⎝ eεR

eεθeεσ

⎞⎠ ,

Dεij =⎛⎝ sinε (ϑε) cosε (θε) sinε (ϑε) sinε (θε) cosε (ϑε)

cosε (ϑε) cosε (θε) cosε (ϑε) sinε (θε) − sinε (ϑε)− sinε (θε) cosε (θε) 0

⎞⎠ ,

Sεj =⎛⎝ eε1

eε2eε3.

⎞⎠ .

In this case, we present

∂ε

∂Rε=

(∂μ

∂R

)ε∂ε

∂με+

(∂η

∂R

)ε∂ε

∂ηε+

(∂σ

∂R

)ε∂ε

∂σ ε

= eεR · ∇ε

= ∇εR,

∂ε

∂ϑε=

(∂μ

∂ϑ

)ε∂ε

∂με+

(∂η

∂ϑ

)ε∂ε

∂ηε+

(∂σ

∂ϑ

)ε∂ε

∂σ ε

= Rεeεϑ · ∇ε

= Rε∇εϑ ,

∂ε

∂θε=

(∂μ

∂θ

)ε∂ε

∂με+

(∂η

∂θ

)ε∂ε

∂ηε+

(∂σ

∂θ

)ε∂ε

∂σ ε

= 1

Rε sinε (ϑε)eεθ · ∇ε

= 1

Rε sinε (ϑε)∇εθ ,

where⎧⎪⎨⎪⎩

∇εR = eεR · ∇ε = ∂ε

∂Rε ,∇εϑ = Rεeεϑ · ∇ε = 1

Rε∂ε

∂ϑε,

∇εθ = Rε sinε (ϑε) eεθ · ∇ε = 1

Rε sinε(ϑε)∂ε

∂θε.

The local fractional gradient operator in the Cantor-type spherical coordinates iswritten as follows:

∇εψ = eεR∇εRψ + eεϑ∇ε

ϑψ + eεθ∇εθψ

= eεR∂εψ

∂Rε+ eεϑ

1

Rε∂εψ

∂ϑε+ eεθ

1

Rε sinε (ϑε)

∂εψ

∂θε.


We obtain the local fractional divergence operator in the Cantor-type sphericalcoordinates

∇ε · r = eεR · ∂εr∂Rε

+ eεϑ · 1

Rε∂εr∂ϑε

+ eεθ · 1

Rε sinε (ϑε)

∂εr∂θε

,

where

eεR · ∂εr∂Rε

= eεR · ∂ε

∂Rε(rReεR + rϑeεϑ + rθeεθ

)

= eεR ·(

eεR∂εrR

∂Rε+ eεϑ

∂εrϑ∂Rε

+ eεθ∂εrθ∂Rε

+ rR∂εeεR∂Rε

+ rϑ∂εeεϑ∂Rε


)

= ∂εrR

∂Rε,

eεϑ · 1

Rε∂εr∂ϑε

= eεϑ · 1

Rε∂ε

∂ϑε

(rReεR + rϑeεϑ + rθ eεθ

)

= eεϑ · 1

Rε

(eεR∂εrR

∂ϑε+ eεϑ

∂εrϑ∂ϑε

+ eεθ∂εrθ∂ϑε

+ rR∂εeεR∂ϑε

+ rϑ∂εeεϑ∂ϑε

+ rθ∂εeεθ∂ϑε

)

= 1

Rε∂εrϑ∂ϑε

+ rR

Rε,

eεθ · 1

Rε sinε (ϑε)

∂εr∂θε

= eεθ · 1

Rε sinε (ϑε)

∂ε

∂θε

(rReεR + rϑeεϑ + rθeεθ

)

= eεθ · 1

Rε sinε (ϑε)

(eεR∂εrR

∂θε+ eεϑ

∂εrϑ∂θε

+ eεθ∂εrθ∂θε

+ rR∂εeεR∂θε

+ rϑ∂εeεϑ∂θε


)

= 1

Rε sinε (ϑε)

(∂εrθ∂θε

+ rϑ cosε(ϑε

)) + rθRε

.

Thus, we rewrite the local fractional gradient operator in the Cantor-type sphericalcoordinates

∇ε · r = ∂εrR

∂Rε+ 2rR

Rε+ 1

Rε∂εrϑ∂ϑε

+ 1

Rε sinε (ϑε)

(∂εrθ∂θε

+ rϑ cosε(ϑε

)).

The local fractional curl operator in the Cantor-type spherical coordinates takesthe form

∇ε × r = eεR × ∂εr∂Rε

+ eεϑ × 1

Rε∂εr∂ϑε

+ eεθ × 1

Rε sinε (ϑε)

∂εr∂θε

,

where

eεR × ∂εr∂Rε

= eεR × ∂ε

∂Rε(rReεR + rϑeεϑ + rθeεθ

)


= eεR ×(

eεR∂εrR

∂Rε+ eεϑ

∂εrϑ∂Rε

+ eεθ∂εrθ∂Rε

+ rR∂εeεR∂Rε

+ rϑ∂εeεϑ∂Rε


)

= eεθ∂εrR

∂Rε− eεϑ

∂εrθ∂Rε

,

eεϑ × 1

Rε∂εr∂ϑε

= eεϑ × 1

Rε∂ε

∂ϑε


)

= eεϑ × 1

Rε

(eεR∂εrR

∂ϑε+ eεϑ

∂εrϑ∂ϑε

+ eεθ∂εrθ∂ϑε

+ rR∂εeεR∂ϑε

+ rϑ∂εeεϑ∂ϑε

+ rθ∂εeεθ∂ϑε

)

= − 1

Rε∂εrR

∂ϑεeεθ + 1

Rε∂εrθ∂ϑε

eεR − rϑRε

eεθ ,

eεθ × 1

Rε sinε (ϑε)

∂εr∂θε

= eεθ × 1

Rε sinε (ϑε)

∂ε

∂θε


)

= eεθ × 1

Rε sinε (ϑε)

(eεR∂εrR

∂θε+ eεϑ

∂εrϑ∂θε

+ eεθ∂εrθ∂θε

+ rR∂εeεR∂θε

+ rϑ∂εeεϑ∂θε


)

= 1

Rε sinε (ϑε)

(eεϑ∂εrR

∂θε− eεR

∂εrϑ∂θε

− sinε(ϑε

)rθeεϑ

+ rθ cosε(ϑε

)eεR

).

Therefore, we obtain the local fractional curl operator in the Cantor-type spherical

coordinates

∇ε × r = eεθ∂εrR

∂Rε− eεϑ

∂εrθ∂Rε

− 1

Rε∂εrR

∂ϑεeεθ + 1

Rε∂εrθ∂ϑε

eεR − rϑRε

eεθ

+ 1

Rε sinε (ϑε)

(eεϑ∂εrR

∂θε− eεR

∂εrϑ∂θε

− sinε(ϑε

)rθeεϑ + rθ cosε

(ϑε

)eεR

)

= eεR

(1

Rε∂εrθ∂ϑε

+ 1

Rε sinε (ϑε)

∂εrϑ∂θε

+ rθ cosε (ϑε)

Rε sinε (ϑε)

)

+ eεϑ

(1

Rε sinε (ϑε)

∂εrR

∂θε− ∂εrθ∂Rε

− rθRε

)+ eεθ

(∂εrR

∂Rε− 1

Rε∂εrR

∂ϑε− rϑ

Rε

)

= eεR1

Rε sinε (ϑε)

(∂ε (rθ sinε (ϑε))

∂ϑε− ∂εrθ∂ϑε

)

+ eεϑ

(1

Rε sinε (ϑε)

∂εrR

∂θε− ∂εrθ∂Rε

− rθRε

)+ eεθ

(∂εrR

∂Rε− 1

Rε∂εrR

∂ϑε− rϑ

Rε

).


The local fractional Laplace operator in Cantor-type spherical coordinates is presentedas follows:

∇2εψ (R, θ , σ) =(

eεR∂ε

∂Rε+ eεϑ

Rε∂ε

∂ϑε+ eεθ

Rε sinε (ϑε)

∂ε

∂θε

)·(

eεR∂εψ

∂Rε+ eεϑ

Rε∂εψ

∂ϑε

+ eεθRε sinε (ϑε)

∂εψ

∂θε

)

= eεR · ∂ε

∂Rε

(eεR∂εψ

∂Rε+ eεϑ

1

Rε∂εψ

∂ϑε+ eεθ

1

Rε sinε (ϑε)

∂εψ

∂θε

)

+ eεϑ · 1

Rε∂ε

∂ϑε

(eεR∂εψ

∂Rε+ eεϑ

1

Rε∂εψ

∂ϑε+ eεθ

1

Rε sinε (ϑε)

∂εψ

∂θε

)

+ eεθ · 1

Rε sinε (ϑε)

∂ε

∂θε

(eεR∂εψ

∂Rε+ eεϑ

1

Rε∂εψ

∂ϑε

+ eεθ1

Rε sinε (ϑε)

∂εψ

∂θε

),

where

eεR · ∂ε

∂Rε

(eεR∂εψ

∂Rε+ eεϑ

1

Rε∂εψ

∂ϑε+ eεθ

1

Rε sinε (ϑε)

∂εψ

∂θε

)= ∂2εψ

∂R2ε,

eεϑ · 1

Rε∂ε

∂ϑε

(eεR∂εψ

∂Rε+ eεϑ

1

Rε∂εψ

∂ϑε+ eεθ

1

Rε sinε (ϑε)

∂εψ

∂θε

)

= 1

Rε∂εψ

∂Rε+ 1

Rε∂ε

∂ϑε

(1

Rε∂εψ

∂ϑε

)

= 1

R2ε

∂2εψ

∂ϑ2ε + 1

Rε∂εψ

∂Rε,

eεθ · 1

Rε sinε (ϑε)

∂ε

∂θε

(eεR∂εψ

∂Rε+ eεϑ

1

Rε∂εψ

∂ϑε+ eεθ

1

Rε sinε (ϑε)

∂εψ

∂θε

)

= 1

Rε∂εψ

∂Rε+ cosε (ϑε)

R2ε sinε (ϑε)

∂εψ

∂ϑε+ 1

R2ε sin2εε (ϑ

ε)

∂2εψ

∂θ2ε .

Thus, the local fractional Laplace operator in Cantor-type spherical coordinatestakes the form

∇2εψ (R, θ , σ) = 1

R2ε

∂2εψ

∂ϑ2ε + 2

Rε∂εψ

∂Rε+ 1

R2ε

∂2εψ

∂ϑ2ε + cosε (ϑε)

R2ε sinε (ϑε)

∂εψ

∂ϑε

+ 1

R2ε sin2εε (ϑ

ε)

∂2εψ

∂θ2ε .

Appendix ETables of local fractional Fouriertransform operators

We present the list of the local fractional Fourier transforms (Table E.1):

� [θ (τ )] = �(ω) = 1

� (1 + ε)

∫ ∞

−∞θ (τ ) Eε

(−iετ εωε)(dτ)ε.

Table E.1 Tables for local fractional Fourier transform operators

Transforms Functions(2π)ε

�(1+ε)δε (ω) 1

1 δε (τ )

Eε

(−iετ ε0 ωε

)δε (τ − τ0)

θ (τ0) Eε

(−iετ ε0 ωε

)δε (τ − τ0) θ (τ )

iεωε δ(ε)ε (τ )

(2π)ε

�(1+ε)δε (ω − ω0) Eε

(iετ εωε

0

)12

(2π)ε

�(1+ε)δε (ω) + 1

iεωε Hε (τ )

2 sinε( ω2 )

ε

ωε rectε (τ )

4 sin2ε(

ω2 )

ε

ω2ε trigε (τ )

2aε

a2ε+ω2ε a.ε (τ )

2iεωε sgnε (τ )

πε2√

1a

�(1+ε)Eε

[− 1

a

(ω2

)2ε]

Eε

(−aτ 2ε)

πε2

�(1+ε)Eε

[− (

ω2

)2ε]

Eε

(−τ 2ε)

(2π)ε

�(1+ε)[δε(ω+a)−δε(ω−a)]

2iε sinε (aετ ε)

(2π)ε

�(1+ε)[δε(ω+a)+δε(ω−a)]

2 cosε (aετ ε)

(2π i)ε

�(1+ε)δ(ε)ε (ω) τ ε

1ωε iε+aε Hε (τ ) Eε (−τ εaε)

Appendix FTables of local fractional Laplacetransform operators

We start with the local fractional Laplace transform of some elementary functions.Find the local fractional Laplace transform of the alternative definition of

analogous rectangular pulse, denoted by rectε (τ , τ0, τ1) = Hε (τ − τ0)−Hε (τ − τ1),namely,

M [rectε (τ , τ0, τ1)] = 1

� (1 + ε)

∫ ∞

0(Hε (τ − τ0)

− Hε (τ − τ1)) Eε

(−τ εsε)(dτ)ε

= 1

� (1 + ε)

∫ τ1

τ0

Eε


= 1

sε

[Eε

(−τ ε0 sε

) − Eε

(−τ ε1 sε

)],

where 0 < τ0 < τ1.Find the local fractional Laplace transform of the analogous Heaviside function,

defined by

Hε (τ − τ0) ={

0, τ ≤ τ0,1, τ > τ0,

that is,

M [Hε (τ )] = 1

� (1 + ε)

∫ ∞

0Hε (τ − τ0) Eε


= 1

� (1 + ε)

∫ ∞

τ0

Eε


= 1

sεEε

(−τ ε0 sε

),

where τ0 > 0.

M

[1

a2 − b2

(cosε

(bτ ε

) − cosε

(aτ ε

))] = sε(s2ε + a2

) (s2ε + b2

) .

226 Appendix F Tables of local fractional Laplace transform operators

Proof. It follows that

M

[1

a2 − b2

(cosε

(bτ ε

) − cosε

(aτ ε

))] = 1

�(1 + ε)

∫ ∞

0

[1

a2 − b2 (cosε(bτ ε)

− cosε(aτ ε))

]Eε


= 1

a2 − b2

[sε

s2ε + b2 − sε

s2ε + a2

]

= sε(s2ε + a2

) (s2ε + b2

) .

M

[1

a2 − b2

(sinhε (aτ ε)

a− sinhε (bτ ε)

b

)]= 1(

s2ε − a2) (

s2ε − b2) .

Proof. We put

M

[1

a2 − b2

(sinhε (aτ ε)


b

)]

= 1

�(1 + ε)

∫ ∞

0

{1

a2 − b2

[sinhε (aτ ε)

a− sinhε (ητ ε)

b

]}Eε


= 1

a2 − b2

[1

s2ε − a2 − 1

s2ε − b2

]

= 1(s2ε − a2

) (s2ε − b2

) .

M

[1

a2 − b2

(coshε

(aτ ε

) − coshε

(bτ ε

))] = sε(s2ε − a2

) (s2ε − b2

) .

Proof. This gives

M

[1

a2 − b2

(coshε

(aτ ε

) − coshε

(bτ ε

))] = 1

�(1 + ε)

∫ ∞

0

[1

a2 − b2

(coshε

(aτ ε

)− coshε

(bτ ε

))]Eε


= 1

a2 − b2

(sε

s2ε − a2 − sε

s2ε − b2

)

= sε(s2ε − a2

) (s2ε − b2

) .

Appendix F Tables of local fractional Laplace transform operators 227

M

[τ ε

� (1 + ε)sinε

(ζ τ ε

)] = ζ sε(s2ε + ζ 2ε

)2 .

Proof. We observe that

M

[τ ε

� (1 + ε)sinε

(ζ τ ε

)] = 1

� (1 + ε)

∫ ∞

0

[τ ε

� (1 + ε)

×Eε (iεζ τ ε) − Eε (−iεζ τ ε)

2iε

]Eε


= 1

2iε

[1

(sε − ζ iε)2 − 1

(sε + ζ iε)2

]

= ζ sε(s2ε + ζ 2ε

)2 .

M

[τ ε

� (1 + ε)cosε

(ζ τ ε

)] = s2ε − ζ 2(s2ε + ζ 2

)2 .

Proof. We have that

M

[τ ε

� (1 + ε)cosε

(ζ τ ε

)] = 1

� (1 + ε)

∫ ∞

0

[τ ε

� (1 + ε)

× Eε (iεζ τ ε) + Eε (−iεζ τ ε)

2

]Eε


= 1

2

[1

(sε − ζ iε)2 + 1

(sε + ζ iε)2

]

= s2ε − ζ 2(s2ε + ζ 2

)2 .

M

[τ ε

� (1 + ε)sinhε

(ζ τ ε

)] = 2ζ sε(s2ε − ζ 2ε

)2 .

Proof. It creates that

M

[τ ε

� (1 + ε)sinhε

(ζ τ ε

)] = 1

� (1 + ε)

∫ ∞

0

[τ ε

� (1 + ε)

× Eε (ζ τ ε) − Eε (−ζ τ ε)

2

]Eε



= 1

2

[1

(sε − ζ )2 − 1

(sε + ζ )2

]

= 2ζ sε(s2ε − ζ 2ε

)2.

M

[τ ε

� (1 + ε)coshε

(ζ τ ε

)] = s2ε + ζ 2(s2ε − ζ 2

)2 .

Proof. It results to

M

[τ ε

� (1 + ε)coshε

(ζ τ ε

)] = 1

� (1 + ε)

∫ ∞

0

[τ ε

� (1 + ε)

× Eε (ζ τ ε) + Eε (−ζ τ ε)

2

]Eε


= 1

2

[1

(sε − ζ )2 + 1

(sε + ζ )2

]

= s2ε + ζ 2(s2ε − ζ 2

)2 .

M

[1

2

(sinε

(ζ τ ε

) − ζ τ ε

� (1 + ε)cosε

(ζ τ ε

))]= ζ 3(

s2ε + ζ 2)2 .

Proof. It is found that

M

[1

2

(sinε

(ζ τ ε

) − ζ τ ε

� (1 + ε)cosε

(ζ τ ε

))]

= 1

� (1 + ε)

∫ ∞

0

[1

2

(sinε

(ζ τ ε

) − ζ τ ε

� (1 + ε)cosε

(ζ τ ε

))]Eε


= 1

2

[ζ

s2ε + ζ 2 − ζ(s2ε − ζ 2

)(s2ε + ζ 2

)2

],

= ζ 3(s2ε + ζ 2

)2 .

M

[1

2

(sinε

(ζ τ ε

) + ζ τ ε

� (1 + ε)cosε

(ζ τ ε

))]= ζ s2ε(

s2ε + ζ 2)2 .


Proof. We can observe that

M

[1

2

(sinε

(ζ τ ε

) + ζ τ ε

� (1 + ε)cosε

(ζ τ ε

))]

= 1

� (1 + ε)

∫ ∞

0

[1

2

(sinε

(ζ τ ε

)

+ ζ τ ε

� (1 + ε)cosε

(ζ τ ε

))]Eε


= 1

2

[ζ

s2ε + ζ 2 + ζ(s2ε − ζ 2

)(s2ε + ζ 2

)2

],

= ζ s2ε(s2ε + ζ 2

)2 .

M

[coshε

(ζ τ ε

) + ζ

2

τ ε

� (1 + ε)sinhε

(ζ τ ε

)] = s3ε(s2ε − ζ 2ε

)2 .

Proof. We can see that

M

[coshε

(ζ τ ε

) + ζ

2

τ ε

� (1 + ε)sinhε

(ζ τ ε

)]

= 1

� (1 + ε)

∫ ∞

0

[coshε

(ζ τ ε

)+ ζ

2

τ ε

� (1 + ε)sinhε

(ζ τ ε

)]Eε


= sε

s2ε − ζ 2 + ζ 2sε(s2ε − ζ 2

)2

= s3ε(s2ε − ζ 2ε

)2.

M

[1

2

(ζ τ ε

� (1 + ε)coshε

(ζ τ ε

) − sinhε

(ζ τ ε

))]= ζ 3(

s2ε − ζ 2)2 .

Proof. It results in

M

[1

2

(ζ τ ε

� (1 + ε)coshε

(ζ τ ε

) − sinhε

(ζ τ ε

))]

= 1

� (1 + ε)

∫ ∞

0

[1

2

(ζ τ ε

� (1 + ε)coshε

(ζ τ ε

) − sinhε

(ζ τ ε

))]Eε



= 1

2

[ζ

(s2ε + ζ 2

)(s2ε − ζ 2

)2 − ζ

s2ε − ζ 2

]

= ζ 3(s2ε − ζ 2

)2.

M [σε (τ − τ0)] = Eε

(−τ ε0 sε

).

Proof.

M [σε (τ − τ0)] = 1

� (1 + ε)

∫ ∞

0σε (τ − τ0) Eε


= Eε

(−τ ε0 sε

).

We present the list of the local fractional Laplace transforms (Table F.1):

M [θ (τ )] = (s) = 1

� (1 + ε)

∫ ∞

0θ (τ ) Eε

(−τ εsε)(dτ)ε.

Table F.1 Tables for local fractional Laplace transform operators

Transforms Functions

1

sε1

1 δε (τ )

Eε

(−τ ε0 sε

)σε (τ − τ0)

1

sε

[Eε

(−τ ε0 sε

) − Eε

(−τ ε1 sε

)]rectε (τ , τ0, τ1)

� (1 + 2ε)

2�3 (1 + ε)

(π

s

) ε

2 τ− ε2

1

sε − aεEε (aετ ε)

sε

s2ε + η2εcosε (ηετ ε)

ηε

s2ε + η2εsinε (ηετ ε)

sε

s2ε − η2εcoshε (ηετ ε)

ηε

s2ε − η2εsinhε (ηετ ε)

1

sε(k+1)

τ kε

� (1 + kε)1

(s − a)(k+1)ε

τ kε

� (1 + kε)Eε (aετ ε)


Table F.1 ContinuedTransforms Functions

(s − a)2ε

(s − a)2ε + η2εEε (aετ ε) cosε (ηετ ε)

ηε

(s − a)2ε + η2εEε (aετ ε) sinε (ηετ ε)

(s − a)ε

(s − a)2ε − η2εEε (aετ ε) coshε (ηετ ε)

ηε

(s − a)2ε − η2εEε (aετ ε) sinhε (ηετ ε)

b − a

(sε + a) (sε + b)Eε (−aτ ε) − Eε (−bτ ε)

a2

(sε + a) s2εEε (−aτ ε) + aτ ε

� (1 + ε)− 1

1

s2ε(s2ε + η2

) 1

η2

τ ε

� (1 + ε)− 1

η3 sinε (ητ ε)

1(s2ε + μ2

) (s2ε + η2

) 1

η2 − μ2

(sinε (μτε)


η

)sε(

s2ε + a2) (

s2ε + b2) 1

a2 − b2 (cosε (bτ ε) − cosε (aτ ε))

1(s2ε − a2

) (s2ε − b2

) 1

a2 − b2

(sinhε (aτ ε)


b

)sε(

s2ε − a2) (

s2ε − b2) 1

a2 − b2 (coshε (aτ ε) − coshε (bτ ε))

1

sεEε

(−τ ε0 sε

)Hε (τ − τ0)

ζ sε(s2ε + ζ 2ε

)2

τ ε

� (1 + ε)sinε (ζ τ ε)

s2ε − ζ 2(s2ε + ζ 2

)2

τ ε

� (1 + ε)cosε (ζ τ ε)

2ζ sε(s2ε − ζ 2ε

)2

τ ε

� (1 + ε)sinhε (ζ τ ε)

s2ε + ζ 2(s2ε − ζ 2

)2

τ ε

� (1 + ε)coshε (ζ τ ε)

ζ 3(s2ε + ζ 2

)2

1

2

(sinε (ζ τ ε) − ζ τ ε

� (1 + ε)cosε (ζ τ ε)

)ζ s2ε(

s2ε + ζ 2)2

1

2

(sinε (ζ τ ε) + ζ τ ε

� (1 + ε)cosε (ζ τ ε)

)s3ε(

s2ε − ζ 2ε)2 coshε (ζ τ ε) + ζ

2

τ ε

� (1 + ε)sinhε (ζ τ ε)

ζ 3(s2ε − ζ 2

)2

1

2

(ζ τ ε

� (1 + ε)coshε (ζ τ ε) − sinhε (ζ τ ε)

)

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Index

Note: Page numbers followed by f indicate figures and t indicate tables.

AAnalogous Dirac distribution

local fractional derivative of, 129via local fractional integral, 125

Analogous rectangular pulseCantor sets, 129Fourier transform, 129–130, 131fLaplace transform, 225

Analogous triangle functionCantor sets, 130Fourier transform, 131–132, 132f

Analogous trigonometric form, 61–62

BBessel inequality, 64Bi-Lipschitz mapping, 13Boussinesq equation, 38Burgers equation, 39

CCanavati fractional derivative, 10Cantor fractal dimension, 34Cantorian coordinate system, 43Cantor sets, 3–4, 78

analogous distributions, 124–136analogous rectangular pulse, 129analogous triangle function, 130complex distribution on, 133cotangent function, 205fractal diffusion equation, 4–5fractal signal processes, 160–170Heaviside function, 129Mittag–Leffler distribution, 134Mittag–Leffler functions, 95, 132, 133signal analysis applications on, 136–139signum distribution on, 134sine and cosine function, 197–198,

199–201, 202–205tangent function, 205trigonometric functions, 197–198,

199–201, 202–205

Cantor-type circular coordinate system, 44diffusion equation in, 48heat-conduction equation, 47Helmholtz equation in, 47homogeneous damped wave equation,

47–48inhomogeneous dissipative wave equation,

48Laplace equation, 46, 47Maxwell’s equations in, 48–49, 53Poisson equation in, 47wave equation, 46

Cantor-type cylindrical coordinates, 43, 44coordinate system of, 213curl operator in, 215–216diffusion equation in, 52, 54divergence operator in, 215gradient operator in, 214heat-conduction equation in, 51, 54Helmholtz equation in, 50homogeneous damped wave equation,

51, 54inhomogeneous dissipative wave

equation, 52Laplace equation in, 50Laplace operator in, 44–45, 216–217Maxwell’s equations in, 52–53Poisson equation in, 50velocity potential of fluid flow, 49–50wave equation for velocity potential, 54

Cantor-type cylindrical symmetrygradient operator in, 45Laplace operator in, 46

Cantor-type spherical coordinates, 43, 45coordinate system of, 217curl operator in, 219–221diffusion equation in, 52, 55divergence operator in, 218–219gradient operator in, 218–221heat-conduction equation in, 51, 54Helmholtz equation in, 51

242 Index

Cantor-type spherical coordinates(Continued)

homogeneous damped wave equation, 51inhomogeneous dissipative wave

equation, 52Laplace equation in, 50Laplace operator in, 45, 219–221Poisson equation in, 50wave equation for, 50, 54

Cantor-type spherical symmetrygradient operator in, 46Laplace operator in, 46

Caputo–Fabrizio fractional derivative, 12Caputo fractional derivative, 8, 13

generalized derivative, 12left-sided derivative, 9, 13right-sided derivative, 9, 13

Cartesian coordinate system, 43Cauchy stress tensor, 42–43Chain rule, 20, 207–208Chen fractional derivative

left-sided derivative, 10right-sided derivative, 10

Coimbra fractional derivative, 12Composition

for generalized local fractional Fouriertransform operator, 117

for local fractional Fourier transformoperator, 105

Concentration-distance curves, 6fConformable fractional derivative, 11

modified left-sided derivative, 11modified right-sided derivative, 11

Conjugategeneralized local fractional Fourier

transform operator, 115local fractional Fourier transform operator,

103Constant signal, Fourier transform, 138Convergence theorem, for Fourier series, 71Convolution, 108, 109–112Convolution theorem

for Fourier transform operator, 111for generalized Fourier transform operator,

123for Laplace transform operator, 157

Coordinate systemCantor-type cylindrical coordinates, 213Cantor-type spherical coordinates, 217

Cosine functionCantor sets, 197–198, 199–201, 202–205on fractal sets, 15

Cossar fractional derivative, 11Cotangent function, Cantor sets, 205Curl operator

Cantor-type cylindrical coordinates,215–216

Cantor-type spherical coordinates,219–221

DDecomposition method

Laplace transform with, 192–196of local fractional operator, 184–187

Differential equationsordinary differential equation, 84–85,

170–176partial differential equation, 86–94,

176–178Diffusion equation

Cantor-type circular coordinate system, 48Cantor-type cylindrical coordinate system,

52, 54Cantor-type spherical coordinate system,

52, 55linear partial differential equation, 37, 41

Dirac distribution, analogouslocal fractional derivative of, 129via local fractional integral, 125

Dirac function, 124–125fractal diffusion equation, 5–6fractal signal processes, 169

Discontinuous space-time, 2fDivergence operator

in Cantor-type cylindrical coordinates, 215in Cantor-type spherical coordinates,

218–219Duality

Fourier transform operator, 104generalized Fourier transform operator,

116

EElementary functions

Laplace transform of, 225–226, 227–231local fractional derivatives of, 207–209Maclaurin’s series of, 211, 212

Index 243

Erdelyi–Kober fractional derivative, 12Euclidean dimensional space, 57Euler’s equation, 43Euler’s Gamma function, 2

FFinal value theorem, 159Fokker–Planck equation, 37Fourier coefficients, 95–96Fourier formula, 99Fourier integral theorem

for Fourier transform operator, 101for generalized Fourier transform operator,

112Fourier series

analogous trigonometric form of, 61–62Bessel inequality for, 64classical mechanism of, 57complex Mittag–Leffler form, 62–63conjugation of, 63convergence theorem for, 71fractal time reversal, 64Hilbert space, 58–59, 60, 61linearity of, 63Mittag–Leffler function, 57, 58, 61of nondifferentiable function, 60–61ordinary differential equation, 84–85partial differential equations, 86–94properties of, 63–64Riemann–Lebesgue theorem, 66, 74–78shift in fractal time, 63signal analysis, 78–84theorems, 64–78of wave equation, 94

Fourier transformanalogous rectangular pulse, 129–130,

131fanalogous triangle function, 131–132,

132fconstant signal, 138with fractal space, 143with variable μ, 143–144

Fourier transform operator, 223, 223tcomposition for, 105conjugate for, 103convolution, 108, 109–112convolution theorem for, 111definitions, 99–101derivation, 95–99

duality for, 104Fourier integral theorem for, 101linearity for, 102Parseval’s theorem for, 112scaling time for, 103shifting time for, 102signal analysis, 136–139theorem, 101, 106, 107, 109, 110translation for, 104

Fractal calculus. See Local fractional calculusFractal diffusion equation, 4–6Fractal geometry, 1, 2Fractal kernel functions

fractal diffusion equation comparisons in,4–6

fractal relaxation equation comparisons in,2–4

Fractal relaxation equationcomparisons in fractal kernel functions,

2–4local fractional ordinary differential

equations, 139Fractal sets

functions defined on, 15–18nondifferentiable functions, 16f

Fractal signal processesFourier series, 78–84Laplace transform operator, 160–170

Fractal space, 143Fractional derivatives

via fractional differences, 7–8with/without singular kernels, 8–13

Fractional differences, 7–8

GGamma function, Euler’s, 2Generalized Caputo fractional derivative, 12Generalized local fractional Fourier

transform operator, 100composition for, 117conjugate for, 115convolution, 120, 121–124convolution theorem for, 123duality for, 116Fourier integral theorem for, 112linearity for, 113Parseval’s theorem for, 124scaling time for, 114shifting time for, 114

244 Index

Generalized local fractional Fouriertransform operator (Continued)

theorem, 117, 118, 119, 121, 122translation for, 116

Generalized Riemann fractional derivative,11–12

Gradient operatorCantor-type cylindrical coordinates,

44–45, 214Cantor-type spherical coordinates, 45,

218–221local fractional partial derivatives, 35

Grünwald–Letnikov derivative, 7Grünwald–Letnikov–Riesz derivative, 7–8

HHadamard fractional derivative, 10Hausdorff measureε-dimensional, 2, 3f , 15fractal derivation via, 1

Heat-conduction equationCantor-type circular coordinate system, 47Cantor-type cylindrical coordinate system,

51, 54Cantor-type spherical coordinate system,

51, 54linear partial differential equation, 40–41

Heat equation, 36Heaviside function

analogous, 225–226, 227–231on Cantor sets, 129

Helmholtz equationCantor-type circular coordinate system, 47Cantor-type cylindrical coordinate

system, 50Cantor-type spherical coordinate system,

51linear partial differential equation, 38, 40

Hilbert spaceconvergent in, 58–59Fourier series from, 60–61interval, 58local fractional integral, 58–59

IInequality theory, 35Initial value theorem, 158Inverse operator

generalized local fractional Fouriertransform, 101, 120

local fractional Fourier transform, 100,143, 144

local fractional Laplace transform, 150,192–194

JJacobian theory, 35

KKlein–Gordon equation

linear partial differential equation, 36nonlinear partial differential equation, 40

Kolwankar-Gangal sense, 20Korteweg–de Vries equation

linear partial differential equation, 37nonlinear partial differential equation,

39–40

LLaplace equation

Cantor-type circular coordinate system,46, 47

Cantor-type cylindrical coordinatesystem, 50

Cantor-type spherical coordinatesystem, 50

in 1 + 1 fractal dimensional space, 143linear partial differential equation, 36, 40nonlinear partial differential equation, 40partial differential equation, 89

Laplace operatorsCantor-type cylindrical coordinates,

44–45, 216–217Cantor-type spherical coordinates, 45,

219–221local fractional partial derivatives, 35

Laplace transformanalogous Heaviside function, 225–226,

227–231analogous rectangular pulse, 170, 225convolution theorem for, 157with decomposition method, 192–196definitions of, 150–151of elementary functions, 225–226,

227–231final value theorem for, 158initial value theorem for, 158

Index 245

inverse operator, 150inverse two-sided Laplace transform

operator, 150linearity for, 151operators, 230tordinary differential equations, 170–176partial differential equations, 176–178properties, 151–160scaling time for, 152shifting time for, 151signal analysis, 160–170theorems, 152, 153, 154, 155, 156translation for, 152two-sided Laplace transform operator, 150with variational iteration method, 188–191

Laplace variational iteration algorithm,188–191

Laurent series, 148Lebesgue–Cantor function, 2–3

fractal diffusion equation, 4on fractal sets, 15

Lighthill–Whitham–Richards equationlinear partial differential equation, 37nonlinear partial differential equation, 40

LinearityFourier transform operator, 102generalized Fourier transform operator,

113Linear partial differential equation

Boussinesq equation, 38compressible Euler’s equation, 43compressible Navier–Stokes equations, 42diffusion equation, 37, 41Fokker–Planck equation, 37heat-conduction equation, 40–41heat equation, 36homogeneous damped wave equation,

38, 41homogeneous Helmholtz equation, 38incompressible Euler’s equation, 43incompressible Navier–Stokes equations,

42–43inhomogeneous dissipative wave equation,

38, 41inhomogeneous Helmholtz equation, 38,

40Klein–Gordon equation, 36Korteweg–de Vries equation, 37Laplace equation, 36, 40

Lighthill–Whitham–Richards equation, 37in mathematical physics, 35–43Maxwell’s equations, 41–42Poisson equation, 37, 40Schrödinger equation, 36, 41transport equation, 37Tricomi equation, 37wave equation, 36, 40

Liouville fractional derivative, 8left-sided fractional derivative, 8right-sided fractional derivative, 9

Lipschitz mapping, 13–14Local fractional calculus, 1Local fractional chain rule, 20, 207–208Local fractional continuity, 13–15, 25Local fractional convolution, 108, 109–112Local fractional curl operator

Cantor-type cylindrical coordinates,215–216

Cantor-type spherical coordinates,219–221

Local fractional derivativeanalogous Dirac distribution, 129definitions of, 1–2, 18–22of elementary functions, 207–209left-hand derivative, 18of nondifferentiable functions, 20, 21tproperties, 22–25right-hand derivative, 18theorems, 22–25using fractal geometry, 1, 2

ε-Local fractional derivative set, 19–20Local fractional differential equations

ordinary differential equation, 84–85,170–176

partial differential equation, 86–94,176–178

Local fractional differentiation rule, fractalsets, 20

Local fractional Dirac function, 5–6,124–125, 169

Local fractional divergence operatorCantor-type cylindrical coordinates, 215Cantor-type spherical coordinates,

218–219Local fractional Fourier coefficients,

95–96Local fractional Fourier formula, 100

for nondifferentiable function, 99

246 Index

Local fractional Fourier seriesanalogous trigonometric form of, 61–62Bessel inequality for, 64classical mechanism of, 57complex Mittag–Leffler form, 62–63conjugation of, 63convergence theorem for, 71fractal time reversal, 64Hilbert space, 58–59, 60, 61linearity of, 63Mittag–Leffler function, 57, 58, 61of nondifferentiable function, 60–61ordinary differential equation, 84–85partial differential equations, 86–94properties of, 63–64Riemann–Lebesgue theorem, 66, 74–78shift in fractal time, 63signal analysis, 78–84theorems of, 64–78

Local fractional Fourier transformanalogous rectangular pulse, 129–130,

131fanalogous triangle function, 131–132,

132fconstant signal, 138with fractal space, 143with variable μ, 143–144

Local fractional Fourier transform operator.See also Generalized local fractionalFourier transform operator

composition for, 105conjugate for, 103convolution, 108, 109–112convolution theorem for, 111definitions, 99–101derivation, 95–99duality for, 104Fourier integral theorem for, 101linearity for, 102Parseval’s theorem for, 112scaling time for, 103shifting time for, 102signal analysis, 136–139theorem, 101, 106, 107, 109, 110translation for, 104

Local fractional gradient operatorCantor-type cylindrical coordinates,

44–45, 214

Cantor-type spherical coordinates, 45,218–221

Local fractional integralsanalogous Dirac distribution via, 125definitions of, 25–26mean value theorem for, 26Newton–Leibniz formula, 28of nondifferentiable functions, 33tproperties and theorems of, 26–29

Local fractional Laplace equationCantor-type circular coordinate system,

46, 47Cantor-type cylindrical coordinate

system, 50Cantor-type spherical coordinate

system, 50in 1 + 1 fractal dimensional space, 143linear partial differential equation,

36, 40nonlinear partial differential equation, 40partial differential equation, 89

Local fractional Laplace operatorCantor-type cylindrical coordinates,

44–45, 216–217Cantor-type spherical coordinates, 45,

219–221partial derivatives, 35

Local fractional Laplace transformanalogous Heaviside function, 225–226,

227–231analogous rectangular pulse, 170, 225convolution theorem for, 157definitions of, 150–151of elementary functions, 225–226,

227–231final value theorem for, 158initial value theorem for, 158inverse operator, 150inverse two-sided Laplace transform

operator, 150linearity for, 151operators, 230tordinary differential equations, 170–176partial differential equations, 176–178properties, 151–160scaling time for, 152shifting time for, 151signal analysis, 160–170theorems, 152, 153, 154, 155, 156

Index 247

translation for, 152two-sided Laplace transform operator, 150

Local fractional Laplace variational iterationalgorithm, 188–191

Local fractional Maclaurin polynomial, 211Local fractional Maclaurin’s series, 32

of elementary functions, 211, 212Local fractional ordinary differential

equationsapplications of, 139–142Fourier series, 84–85Laplace transform operators, 170–176with positive parameter p, 141

Local fractional partial derivativesgradient and Laplace operators, 35Jacobian and inequality theory, 35in mathematical physics, 34–35operator to coordinate systems, 43–46

Local fractional partial differential equations,179

alternative observations of, 46–55applications, 142–145decomposition method, 192–196Laplace decomposition method, 192–196Laplace variational iteration method,

188–191variational iteration method, 188–191

Local fractional Rolle’s theorem, 22, 23–24Local fractional Taylor’s theorem

for elementary functions, 31–32for nondifferentiable functions, 29–31

Local fractional variational iterationalgorithm, 180–184

MMaclaurin polynomial, 211Maclaurin’s series, 32

of elementary functions, 211, 212Marchaud fractional derivative, 9

left-sided derivative, 10right-sided derivative, 10

Mathematical physicslinear/nonlinear PDEs in, 35–43partial derivatives in, 34–35

Maxwell’s equationsCantor-type circular coordinate system,

48–49, 53Cantor-type cylindrical coordinate system,

52–53

linear partial differential equation, 41–42Mittag–Leffler distribution, 134Mittag–Leffler function

on Cantor sets, 95, 132, 133Fourier series, 57, 58, 61, 62–63on fractal sets, 15, 16–17, 43nondifferentiable functions via, 32, 33t

Modified conformable fractional derivativeleft-sided derivative, 11right-sided derivative, 11

Modified Riemann–Liouville fractionalderivative, 11

NNavier–Stokes equations, 42–43Newton–Leibniz formula, 28Nondifferentiable functions

comparisons of, 16–17, 16f , 17fconcentration-distance curves for, 6fFourier formula, 99local fractional derivative of, 20, 21tlocal fractional differentiation rules of, 20local fractional integral of, 33tMaclaurin’s series, 211, 212Taylor’s theorem for, 29–31

Nonlinear partial differential equationBurgers equation, 39forced Burgers equation, 39generalized Korteweg–de Vries equation,

40inviscid Burgers equation, 39Klein–Gordon equation, 40Korteweg–de Vries equation, 39Laplace equation, 40Lighthill–Whitham–Richards equation, 40in mathematical physics, 35–43modified Korteweg–de Vries equation,

39–40Poisson equation, 40transport equation, 39velocity potential of fluid flow, 41wave equation, 39

OOrdinary differential equations (ODEs)

applications of, 139–142Fourier series, 84–85Laplace transform operators, 170–176with positive parameter p, 141

248 Index

PParseval’s theorem


124Partial derivatives

gradient operators, 35Jacobian and inequality theory, 35Laplace operators, 35in mathematical physics, 34–35operator to coordinate systems, 43–46

Partial differential equations (PDEs),179

alternative observations of, 46–55applications, 142–145decomposition method,

192–196Fourier series, 86–94Laplace decomposition method, 192–196Laplace transform operators, 176–178Laplace variational iteration method,

188–191linear. See (Linear partial differential

equation)nonlinear. See (Nonlinear partial

differential equation)variational iteration method, 188–191

Periodic functions, 57Poisson equation

Cantor-type circular coordinate system, 47Cantor-type cylindrical coordinate

system, 50Cantor-type spherical coordinate

system, 50linear partial differential equation,

37, 40nonlinear partial differential equation, 40

RRectangular pulse, analogous

Cantor sets, 129Fourier transform, 129–130, 131fLaplace transform, 225

Riemann fractional derivative, 8, 11–12Riemann integral, 25Riemann–Lebesgue theorem, 66, 74–78Riemann–Liouville fractional derivative, 1

left-sided fractional derivative, 9, 12modified derivative, 11

right-sided fractional derivative, 9, 12Riesz fractional derivative, 10Rolle’s theorem, 22, 23–24

SScaling time


114Schrödinger equation, 36, 41Shifting time


114Signal analysis

on Cantor sets, 136–139Fourier series, 78–84Fourier transform, 136–139Laplace transform, 160–170

Signum distribution, on Cantor sets, 134Sine function

Cantor sets, 197–198, 199–201, 202–205on fractal sets, 15

Singular kernels, fractional derivativeswith/without, 8–13

TTangent function, Cantor sets, 205Taylor’s theorem

for elementary functions, 31–32for nondifferentiable functions, 29–31

Tempered left-sided fractional derivative, 11Translation


116Transport equation

linear partial differential equation, 37nonlinear partial differential equation, 39

Triangle function, analogousCantor sets, 130Fourier transform, 131–132, 132f

Tricomi equation, 37Trigonometric functions, on Cantor sets,

197–198, 199–201, 202–205

VVariational iteration algorithm, 180–184Variational iteration method

Index 249

Laplace transform with, 188–191of local fractional operator, 179–184

Velocity potential, 39, 40, 41, 46, 49–50, 54

WWave equation, 38, 41

Cantor-type circular coordinate system,46, 47–48

Cantor-type cylindrical coordinate system,51, 52, 54

Cantor-type spherical coordinate system,50, 51, 52, 54

Fourier solution of, 94linear partial differential equation, 36, 38,

40, 41nonlinear partial differential equation, 39

Weyl fractional derivative, 9

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Local Fractional Integral Transforms and their Applications