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Counterexamples on Jumarie’s two basic fractional calculus formulae

Cheng-shi Liu

PII: S1007-5704(14)00355-4DOI: http://dx.doi.org/10.1016/j.cnsns.2014.07.022Reference: CNSNS 3289

To appear in: Communications in Nonlinear Science and Numer-ical Simulation

Received Date: 24 February 2014Revised Date: 18 July 2014Accepted Date: 25 July 2014

Please cite this article as: Liu, C-s., Counterexamples on Jumarie’s two basic fractional calculus formulae,Communications in Nonlinear Science and Numerical Simulation (2014), doi: http://dx.doi.org/10.1016/j.cnsns.2014.07.022

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Counterexamples on Jumarie’s two basic

fractional calculus formulae

Cheng-shi Liu

Department of Mathematics

Northeast Petroleum University

Daqing 163318, China

Email: chengshiliu-68@126.com

August 5, 2014

Abstract

Juamrie proposed a modified Riemann-Liouville derivative definitionand gave two basic fractional calculus formulae (u(t)v(t))(α) = u(α)(t)v(t)+u(t)v(α)(t) and (f(u(t)))(α) = f ′

uu(α)(t). We give two counterexamples to

show that Jumarie’s two formulae are not true. Respectively, all resultsobtained in references by using Jumarie’s these two formulae are incorrect.In the end, we give the corresponding formulae.

Keywords: counterexample, fractional calculus, modified Riemann-Liouville’s derivative

1 Introduction

Jumarie proposed the following modified Riemann-Liouville fractional derivative[1-5]:

f (α)(t) =1

Γ(1 − α)

d

dt

∫ t

0

(t − x)−α(f(x) − f(0))dx. (1)

and gave some basic fractional calculus formulae, for example, formulae (4.12)and (4.13) in [4]:

(u(t)v(t))(α) = u(α)(t)v(t) + u(t)v(α)(t), (2)

(f(u(t)))(α) = f ′uu(α)(t). (3)

The last formula (3) has been applied to solve the exact solutions to somenonlinear fractional order differential equations[6-9]. If this formula were true,then we could take the transformation ξ = x − ktα

Γ(1+α) , and reduce the partial

derivative ∂αU(x,t)∂tα to U ′(ξ). Therefore the corresponding fractional differential

equations become the ordinary differential equations which are easy to study.

1

But we must point out that Jumarie’s basic formulae (2) and (3) are notcorrect, and therefore the corresponding results on differential equations arenot true. We will give two counterexamples on these two formulae. Our ex-amples show that Jumarie’s formulae are incorrect. In Section 3, we give thecorresponding formulae.

2 Counterexamples

Before giving counterexamples, we need to compute exact values of some frac-tional order derivatives under Jumarie’s definition. Of course, these specialexact solutions can also be found in [10] without details. In order that thepaper is self-contained, we give the computation details.

We need the 12−order derivatives of the following three functions f(t) =

t, f(t) =√

t and f(t) = t2. Since f(0) = 0 for these three functions, Jumarie’sfractional derivative is just the routine Riemann-Liouville derivative. Accordingto definition (1), taking the variable transformation x = t − y2 and noting thatΓ( 1

2 ) =√

π, we have

f (1/2)(t) =2√π

d

dt

∫

√t

0

(f(t − y2) − f(0))dy. (4)

Taking the functions f(t) = t, f(t) =√

t and f(t) = t2, we have respectively

(t)(1/2) = 2

√

t

π, (5)

(√

t)(1/2) =

√π

2, (6)

(t2)(1/2) =8t3/2

3√

π. (7)

Counterexample 1 (the counterexample of formula (2)). Take u(t) =v(t) = t1/2 and α = 1

2 . From formula (5), the left of Jumarie’s formula (2) is

(u(t)v(t))(1/2) = t(1/2) = 2

√

t

π. (8)

On the other hand, computing the right side of Jumarie’s formula (2) and using(6), we have

u(1/2)(t)v(t) + u(t)v(1/2)(t) = 2√

t(√

t)(1/2) =√

πt. (9)

By the above two expressions, we obtain:

(u(t)v(t))(1/2) 6= u(1/2)(t)v(t) + u(t)v(1/2)(t). (10)

This example shows that Jumarie’s formula (2) is not true.

2

Counterexample 2 (the counterexample of formula (3)). Take u(t) =t2, f(u) = u1/2 and α = 1

2 . First, we directly compute

(f(u(t)))(1/2) = (t)(1/2) = 2

√

t

π. (11)

Second, the right side of formula (3) is

f ′uu(1/2)(t) =

1

2√

u(t2)(1/2) =

1

2t

8t3/2

3√

π=

4t1/2

3√

π. (12)

So we find(f(u(t)))(1/2) 6= f ′

uu(1/2)(t). (13)

This means that Jumarie’s formula (3) is also incorrect.By the above two counterexamples, we show that Jumarie’s two basic frac-

tional calculus formulae are not true, and then all results obtained in [6-9] byusing Jumarie’s formulae are incorrect.

Remark: In [4] and [5], Jumarie gives another formula

(f(u(t)))(α) = f (α)(u)(u′(t))α. (14)

According to the counterexample 2, we know that this formula is also incorrect.In fact, under the assumptions of counterexample 2, the right side of the formulais

f (α)(u)(u′(t))α = (√

u)(1/2)√

(t2)′ =

√

πt

2. (15)

This shows that the formula (14) is incorrect, i.e.,

(f(u(t)))(α) 6= f (α)(u)(u′(t))α. (16)

Moreover, from (12), we also have

f ′uu(α)(t) 6= f (α)(u)(u′(t))α. (17)

3 Discussion and conclusion

In [4], Jumarie gives a proof on his formulae (2) and (3)(see the corollary 4.3 andits proof in [4]). By our counterexamples, we can conclude that there exist sometheoretical mistakes in Jumarie’s proof or his theory. For the Riemann-Liouvillefractional derivative, the right formulae on (u(t)v(t))(α) and (f(u(t)))(α) can befound in [10](see sections 5.5 and 5.6 in book [10]). It is not difficult to generalizethese formulae to Jumarie’s modified Riemann-Liouville fractional derivative. Infact, from formula (1) of Jumarie’s definition, we have

(f(x)g(x))(α)J = (f(x)g(x))

(α)R−L − (f(0)g(0))

(α)R−L, (18)

(f(g(x)))(α)J = (f(g(x)))

(α)R−L − (f(g(0)))

(α)R−L, (19)

3

where the lower index J and R − L denote respectively the Jumarie’s andRiemann-Liouville’s fractional calculus. According to the corresponding for-mulae of Riemann-Liouville fractional calculus [10], we have

(f(x)g(x))(α)J =

+∞∑

j=0

(

α

j

)

f (j)(x)g(α−j)R−L (x) − f(0)g(0)

xαΓ(1 − α), (20)

(f(g(x)))(α)J =

+∞∑

j=1

(

α

j

)

xj−αj!

Γ(j − α + 1)

j∑

m=1

f (m)(g)∑

j∏

k=1

1

Pk!(g(k)

k!)Pk

+f(g(x)) − f(g(0))

xαΓ(1 − α), (21)

where∑

extends over all combinations of nonnegative integer values of P1, · · · , Pn

such that∑n

k=1 kPk = n and∑n

k=1 Pk = m.Acknowledgement. Thanks to reviewers for their helpful suggestions.

This project is supported by Natural Science Foundation of Heilongjiang Provinceof China under Grant No.A201308.

References

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[2] Jumarie G. On the solution of the stochastic differential equation of expo-nential growth driven by fractional Brownian motion. Applied MathematicsLetters, 2005, 18(7): 817-826.

[3] Jumarie G. New stochastic fractional models for Malthusian growth, thePoissonian birth process and optimal management of populations. Mathe-matical and computer modelling, 2006, 44(3): 231-254.

[4] Jumarie G. Modified Riemann-Liouville derivative and fractional Taylor se-ries of nondifferentiable functions further results. Computers and Mathe-matics with Applications, 2006, 51(9): 1367-1376.

[5] Jumarie G. Table of some basic fractional calculus formulae derived from amodified Riemann-Liouville derivative for non-differentiable functions. Ap-plied Mathematics Letters, 2009, 22(3): 378-385.

[6] Pandir Y, Gurefe Y, Misirli E. The Extended Trial Equation Methodfor Some Time Fractional Differential Equations. Discrete Dynam-ics in Nature and Society, 2013,Volume 2013, Article ID 491359.http://dx.doi.org/10.1155/2013/491359.

4

[7] Bulut H, Baskonus H M, Pandir Y. The Modified Trial Equation Method forFractional Wave Equation and Time Fractional Generalized Burgers Equa-tion. Abstract and Applied Analysis. 2013, Volume 2013, Article ID 636802.http://dx.doi.org/10.1155/2013/636802.

[8] Li Z B, Zhu W H, He J H. Exact solutions of time-fractional heat conductionequation by the fractional complex transform. Thermal Science, 2012, 16(2):335-338.

[9] Ghany H A, El Bab A S O, Zabel A M and Hyder A A. The fractional cou-pled KdV equations: Exact solutions and white noise functional approach.Chinese Physics B, 2013, 22(8): 302-308.

[10] Oldham K B, Spanier J. The fractional calculus:Theory and appkicationsof Differentiation and integration to arbitary order. New York: Academicpress, 1974.

5

We give two counterexamples of Jumarie’s two basic fractional

calculus formulae

$(u(t)v(t))^{(\alpha)}=u^{(\alpha)}(t)v(t)+u(t)v^{(\alpha)}(t)$and$

(f(u(t)))^{(\alpha)}=f'_uu^{(\alpha)}(t)$ on modified

Riemann-Liouville derivative definition. Then all results

obtained in references by using Jumarie's these two formulae

are incorrect. We also give the corresponding formulae.