Nonlocal Telegraph Equation in Frame of the …downloads.hindawi.com/journals/amp/2019/7528937.pdfAdvancesinMathematicalPhysics Proposition . Fr =1and =0the formula coincideswiththefundamentalsolutionoftheclassical

Research ArticleNonlocal Telegraph Equation in Frame of the ConformableTime-Fractional Derivative

Mohamed Bouaouid Khalid Hilal and Said Melliani

Department of Mathematics Sultan Moulay Slimane University BP 523 23000 Beni Mellal Morocco

Correspondence should be addressed to Mohamed Bouaouid bouaouidfstgmailcom

Received 15 August 2018 Revised 23 December 2018 Accepted 31 January 2019 Published 3 March 2019

Academic Editor Sergey Shmarev

Copyright copy 2019 Mohamed Bouaouid et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We use the cosine family of linear operators to prove the existence uniqueness and stability of the integral solution of a nonlocaltelegraph equation in frame of the conformable time-fractional derivative Moreover we give its implicit fundamental solution interms of the classical trigonometric functions

1 Introduction and Statement of the Problem

The telegraph equation is better than the heat equation inmodeling of physical phenomena which have a parabolicbehavior [1] The one-dimensional telegraph equation can bewritten as follows

1205972119906 (119905 ℓ)1205971199052 = 1

1198711198621205972119906 (119905 ℓ)

120597ℓ2 minus 119877119866119871119862119906 (119905 ℓ)

minus (119877119871 + 119866

119862) 120597119906 (119905 ℓ)120597119905

(1)

where 119877 and 119866 are respectively the resistance and the con-ductance of resistor119862 is the capacitance of capacitor and 119871 isthe inductance of coil Many concrete applications amount toreplacing the time derivative in the telegraph equation witha fractional derivative For example in the works [2ndash8] theauthors have extensively studied the time-fractional telegraphequation with Caputo fractional derivative For more detailsabout the good effect of the fractional derivative we refer tomonographs [9ndash13]

Recently a new definition of fractional derivative namedldquofractional conformable derivativerdquo is introduce by Khalilet al [14] This novel fractional derivative is compatiblewith the classical derivative and it is excellent for studynonregular solutions Since the subject of the fractionalconformable derivative has attracted the attention of manyauthors in domains such as mechanics [15] electronic [16]

and anomalous diffusion [17]We are interested in studying inthis paper the telegraph model (1) in framework of the time-fractional conformable derivative Precisely we will proposethe following transformations

12059721205971199052 997888rarr

120597120572120597119905120572

120597120572120597119905120572 0 lt 120572 lt 1 (2)

120597120597119905 997888rarr 120597120574

120597119905120574 0 lt 120574 le 120572 lt 1 (3)

where 120597120572120597119905120572 and 120597120574120597119905120574 are the time-fractional conformablederivative operators [14] Then we get the fractional con-formable telegraph model associated with the transforma-tions (2) and (3) as follows

120597120572120597119905120572

120597120572119906 (119905 ℓ)120597119905120572 = 1

1198711198621205972119906 (119905 ℓ)

120597ℓ2 minus 119877119866119871119862119906 (119905 ℓ)

minus (119877119871 + 119866

119862) 120597120574119906 (119905 ℓ)120597119905120574

(4)

where the time-parameter 119905 belongs to an interval [0 120591] with120591 is a fixed positive real number The spatial parameter ℓbelongs to the interval [0 120587]

HindawiAdvances in Mathematical PhysicsVolume 2019 Article ID 7528937 7 pageshttpsdoiorg10115520197528937

2 Advances in Mathematical Physics

We associate to (4) the boundary and the nonlocal initialconditions

119906 (119905 0) = 0 119905 isin [0 120591] (5)

119906 (119905 120587) = 0 119905 isin [0 120591] (6)

119906 (0 ℓ) = 1199090 ℓ isin [0 120587] (7)

120597120572119906 (0 ℓ)120597119905120572 = 1199091 +

119901sum119894=1

120576119894119906 (119905119894 ℓ) ℓ isin [0 120587] (8)

where 119901 is a nonnull fixed integer and 0 lt 1199051 lt sdot sdot sdot lt 119905119901 lt 120591The quantities 1199090 1199091 and 1205761 120576119901 are physical measuresThe condition appearing in (8) means the nonlocal condition[18] For physical interpretations of this condition we referto works [19 20] For example in [19] the author used anonlocal condition of the form (8) to describe the diffusionphenomenon of a small amount of gas in a transparent tube

Wenote that it is not easy to find the fundamental solutionof (4) by using the Laplace transformmethod if120572 = 120574 For thisreasonwewill propose the integral solution concept based onan operator theory approach When 120572 = 120574 we investigate animplicit fundamental solution

The content of this paper is organized as follows InSection 2 we recall some needed results of the conformablefractional derivative and cosine family of linear operators InSection 3 we prove the existence uniqueness and stabilityof the integral solution of (4) by using of an operator theoryapproach Section 4 is devoted to an implicit fundamentalsolution of (4) in terms of the classical trigonometric func-tions

2 Preliminaries

We start recalling some concepts on the conformable frac-tional calculus [14]

The conformable fractional derivative of a function 119909 oforder 120572 at 119905 gt 0 is defined by the following limit

119889120572119909 (119905)119889119905120572 = lim

120576997888rarr0

119909 (119905 + 1205761199051minus120572) minus 119909 (119905)120576 (9)

When this limit exists we say that 119909 is (120572)-differentiable at 119905If 119909 is (120572)-differentiable at some 119905 sim 0 and the limit

lim119905997888rarr0+(119889120572119909(119905)119889119905120572) existsThen we define the conformablefractional derivative of 119909 at 0 by

119889120572119909 (0)119889119905120572 = lim

119905997888rarr0+

119889120572119909 (119905)119889119905120572 (10)

The (120572)-fractional integral of a function 119909 is defined by

119868120572 (119909) (119905) = int1199050119904120572minus1119909 (119904) 119889119904 (11)

If119909 is a continuous function in the domain of 119868120572 thenwe have119889120572 (119868120572 (119909) (119905))

119889119905120572 = 119909 (119905) (12)

According to [21] if 119909 is differentiable then we have

119868120572 (119889120572119909119889119905120572 ) (119905) = 119909 (119905) minus 119909 (0) (13)

We remark that the classical Laplace transform is not com-patible with the conformable fractional derivative For thisreason the adapted transform is defined as [21]

L120572 (119909 (119905)) (120582) fl int+infin0

119905120572minus1119890minus120582119905120572120572119909 (119905) 119889119905 (14)

The above transformation is called fractional Laplace trans-form of order 120572 of the function 119909

If 119909 is differentiable then the action of the fractionalLaplace transform on the conformable fractional derivativeis given as follows

L120572 (119889120572119909 (119905)119889119905120572 ) (120582) = 120582L120572 (119909 (119905)) (120582) minus 119909 (0) (15)

Now we introduce some results concerning the cosine familytheory [22]

A one-parameter family (119862(119905))119905isinR of bounded linearoperators on the Banach space 119883 is called a strongly contin-uous cosine family if and only if

(1) 119862(0) = 119868 (I is the identity operator)(2) 119862(119904 + 119905) + 119862(119904 minus 119905) = 2119862(119904)119862(119905) for all 119905 119904 isin R(3) 119905 997891997888rarr 119862(119905)119909 is continuous for each fixed 119909 isin 119883

We define also the sine family by

119878 (119905) fl int1199050119862 (119904) 119889119904 (16)

The infinitesimal generator119860 of a strongly continuous cosinefamily ((119862(119905))119905isinR (119878(119905))119905isinR) on119883 is defined by

119863(119860) = 119909 isin 119883 119905 997891997888rarr 119862 (119905)sdot 119909 is a twice continuously differentiable function

119860 = 1198892119862 (119905)1198891199052 rfloor

119905=0

(17)

If 119860 is the infinitesimal generator of a strongly cosine family((119862(119905))119905isinR (119878(119905))119905isinR) on119883 then there exists a constant 120596 ge 0such that for all 120582 with 119877119890(120582) gt 120596 we have

1205822 isin 120588 (119860) (120588 (119860) is the resolvent set of 119860)

120582 (1205822119868 minus 119860)minus1 = int+infin0

119890minus120582119905119862 (119905) 119889119905(1205822119868 minus 119860)minus1 = int+infin

0119890minus120582119905119878 (119905) 119889119905

(18)

where 119877119890(120582) is the real part of the complex number 120582

Advances in Mathematical Physics 3

Before presenting the main results we introduce thefollowing notations

119886 = 119877119866119871119862

119887 = 119877119871 + 119866

119862119888 = 1

119871119862and 120576 = 119901sum

119894=1

10038161003816100381610038161205761198941003816100381610038161003816

(19)

3 Integral Solution by Using an OperatorTheory Approach

Define the operator 119860 1198712([0 120587] 119889ℓ) 997888rarr 1198712([0 120587] 119889ℓ) by

119860 = 1198881205972 ()120597ℓ2and 119863(119860)

= 119906 isin 1198672 ((0 120587) 119889ℓ) 119906 (0) = 119906 (120587) = 0 (20)

According to [23] the operator 119860 generates a cosine family((119862(119905))119905isinR (119878(119905))119905isinR) on 1198712([0 120587] 119889ℓ) Moreover |119862(119905)| le 1and |119878(119905)| le 1 for all 119905 isin [0 120591]

Next we consider the following transformations

119909 (119905) (ℓ) = 119906 (119905 ℓ) 119891 (119905 119909 (119905) 119889120574119909 (119905)

119889119905120574 ) = minus119886119909 (119905) minus 119887119889120574119909 (119905)119889119905120574

119892 (119909) = 119901sum119894=1

120576119894119909 (119905119894) (21)

Then we get the following nonlocal fractional ordinarydifferential equation

119889120572119889119905120572

119889120572119909 (119905)119889119905120572 = 119860119909 (119905) + 119891(119905 119909 (119905) 119889120574119909 (119905)

119889119905120574 ) (22)

119909 (0) = 1199090 (23)

119889120572119909 (0)119889119905120572 = 1199091 + 119892 (119909) (24)

We denote C120572 the Banach space of continuously (120572)-differentiable functions from [0 120591] into1198712([0 120587] 119889ℓ)with thenorm |119909| = sup119905isin[0120591]119909(119905) + sup119905isin[0120591]119889120572119909(119905)119889119905120572 Here is the classical norm in the space 1198712([0 120587] 119889ℓ)

31 Existence and Uniqueness of the Integral Solution Toexplain integral Duhamelrsquos formula we apply the fractionalLaplace transform to (22) obtaining

119909 (119905)= 119862(119905120572

120572 )1199090 + 119878(119905120572120572 ) [1199091 + 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )119889119904

(25)

We remark that for 120572 = 120574 = 1 we have classical Duhamelrsquosformula [22] Hence we can introduce the following defini-tion

Definition 1 We say that 119909 isin C120572 is an integral solution of theequation (22) if the following assertion is true

119909 (119905)= 119862(119905120572

120572 )1199090 + 119878(119905120572120572 ) [1199091 + 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )119889119904119905 isin [0 120591]

(26)

Theorem 2 e Cauchy problem (22) has a unique integralsolution provided that

120576 +max(119886120591120572120572 1198871205912120572minus1205742120572 minus 120574) lt 1

2 (27)

Proof Define the operator Γ C120572 997888rarr C120572 by

Γ (119909) (119905)= 119862(119905120572

120572 )1199090 + 119878(119905120572120572 ) [1199091 + 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )119889119904

(28)

Let 119909 119910 isin C120572 then we have

Γ (119910) (119905) minus Γ (119909) (119905) = 119878 (119905120572120572 ) [119892 (119910) minus 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )[119891(119904 119910 (119904) 119889120574119910 (119904)119889119905120574 )

minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904

119889120572119889119905120572 [Γ (119910) (119905) minus Γ (119909) (119905)] = 119862(119905120572

120572 ) [119892 (119910) minus 119892 (119909)]

+ int1199050119904120572minus1119862(119905120572 minus 119904120572

120572 )[119891(119904 119910 (119904) 119889120574119910 (119904)119889119905120574 )

minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904

(29)


Accordingly we obtain

1003817100381710038171003817Γ (119910) (119905) minus Γ (119909) (119905)1003817100381710038171003817 +10038171003817100381710038171003817100381710038171003817119889120572119889119905120572 [Γ (119910) (119905) minus Γ (119909) (119905)]10038171003817100381710038171003817100381710038171003817

le 2 [120576 +max(119886120591120572120572 1198871205912120572minus1205742120572 minus 120574)] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816

(30)

Then we get

1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816le 2 [120576 +max(119886120591120572

120572 1198871205912120572minus1205742120572 minus 120574)] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816 (31)

Finally Γ has an unique fixed point 119909 in (C120572 ||) which is theintegral solution of the equation (22)

Now we give a result that is better than the previous one


120576 lt 12 (32)


Γ (119909) (119905)= 119862(119905120572

120572 )1199090 + 119878(119905120572120572 ) [1199091 + 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )119889119904

119905 isin [0 120591]

(33)

Next we define a new norm ||120572 inC120572 by

|119909|120572 =10038161003816100381610038161003816100381610038161003816exp(minus120579 ()120572

120572 )11990910038161003816100381610038161003816100381610038161003816 (34)

where

120579 = 3max (119886 119887120591120572minus120574)1 minus 2120576 (35)

For 119909 119910 isin C120572 and 119905 isin [0 120591] we haveΓ (119910) (119905) minus Γ (119909) (119905) = 119878 (119905120572

120572 ) [119892 (119910) minus 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )[119891(119904 119910 (119904) 119889120574119910 (119904)119889119905120574 )

minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904

(36)

Therefore we obtain

1003817100381710038171003817Γ (119910) (t) minus Γ (119909) (119905)1003817100381710038171003817 le [120576 exp(120579119905120572120572 )

+max (119886 119887120591120572minus120574) int1199050119904120572minus1 exp(120579119904120572

120572 )119889119904] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572 10038171003817100381710038171003817100381710038171003817119889120572119889119905120572 [Γ (119910) (119905) minus Γ (119909) (119905)]10038171003817100381710038171003817100381710038171003817 le [120576 exp(120579119905120572

120572 )


120572 )119889119904] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572

(37)

Accordingly we get

1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816120572 le 2 [120576 + max (119886 119887120591120572minus120574)120579 ] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572 (38)

Hence we conclude that

1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816120572 le 2 (120576 + 1)3 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572 (39)

The fact 2(120576 + 1)3 lt 1 proves that Γ has an unique fixedpoint 119909 in (C120572 ||120572) which is the integral solution of equation(22)

32 Stability of the Integral Solution Here we give a resultconcerning the nonlocal-condition effect on the stability ofthe integral solution

Theorem 4 Let 119909 and 119910 be solutions associated with (1199090 1199091)and (1199090 1199101) respectivelyen we have the following estimate

1003816100381610038161003816119910 minus 1199091003816100381610038161003816le 2

1 minus 2 (120576 +max (119886120591120572120572 1198871205912120572minus120574 (2120572 minus 120574))) 10038171003817100381710038171199101minus 11990911003817100381710038171003817

(40)

provided that


2 (41)


Proof We have

119910 (119905) minus 119909 (119905) = 119878 (119905120572120572 ) [1199101 minus 1199091 + 119892 (119910) minus 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )[119891(119904 119910 (119904) 119889120574119910 (119904)119889119905120574 )

minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904

119889120572119889119905120572 [119910 (119905) minus 119909 (119905)] = 119862(119905120572

120572 ) [1199101 minus 1199091 + 119892 (119910)

minus 119892 (119909)] + int1199050119904120572minus1119862(119905120572 minus 119904120572

120572 )

sdot [119891(119904 119910 (119904) 119889120574119910 (119904)119889119905120574 )

minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904

(42)

Consequently we get

1003816100381610038161003816119910 minus 1199091003816100381610038161003816 le 2(120576 +max(119886120591120572120572 1198871205912120572minus1205742120572 minus 120574)) 1003816100381610038161003816119910 minus 1199091003816100381610038161003816

+ 2 10038171003817100381710038171199101 minus 11990911003817100381710038171003817 (43)

Finally we obtain the following estimation1003816100381610038161003816119910 minus 1199091003816100381610038161003816

le 21 minus 2 (120576 +max (119886120591120572120572 1198871205912120572minus120574 (2120572 minus 120574))) 10038171003817100381710038171199101

minus 11990911003817100381710038171003817 (44)

4 Implicit Fundamental Solution inthe Case When 120574=120572

Here we give the implicit fundamental solution of (4) byusing the separating variables method To do so let 119906(119905 ℓ) =119909(119905)119910(ℓ) Then (4) becomes as follows

119886119909 (119905) 119910 (ℓ) + 119887119910 (ℓ) 119889120572119909 (119905)119889119905120572 + 119910 (ℓ) 119889120572

119889119905120572119889120572119909 (119905)119889119905120572

= 119888119909 (119905) 1198892119910 (ℓ)119889ℓ2

(45)

Then there exists a constant 1205822 such as

(119886 + 1205822) 119909 (119905) + 119887119889120572119909 (119905)119889119905120572 + 119889120572

119889119905120572119889120572119909 (119905)119889119905120572 = 0 (46)

1198881198892119910 (ℓ)119889ℓ2 = minus1205822119910 (ℓ) (47)

According to (5) and (6) the solution of (47) is given by

119910119899 (ℓ) = sin (119899ℓ) (48)

Moreover the solution of (4) can written as

119906 (119905 ℓ) = +infinsum119899=0

119909119899 (119905) 119910119899 (ℓ) (49)

where 119910119899(ℓ) = sin(119899ℓ) and 119909119899(119905) is the solution of thefollowing ordinary differential equation

(119886 + 1198881198992) 119909119899 (119905) + 119887119889120572119909119899 (119905)119889119905120572 + 119889120572119889119905120572

119889120572119909119899 (119905)119889119905120572 = 0 (50)

with the nonlocal initial conditions

119909119899 (0) = 2120587 int1205870sin (119899119904) 119906 (0 119904) 119889119904

119889120572119909119899 (0)119889119905120572 = 2120587 int1205870sin (119899119904) 120597120572119906 (0 119904)

120597119905120572 119889119904(51)

By using the fractional Laplace transform in (50) we get

119909119899 (119905) = 2120587 exp(minus119887119905120572

2120572 )int1205870cos(radic41198881198992 + 4119886 minus 1198872

2120572 119905120572)

sdot 1199090 sin (119899119904) 119889119904 + 2119887120587 exp(minus119887119905120572

2120572 )

sdot int1205870

sin ((radic41198881198992 + 4119886 minus 11988722120572) 119905120572)radic41198881198992 + 4119886 minus 1198872 1199090 sin (119899119904) 119889119904

+ 4120587 exp(minus119887119905120572

2120572 )

sdot 119901sum119894=1

int1205870

sin ((radic41198881198992 + 4119886 minus 11988722120572) 119905120572)radic41198881198992 + 4119886 minus 1198872 [1199091119901

+ 120576119894119906 (119905119894 119904)] sin (119899119904) 119889119904

(52)

Finally replacing 119909119899(119905) in (49) we get

119906 (119905 ℓ) = 2120587 exp(minus119887119905120572

2120572 ) +infinsum119899=0

int1205870cos(radic41198881198992 + 4119886 minus 1198872

2120572sdot 119905120572)1199090 sin (119899ℓ) sin (119899119904) 119889119904 + 2119887

120587 exp(minus1198871199051205722120572 )

sdot +infinsum119899=0

int1205870

sin ((radic41198881198992 + 4119886 minus 11988722120572) 119905120572)radic41198881198992 + 4119886 minus 1198872 1199090 sin (119899ℓ)

sdot sin (119899119904) 119889119904 + 4120587 exp(minus119887119905120572

2120572 )


119901sum119894=1

int1205870


+ 120576119894119906 (119905119894 119904)] sin (119899ℓ) sin (119899119904) 119889119904

(53)


Proposition 5 For 120572 = 1 and 120576 = 0 the formula(53) coincides with the fundamental solution of the classicaltelegraph equation (1)

Proof Based on the separating variables method in (1) weobtain

119906 (119905 ℓ) = +infinsum119899=0

119909119899 (119905) sin (119899ℓ) (54)

where 119909119899(119905) is the solution of the following ordinary differen-tial equation

(119886 + 1198881198992) 119909119899 (119905) + 119887d119909119899 (119905)119889119905 + 1198892119909119899 (119905)1198891199052 = 0 (55)

with the initial conditions

119909119899 (0) = 2120587 int1205870sin (119899119904) 119906 (0 119904) 119889119904

119889119909119899 (0)119889119905 = 2120587 int1205870sin (119899119904) 120597119906 (0 119904)

120597119905 119889119904(56)

By using the classical Laplace transform in (55) we get

119909119899 (119905) = 2120587exp(minus119887

2119905)

sdot int1205870cos(radic41198881198992 + 4119886 minus 1198872

2 119905) 1199090 sin (119899119904) 119889119904 + 2119887120587

sdot exp(minus1198872119905)

sdot int1205870


+ 4120587 exp(minus119887

2119905)

sdot int1205870


(57)

Replacing 119909119899(119905) in (54) we find the fundamental solution ofthe classical telegraph equation (1) as follows

119906 (119905 ℓ) = 2120587 exp(minus119887

2119905)



2 119905) 1199090 sin (119899ℓ)


2119905)


int1205870


sdot sin (119899119904) 119889119904 + 4120587exp(minus119887

2119905)


int1205870


sdot sin (119899119904) 119889119904(58)

Remark 6 If we consider the fractional derivative in Caputorsquossense the implicit fundamental solution of (4) can be writtenin terms of the Mittag-Leffler function However in our casewe have found the implicit fundamental solution in terms ofthe classical trigonometric functions

Remark 7 The integral solution does not impose any con-straint concerning the choice of the derivation parametersThis provides more freedom concerning the choice of sensi-tive parameters 120572 and 120574 in practice situations for modelingnaturel phenomena

5 Conclusion

We have studied a time-conformable fractional telegraphequation with nonlocal condition In the case when 120572 = 120574we have given the implicit fundamental solution in terms ofthe classical trigonometric functions In the general case wehave established the existence uniqueness and stability of theintegral solution

As for future work we intend to give the fundamentalsolution for all values of 120572 and 120574Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] E C Eckstein J AGoldstein andM Leggas ldquoThemathematicsof suspensions Kacwalks and asymptotic analyticityrdquoElectronicJournal of Differential Equations vol 3 pp 39ndash50 1999

[2] R C Cascaval E C Eckstein C L Frota and J A GoldsteinldquoFractional telegraph equationsrdquo Journal of Mathematical Anal-ysis and Applications vol 276 no 1 pp 145ndash159 2002

[3] J Chen F Liu and V Anh ldquoAnalytical solution for the time-fractional telegraph equation by the method of separatingvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 338 no 2 pp 1364ndash1377 2008

[4] S Das K Vishal P K Gupta and A Yildirim ldquoAn approxi-mate analytical solution of time-fractional telegraph equationrdquoApplied Mathematics and Computation vol 217 no 18 pp7405ndash7411 2011

[5] V RHosseiniWChen andZAvazzadeh ldquoNumerical solutionof fractional telegraph equation by using radial basis functionsrdquo


Engineering Analysis with Boundary Elements vol 38 no 12 pp31ndash39 2014

[6] S Kumar ldquoA new analytical modelling for fractional telegraphequation via Laplace transformrdquo Applied Mathematical Mod-elling vol 38 no 13 pp 3154ndash3163 2014

[7] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005

[8] V K Srivastava M K Awasthi and S Kumar ldquoAnalyticalapproximations of two and three dimensional time-fractionaltelegraphic equation by reduced differential transformmethodrdquoEgyptian Journal of Basic and Applied Sciences vol 1 no 1 pp60ndash66 2014

[9] A A Kilbas H M Srivastava and J J Trujillo eory andApplications of Fractional Differential Equations North HollandMathematics Studies 204 Elsevier New York NY USA 2006

[10] K S Miller An Introduction to the Fractional Calculus andFractional Differential Equations John Wiley and Sons 1993

[11] K B Oldham and J Spaniere Fractional Calculus AcademicPress New York NY USA 1974

[12] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives eory and Applications Gordon andBreach Yverdon Switzerland 1993

[14] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[15] W S Chung ldquoFractional Newton mechanics with conformablefractional derivativerdquo Journal of Computational and AppliedMathematics vol 290 pp 150ndash158 2015

[16] L Martınez J Rosales C Carreno and J Lozano ldquoElectricalcircuits described by fractional conformable derivativerdquo Inter-national Journal of Circuit eory and Applications vol 46 no5 pp 1091ndash1100 2018

[17] H W Zhou S Yang and S Q Zhang ldquoConformable deriva-tive approach to anomalous diffusionrdquo Physica A StatisticalMechanics and its Applications vol 491 pp 1001ndash1013 2018

[18] L Byszewski ldquoTheorems about the existence and uniqueness ofsolutions of a semilinear evolution nonlocal Cauchy problemrdquoJournal of Mathematical Analysis and Applications vol 162 no2 pp 494ndash505 1991

[19] K Deng ldquoExponential decay of solutions of semilinearparabolic equations with nonlocal initial conditionsrdquo Journal ofMathematical Analysis and Applications vol 179 no 2 pp 630ndash637 1993

[20] W E Olmstead and C A Roberts ldquoThe one-dimensional heatequation with a nonlocal initial conditionrdquo Applied Mathemat-ics Letters vol 10 no 3 pp 89ndash94 1997

[21] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015

[22] C C Travis and G F Webb ldquoCosine families and abstract non-linear second order differential equationsrdquo Acta MathematicaHungarica vol 32 no 1-2 pp 75ndash96 1978

[23] M E Hernandez ldquoExistence of solutions to a second order par-tial differential equation with nonlocal conditionsrdquo ElectronicJournal of Differential Equations vol 2003 pp 1ndash10 2003

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We associate to (4) the boundary and the nonlocal initialconditions

119906 (119905 0) = 0 119905 isin [0 120591] (5)

119906 (119905 120587) = 0 119905 isin [0 120591] (6)

119906 (0 ℓ) = 1199090 ℓ isin [0 120587] (7)

120597120572119906 (0 ℓ)120597119905120572 = 1199091 +

119901sum119894=1

120576119894119906 (119905119894 ℓ) ℓ isin [0 120587] (8)

where 119901 is a nonnull fixed integer and 0 lt 1199051 lt sdot sdot sdot lt 119905119901 lt 120591The quantities 1199090 1199091 and 1205761 120576119901 are physical measuresThe condition appearing in (8) means the nonlocal condition[18] For physical interpretations of this condition we referto works [19 20] For example in [19] the author used anonlocal condition of the form (8) to describe the diffusionphenomenon of a small amount of gas in a transparent tube

Wenote that it is not easy to find the fundamental solutionof (4) by using the Laplace transformmethod if120572 = 120574 For thisreasonwewill propose the integral solution concept based onan operator theory approach When 120572 = 120574 we investigate animplicit fundamental solution

The content of this paper is organized as follows InSection 2 we recall some needed results of the conformablefractional derivative and cosine family of linear operators InSection 3 we prove the existence uniqueness and stabilityof the integral solution of (4) by using of an operator theoryapproach Section 4 is devoted to an implicit fundamentalsolution of (4) in terms of the classical trigonometric func-tions

2 Preliminaries

We start recalling some concepts on the conformable frac-tional calculus [14]

The conformable fractional derivative of a function 119909 oforder 120572 at 119905 gt 0 is defined by the following limit

119889120572119909 (119905)119889119905120572 = lim

120576997888rarr0

119909 (119905 + 1205761199051minus120572) minus 119909 (119905)120576 (9)

When this limit exists we say that 119909 is (120572)-differentiable at 119905If 119909 is (120572)-differentiable at some 119905 sim 0 and the limit

lim119905997888rarr0+(119889120572119909(119905)119889119905120572) existsThen we define the conformablefractional derivative of 119909 at 0 by

119889120572119909 (0)119889119905120572 = lim

119905997888rarr0+

119889120572119909 (119905)119889119905120572 (10)

The (120572)-fractional integral of a function 119909 is defined by

119868120572 (119909) (119905) = int1199050119904120572minus1119909 (119904) 119889119904 (11)

If119909 is a continuous function in the domain of 119868120572 thenwe have119889120572 (119868120572 (119909) (119905))

119889119905120572 = 119909 (119905) (12)

According to [21] if 119909 is differentiable then we have

119868120572 (119889120572119909119889119905120572 ) (119905) = 119909 (119905) minus 119909 (0) (13)

We remark that the classical Laplace transform is not com-patible with the conformable fractional derivative For thisreason the adapted transform is defined as [21]

L120572 (119909 (119905)) (120582) fl int+infin0

119905120572minus1119890minus120582119905120572120572119909 (119905) 119889119905 (14)

The above transformation is called fractional Laplace trans-form of order 120572 of the function 119909

If 119909 is differentiable then the action of the fractionalLaplace transform on the conformable fractional derivativeis given as follows

L120572 (119889120572119909 (119905)119889119905120572 ) (120582) = 120582L120572 (119909 (119905)) (120582) minus 119909 (0) (15)

Now we introduce some results concerning the cosine familytheory [22]

A one-parameter family (119862(119905))119905isinR of bounded linearoperators on the Banach space 119883 is called a strongly contin-uous cosine family if and only if

(1) 119862(0) = 119868 (I is the identity operator)(2) 119862(119904 + 119905) + 119862(119904 minus 119905) = 2119862(119904)119862(119905) for all 119905 119904 isin R(3) 119905 997891997888rarr 119862(119905)119909 is continuous for each fixed 119909 isin 119883

We define also the sine family by

119878 (119905) fl int1199050119862 (119904) 119889119904 (16)

The infinitesimal generator119860 of a strongly continuous cosinefamily ((119862(119905))119905isinR (119878(119905))119905isinR) on119883 is defined by

119863(119860) = 119909 isin 119883 119905 997891997888rarr 119862 (119905)sdot 119909 is a twice continuously differentiable function

119860 = 1198892119862 (119905)1198891199052 rfloor

119905=0

(17)

If 119860 is the infinitesimal generator of a strongly cosine family((119862(119905))119905isinR (119878(119905))119905isinR) on119883 then there exists a constant 120596 ge 0such that for all 120582 with 119877119890(120582) gt 120596 we have

1205822 isin 120588 (119860) (120588 (119860) is the resolvent set of 119860)

120582 (1205822119868 minus 119860)minus1 = int+infin0

119890minus120582119905119862 (119905) 119889119905(1205822119868 minus 119860)minus1 = int+infin

0119890minus120582119905119878 (119905) 119889119905

(18)

where 119877119890(120582) is the real part of the complex number 120582



119886 = 119877119866119871119862

119887 = 119877119871 + 119866

119862119888 = 1

119871119862and 120576 = 119901sum

119894=1

10038161003816100381610038161205761198941003816100381610038161003816

(19)



119860 = 1198881205972 ()120597ℓ2and 119863(119860)

= 119906 isin 1198672 ((0 120587) 119889ℓ) 119906 (0) = 119906 (120587) = 0 (20)



119909 (119905) (ℓ) = 119906 (119905 ℓ) 119891 (119905 119909 (119905) 119889120574119909 (119905)

119889119905120574 ) = minus119886119909 (119905) minus 119887119889120574119909 (119905)119889119905120574

119892 (119909) = 119901sum119894=1

120576119894119909 (119905119894) (21)


119889120572119889119905120572

119889120572119909 (119905)119889119905120572 = 119860119909 (119905) + 119891(119905 119909 (119905) 119889120574119909 (119905)

119889119905120574 ) (22)

119909 (0) = 1199090 (23)

119889120572119909 (0)119889119905120572 = 1199091 + 119892 (119909) (24)



119909 (119905)= 119862(119905120572

120572 )1199090 + 119878(119905120572120572 ) [1199091 + 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )119889119904

(25)



119909 (119905)= 119862(119905120572

120572 )1199090 + 119878(119905120572120572 ) [1199091 + 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )119889119904119905 isin [0 120591]

(26)



2 (27)


Γ (119909) (119905)= 119862(119905120572

120572 )1199090 + 119878(119905120572120572 ) [1199091 + 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )119889119904

(28)


Γ (119910) (119905) minus Γ (119909) (119905) = 119878 (119905120572120572 ) [119892 (119910) minus 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )[119891(119904 119910 (119904) 119889120574119910 (119904)119889119905120574 )

minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904

119889120572119889119905120572 [Γ (119910) (119905) minus Γ (119909) (119905)] = 119862(119905120572

120572 ) [119892 (119910) minus 119892 (119909)]

+ int1199050119904120572minus1119862(119905120572 minus 119904120572

120572 )[119891(119904 119910 (119904) 119889120574119910 (119904)119889119905120574 )

minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904

(29)



1003817100381710038171003817Γ (119910) (119905) minus Γ (119909) (119905)1003817100381710038171003817 +10038171003817100381710038171003817100381710038171003817119889120572119889119905120572 [Γ (119910) (119905) minus Γ (119909) (119905)]10038171003817100381710038171003817100381710038171003817


(30)

Then we get

1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816le 2 [120576 +max(119886120591120572





120576 lt 12 (32)


Γ (119909) (119905)= 119862(119905120572

120572 )1199090 + 119878(119905120572120572 ) [1199091 + 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )119889119904

119905 isin [0 120591]

(33)


|119909|120572 =10038161003816100381610038161003816100381610038161003816exp(minus120579 ()120572

120572 )11990910038161003816100381610038161003816100381610038161003816 (34)

where

120579 = 3max (119886 119887120591120572minus120574)1 minus 2120576 (35)


120572 ) [119892 (119910) minus 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )[119891(119904 119910 (119904) 119889120574119910 (119904)119889119905120574 )

minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904

(36)

Therefore we obtain

1003817100381710038171003817Γ (119910) (t) minus Γ (119909) (119905)1003817100381710038171003817 le [120576 exp(120579119905120572120572 )



120572 )


120572 )119889119904] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572

(37)

Accordingly we get



1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816120572 le 2 (120576 + 1)3 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572 (39)




1003816100381610038161003816119910 minus 1199091003816100381610038161003816le 2


(40)

provided that


2 (41)


Proof We have


+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )[119891(119904 119910 (119904) 119889120574119910 (119904)119889119905120574 )

minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904

119889120572119889119905120572 [119910 (119905) minus 119909 (119905)] = 119862(119905120572

120572 ) [1199101 minus 1199091 + 119892 (119910)


120572 )

sdot [119891(119904 119910 (119904) 119889120574119910 (119904)119889119905120574 )

minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904

(42)

Consequently we get


+ 2 10038171003817100381710038171199101 minus 11990911003817100381710038171003817 (43)



minus 11990911003817100381710038171003817 (44)



119886119909 (119905) 119910 (ℓ) + 119887119910 (ℓ) 119889120572119909 (119905)119889119905120572 + 119910 (ℓ) 119889120572

119889119905120572119889120572119909 (119905)119889119905120572

= 119888119909 (119905) 1198892119910 (ℓ)119889ℓ2

(45)


(119886 + 1205822) 119909 (119905) + 119887119889120572119909 (119905)119889119905120572 + 119889120572

119889119905120572119889120572119909 (119905)119889119905120572 = 0 (46)

1198881198892119910 (ℓ)119889ℓ2 = minus1205822119910 (ℓ) (47)


119910119899 (ℓ) = sin (119899ℓ) (48)


119906 (119905 ℓ) = +infinsum119899=0

119909119899 (119905) 119910119899 (ℓ) (49)


(119886 + 1198881198992) 119909119899 (119905) + 119887119889120572119909119899 (119905)119889119905120572 + 119889120572119889119905120572

119889120572119909119899 (119905)119889119905120572 = 0 (50)


119909119899 (0) = 2120587 int1205870sin (119899119904) 119906 (0 119904) 119889119904

119889120572119909119899 (0)119889119905120572 = 2120587 int1205870sin (119899119904) 120597120572119906 (0 119904)

120597119905120572 119889119904(51)


119909119899 (119905) = 2120587 exp(minus119887119905120572


2120572 119905120572)


2120572 )

sdot int1205870


+ 4120587 exp(minus119887119905120572

2120572 )

sdot 119901sum119894=1

int1205870


+ 120576119894119906 (119905119894 119904)] sin (119899119904) 119889119904

(52)


119906 (119905 ℓ) = 2120587 exp(minus119887119905120572

2120572 ) +infinsum119899=0


2120572sdot 119905120572)1199090 sin (119899ℓ) sin (119899119904) 119889119904 + 2119887

120587 exp(minus1198871199051205722120572 )


int1205870



2120572 )


119901sum119894=1

int1205870


+ 120576119894119906 (119905119894 119904)] sin (119899ℓ) sin (119899119904) 119889119904

(53)




119906 (119905 ℓ) = +infinsum119899=0

119909119899 (119905) sin (119899ℓ) (54)


(119886 + 1198881198992) 119909119899 (119905) + 119887d119909119899 (119905)119889119905 + 1198892119909119899 (119905)1198891199052 = 0 (55)


119909119899 (0) = 2120587 int1205870sin (119899119904) 119906 (0 119904) 119889119904

119889119909119899 (0)119889119905 = 2120587 int1205870sin (119899119904) 120597119906 (0 119904)

120597119905 119889119904(56)


119909119899 (119905) = 2120587exp(minus119887

2119905)


2 119905) 1199090 sin (119899119904) 119889119904 + 2119887120587


sdot int1205870


+ 4120587 exp(minus119887

2119905)

sdot int1205870


(57)


119906 (119905 ℓ) = 2120587 exp(minus119887

2119905)



2 119905) 1199090 sin (119899ℓ)


2119905)


int1205870



2119905)


int1205870


sdot sin (119899119904) 119889119904(58)



5 Conclusion






References

































Journal of












Journal of







Volume 2018






Volume 2018










119886 = 119877119866119871119862

119887 = 119877119871 + 119866

119862119888 = 1

119871119862and 120576 = 119901sum

119894=1

10038161003816100381610038161205761198941003816100381610038161003816

(19)



119860 = 1198881205972 ()120597ℓ2and 119863(119860)

= 119906 isin 1198672 ((0 120587) 119889ℓ) 119906 (0) = 119906 (120587) = 0 (20)



119909 (119905) (ℓ) = 119906 (119905 ℓ) 119891 (119905 119909 (119905) 119889120574119909 (119905)

119889119905120574 ) = minus119886119909 (119905) minus 119887119889120574119909 (119905)119889119905120574

119892 (119909) = 119901sum119894=1

120576119894119909 (119905119894) (21)


119889120572119889119905120572

119889120572119909 (119905)119889119905120572 = 119860119909 (119905) + 119891(119905 119909 (119905) 119889120574119909 (119905)

119889119905120574 ) (22)

119909 (0) = 1199090 (23)

119889120572119909 (0)119889119905120572 = 1199091 + 119892 (119909) (24)



119909 (119905)= 119862(119905120572

120572 )1199090 + 119878(119905120572120572 ) [1199091 + 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )119889119904

(25)



119909 (119905)= 119862(119905120572

120572 )1199090 + 119878(119905120572120572 ) [1199091 + 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )119889119904119905 isin [0 120591]

(26)



2 (27)


Γ (119909) (119905)= 119862(119905120572

120572 )1199090 + 119878(119905120572120572 ) [1199091 + 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )119889119904

(28)


Γ (119910) (119905) minus Γ (119909) (119905) = 119878 (119905120572120572 ) [119892 (119910) minus 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )[119891(119904 119910 (119904) 119889120574119910 (119904)119889119905120574 )

minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904

119889120572119889119905120572 [Γ (119910) (119905) minus Γ (119909) (119905)] = 119862(119905120572

120572 ) [119892 (119910) minus 119892 (119909)]

+ int1199050119904120572minus1119862(119905120572 minus 119904120572

120572 )[119891(119904 119910 (119904) 119889120574119910 (119904)119889119905120574 )

minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904

(29)



1003817100381710038171003817Γ (119910) (119905) minus Γ (119909) (119905)1003817100381710038171003817 +10038171003817100381710038171003817100381710038171003817119889120572119889119905120572 [Γ (119910) (119905) minus Γ (119909) (119905)]10038171003817100381710038171003817100381710038171003817


(30)

Then we get

1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816le 2 [120576 +max(119886120591120572





120576 lt 12 (32)


Γ (119909) (119905)= 119862(119905120572

120572 )1199090 + 119878(119905120572120572 ) [1199091 + 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )119889119904

119905 isin [0 120591]

(33)


|119909|120572 =10038161003816100381610038161003816100381610038161003816exp(minus120579 ()120572

120572 )11990910038161003816100381610038161003816100381610038161003816 (34)

where

120579 = 3max (119886 119887120591120572minus120574)1 minus 2120576 (35)


120572 ) [119892 (119910) minus 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )[119891(119904 119910 (119904) 119889120574119910 (119904)119889119905120574 )

minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904

(36)

Therefore we obtain

1003817100381710038171003817Γ (119910) (t) minus Γ (119909) (119905)1003817100381710038171003817 le [120576 exp(120579119905120572120572 )



120572 )


120572 )119889119904] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572

(37)

Accordingly we get



1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816120572 le 2 (120576 + 1)3 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572 (39)




1003816100381610038161003816119910 minus 1199091003816100381610038161003816le 2


(40)

provided that


2 (41)


Proof We have


+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )[119891(119904 119910 (119904) 119889120574119910 (119904)119889119905120574 )

minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904

119889120572119889119905120572 [119910 (119905) minus 119909 (119905)] = 119862(119905120572

120572 ) [1199101 minus 1199091 + 119892 (119910)


120572 )

sdot [119891(119904 119910 (119904) 119889120574119910 (119904)119889119905120574 )

minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904

(42)

Consequently we get


+ 2 10038171003817100381710038171199101 minus 11990911003817100381710038171003817 (43)



minus 11990911003817100381710038171003817 (44)



119886119909 (119905) 119910 (ℓ) + 119887119910 (ℓ) 119889120572119909 (119905)119889119905120572 + 119910 (ℓ) 119889120572

119889119905120572119889120572119909 (119905)119889119905120572

= 119888119909 (119905) 1198892119910 (ℓ)119889ℓ2

(45)


(119886 + 1205822) 119909 (119905) + 119887119889120572119909 (119905)119889119905120572 + 119889120572

119889119905120572119889120572119909 (119905)119889119905120572 = 0 (46)

1198881198892119910 (ℓ)119889ℓ2 = minus1205822119910 (ℓ) (47)


119910119899 (ℓ) = sin (119899ℓ) (48)


119906 (119905 ℓ) = +infinsum119899=0

119909119899 (119905) 119910119899 (ℓ) (49)


(119886 + 1198881198992) 119909119899 (119905) + 119887119889120572119909119899 (119905)119889119905120572 + 119889120572119889119905120572

119889120572119909119899 (119905)119889119905120572 = 0 (50)


119909119899 (0) = 2120587 int1205870sin (119899119904) 119906 (0 119904) 119889119904

119889120572119909119899 (0)119889119905120572 = 2120587 int1205870sin (119899119904) 120597120572119906 (0 119904)

120597119905120572 119889119904(51)


119909119899 (119905) = 2120587 exp(minus119887119905120572


2120572 119905120572)


2120572 )

sdot int1205870


+ 4120587 exp(minus119887119905120572

2120572 )

sdot 119901sum119894=1

int1205870


+ 120576119894119906 (119905119894 119904)] sin (119899119904) 119889119904

(52)


119906 (119905 ℓ) = 2120587 exp(minus119887119905120572

2120572 ) +infinsum119899=0


2120572sdot 119905120572)1199090 sin (119899ℓ) sin (119899119904) 119889119904 + 2119887

120587 exp(minus1198871199051205722120572 )


int1205870



2120572 )


119901sum119894=1

int1205870


+ 120576119894119906 (119905119894 119904)] sin (119899ℓ) sin (119899119904) 119889119904

(53)




119906 (119905 ℓ) = +infinsum119899=0

119909119899 (119905) sin (119899ℓ) (54)


(119886 + 1198881198992) 119909119899 (119905) + 119887d119909119899 (119905)119889119905 + 1198892119909119899 (119905)1198891199052 = 0 (55)


119909119899 (0) = 2120587 int1205870sin (119899119904) 119906 (0 119904) 119889119904

119889119909119899 (0)119889119905 = 2120587 int1205870sin (119899119904) 120597119906 (0 119904)

120597119905 119889119904(56)


119909119899 (119905) = 2120587exp(minus119887

2119905)


2 119905) 1199090 sin (119899119904) 119889119904 + 2119887120587


sdot int1205870


+ 4120587 exp(minus119887

2119905)

sdot int1205870


(57)


119906 (119905 ℓ) = 2120587 exp(minus119887

2119905)



2 119905) 1199090 sin (119899ℓ)


2119905)


int1205870



2119905)


int1205870


sdot sin (119899119904) 119889119904(58)



5 Conclusion






References

































Journal of












Journal of







Volume 2018






Volume 2018










1003817100381710038171003817Γ (119910) (119905) minus Γ (119909) (119905)1003817100381710038171003817 +10038171003817100381710038171003817100381710038171003817119889120572119889119905120572 [Γ (119910) (119905) minus Γ (119909) (119905)]10038171003817100381710038171003817100381710038171003817


(30)

Then we get

1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816le 2 [120576 +max(119886120591120572





120576 lt 12 (32)


Γ (119909) (119905)= 119862(119905120572

120572 )1199090 + 119878(119905120572120572 ) [1199091 + 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )119889119904

119905 isin [0 120591]

(33)


|119909|120572 =10038161003816100381610038161003816100381610038161003816exp(minus120579 ()120572

120572 )11990910038161003816100381610038161003816100381610038161003816 (34)

where

120579 = 3max (119886 119887120591120572minus120574)1 minus 2120576 (35)


120572 ) [119892 (119910) minus 119892 (119909)]

+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )[119891(119904 119910 (119904) 119889120574119910 (119904)119889119905120574 )

minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904

(36)

Therefore we obtain

1003817100381710038171003817Γ (119910) (t) minus Γ (119909) (119905)1003817100381710038171003817 le [120576 exp(120579119905120572120572 )



120572 )


120572 )119889119904] 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572

(37)

Accordingly we get



1003816100381610038161003816Γ (119910) minus Γ (119909)1003816100381610038161003816120572 le 2 (120576 + 1)3 1003816100381610038161003816119910 minus 1199091003816100381610038161003816120572 (39)




1003816100381610038161003816119910 minus 1199091003816100381610038161003816le 2


(40)

provided that


2 (41)


Proof We have


+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )[119891(119904 119910 (119904) 119889120574119910 (119904)119889119905120574 )

minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904

119889120572119889119905120572 [119910 (119905) minus 119909 (119905)] = 119862(119905120572

120572 ) [1199101 minus 1199091 + 119892 (119910)


120572 )

sdot [119891(119904 119910 (119904) 119889120574119910 (119904)119889119905120574 )

minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904

(42)

Consequently we get


+ 2 10038171003817100381710038171199101 minus 11990911003817100381710038171003817 (43)



minus 11990911003817100381710038171003817 (44)



119886119909 (119905) 119910 (ℓ) + 119887119910 (ℓ) 119889120572119909 (119905)119889119905120572 + 119910 (ℓ) 119889120572

119889119905120572119889120572119909 (119905)119889119905120572

= 119888119909 (119905) 1198892119910 (ℓ)119889ℓ2

(45)


(119886 + 1205822) 119909 (119905) + 119887119889120572119909 (119905)119889119905120572 + 119889120572

119889119905120572119889120572119909 (119905)119889119905120572 = 0 (46)

1198881198892119910 (ℓ)119889ℓ2 = minus1205822119910 (ℓ) (47)


119910119899 (ℓ) = sin (119899ℓ) (48)


119906 (119905 ℓ) = +infinsum119899=0

119909119899 (119905) 119910119899 (ℓ) (49)


(119886 + 1198881198992) 119909119899 (119905) + 119887119889120572119909119899 (119905)119889119905120572 + 119889120572119889119905120572

119889120572119909119899 (119905)119889119905120572 = 0 (50)


119909119899 (0) = 2120587 int1205870sin (119899119904) 119906 (0 119904) 119889119904

119889120572119909119899 (0)119889119905120572 = 2120587 int1205870sin (119899119904) 120597120572119906 (0 119904)

120597119905120572 119889119904(51)


119909119899 (119905) = 2120587 exp(minus119887119905120572


2120572 119905120572)


2120572 )

sdot int1205870


+ 4120587 exp(minus119887119905120572

2120572 )

sdot 119901sum119894=1

int1205870


+ 120576119894119906 (119905119894 119904)] sin (119899119904) 119889119904

(52)


119906 (119905 ℓ) = 2120587 exp(minus119887119905120572

2120572 ) +infinsum119899=0


2120572sdot 119905120572)1199090 sin (119899ℓ) sin (119899119904) 119889119904 + 2119887

120587 exp(minus1198871199051205722120572 )


int1205870



2120572 )


119901sum119894=1

int1205870


+ 120576119894119906 (119905119894 119904)] sin (119899ℓ) sin (119899119904) 119889119904

(53)




119906 (119905 ℓ) = +infinsum119899=0

119909119899 (119905) sin (119899ℓ) (54)


(119886 + 1198881198992) 119909119899 (119905) + 119887d119909119899 (119905)119889119905 + 1198892119909119899 (119905)1198891199052 = 0 (55)


119909119899 (0) = 2120587 int1205870sin (119899119904) 119906 (0 119904) 119889119904

119889119909119899 (0)119889119905 = 2120587 int1205870sin (119899119904) 120597119906 (0 119904)

120597119905 119889119904(56)


119909119899 (119905) = 2120587exp(minus119887

2119905)


2 119905) 1199090 sin (119899119904) 119889119904 + 2119887120587


sdot int1205870


+ 4120587 exp(minus119887

2119905)

sdot int1205870


(57)


119906 (119905 ℓ) = 2120587 exp(minus119887

2119905)



2 119905) 1199090 sin (119899ℓ)


2119905)


int1205870



2119905)


int1205870


sdot sin (119899119904) 119889119904(58)



5 Conclusion






References

































Journal of












Journal of







Volume 2018






Volume 2018









Proof We have


+ int1199050119904120572minus1119878 (119905120572 minus 119904120572

120572 )[119891(119904 119910 (119904) 119889120574119910 (119904)119889119905120574 )

minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904

119889120572119889119905120572 [119910 (119905) minus 119909 (119905)] = 119862(119905120572

120572 ) [1199101 minus 1199091 + 119892 (119910)


120572 )

sdot [119891(119904 119910 (119904) 119889120574119910 (119904)119889119905120574 )

minus 119891(119904 119909 (119904) 119889120574119909 (119904)119889119905120574 )]119889119904

(42)

Consequently we get


+ 2 10038171003817100381710038171199101 minus 11990911003817100381710038171003817 (43)



minus 11990911003817100381710038171003817 (44)



119886119909 (119905) 119910 (ℓ) + 119887119910 (ℓ) 119889120572119909 (119905)119889119905120572 + 119910 (ℓ) 119889120572

119889119905120572119889120572119909 (119905)119889119905120572

= 119888119909 (119905) 1198892119910 (ℓ)119889ℓ2

(45)


(119886 + 1205822) 119909 (119905) + 119887119889120572119909 (119905)119889119905120572 + 119889120572

119889119905120572119889120572119909 (119905)119889119905120572 = 0 (46)

1198881198892119910 (ℓ)119889ℓ2 = minus1205822119910 (ℓ) (47)


119910119899 (ℓ) = sin (119899ℓ) (48)


119906 (119905 ℓ) = +infinsum119899=0

119909119899 (119905) 119910119899 (ℓ) (49)


(119886 + 1198881198992) 119909119899 (119905) + 119887119889120572119909119899 (119905)119889119905120572 + 119889120572119889119905120572

119889120572119909119899 (119905)119889119905120572 = 0 (50)


119909119899 (0) = 2120587 int1205870sin (119899119904) 119906 (0 119904) 119889119904

119889120572119909119899 (0)119889119905120572 = 2120587 int1205870sin (119899119904) 120597120572119906 (0 119904)

120597119905120572 119889119904(51)


119909119899 (119905) = 2120587 exp(minus119887119905120572


2120572 119905120572)


2120572 )

sdot int1205870


+ 4120587 exp(minus119887119905120572

2120572 )

sdot 119901sum119894=1

int1205870


+ 120576119894119906 (119905119894 119904)] sin (119899119904) 119889119904

(52)


119906 (119905 ℓ) = 2120587 exp(minus119887119905120572

2120572 ) +infinsum119899=0


2120572sdot 119905120572)1199090 sin (119899ℓ) sin (119899119904) 119889119904 + 2119887

120587 exp(minus1198871199051205722120572 )


int1205870



2120572 )


119901sum119894=1

int1205870


+ 120576119894119906 (119905119894 119904)] sin (119899ℓ) sin (119899119904) 119889119904

(53)




119906 (119905 ℓ) = +infinsum119899=0

119909119899 (119905) sin (119899ℓ) (54)


(119886 + 1198881198992) 119909119899 (119905) + 119887d119909119899 (119905)119889119905 + 1198892119909119899 (119905)1198891199052 = 0 (55)


119909119899 (0) = 2120587 int1205870sin (119899119904) 119906 (0 119904) 119889119904

119889119909119899 (0)119889119905 = 2120587 int1205870sin (119899119904) 120597119906 (0 119904)

120597119905 119889119904(56)


119909119899 (119905) = 2120587exp(minus119887

2119905)


2 119905) 1199090 sin (119899119904) 119889119904 + 2119887120587


sdot int1205870


+ 4120587 exp(minus119887

2119905)

sdot int1205870


(57)


119906 (119905 ℓ) = 2120587 exp(minus119887

2119905)



2 119905) 1199090 sin (119899ℓ)


2119905)


int1205870



2119905)


int1205870


sdot sin (119899119904) 119889119904(58)



5 Conclusion






References

































Journal of












Journal of







Volume 2018






Volume 2018











119906 (119905 ℓ) = +infinsum119899=0

119909119899 (119905) sin (119899ℓ) (54)


(119886 + 1198881198992) 119909119899 (119905) + 119887d119909119899 (119905)119889119905 + 1198892119909119899 (119905)1198891199052 = 0 (55)


119909119899 (0) = 2120587 int1205870sin (119899119904) 119906 (0 119904) 119889119904

119889119909119899 (0)119889119905 = 2120587 int1205870sin (119899119904) 120597119906 (0 119904)

120597119905 119889119904(56)


119909119899 (119905) = 2120587exp(minus119887

2119905)


2 119905) 1199090 sin (119899119904) 119889119904 + 2119887120587


sdot int1205870


+ 4120587 exp(minus119887

2119905)

sdot int1205870


(57)


119906 (119905 ℓ) = 2120587 exp(minus119887

2119905)



2 119905) 1199090 sin (119899ℓ)


2119905)


int1205870



2119905)


int1205870


sdot sin (119899119904) 119889119904(58)



5 Conclusion






References

































Journal of












Journal of







Volume 2018






Volume 2018



































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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Nonlocal Telegraph Equation in Frame of the …downloads.hindawi.com/journals/amp/2019/7528937.pdfAdvancesinMathematicalPhysics Proposition . Fr =1and =0the formula coincideswiththefundamentalsolutionoftheclassical