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Full Terms & Conditions of access and use can be found at

http://www.tandfonline.com/action/journalInformation?journalCode=gitr20

Download by: [Gustavo Dorrego] Date: 10 October 2017, At: 03:43

Integral Transforms and Special Functions

ISSN: 1065-2469 (Print) 1476-8291 (Online) Journal homepage: http://www.tandfonline.com/loi/gitr20

The Mittag–Leffler function and its application to

the ultra-hyperbolic time-fractional diffusion-wave

equation

Gustavo A. Dorrego

To cite this article: Gustavo A. Dorrego (2016) The Mittag–Leffler function and its application to the ultra-hyperbolic time-fractional diffusion-wave equation, Integral Transforms and Special Functions, 27:5, 392-404, DOI: 10.1080/10652469.2016.1144185

To link to this article: http://dx.doi.org/10.1080/10652469.2016.1144185

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VOL. 27, NO. 5, 392–404

http://dx.doi.org/10.1080/10652469.2016.1144185

The Mittag–Leffler function and its application to the

ultra-hyperbolic time-fractional diffusion-wave equation

Gustavo A. Dorregoa

aDepartment of Mathematics, Faculty of Exacts Sciences, Northeast National University, Corrientes, Argentina

ABSTRACT

In this paper we study an n-dimensional generalization of time-fractional diffusion-wave equation, where the Laplacian operator is replaced by the ultra-hyperbolic operator and the time-fractional derivative is taken in the Hilfer sense. The analytical solution is obtained in terms of the Fox’s H-function, for which the inverse Fourier transform of a Mittag–Leffler-type function that contains in its argument a positive-definite quadratic form is calculated.

ARTICLE HISTORY

Received 9 October 2015 Accepted 17 January 2016

KEYWORDS

Fractional differential equation; Hilfer fractional derivative; Caputo fractional derivative;

Riemann–Liouville fractional derivative;

Mittag–Leffler-type function; Fox’s H-function; integrals transforms; ultra-hyperbolic operator

AMS SUBJECT CLASSIFICATION

26A33; 33E12; 33E20; 35R11

1. Introduction

The diffusion-wave equation has been studied by many authors in the fractional cal-culus context since it was observed that it could interpolate between heat and waves diffusion processes. Pioneering work on this type of equation may be mentioned: Fujita,[1] Nigmatullin,[2] Wyss,[3] Schnaider and Wyss [4] and Mainardi.[5] Fundamen-tally, according author’s knowledge, the generalizations of the heat and wave equations have been studied in which the time derivative has been replaced by a fractional deriva-tive of real order following Riemann–Liouville definitions and Caputo definition,[6,7] or the Hilfer definition, which contains as a particular case both.[8,9] Also one-dimensional generalizations have been studied in which the ordinary derivative in the space variable has been replaced by the Riesz derivative (see, e.g. [10]). Then-dimensional generaliza-tions have been proposed considering fractional power of the Laplace operator on the space variable.[11] Othern-dimensional generalization, where the Laplacian operator was replaced by another of elliptic type,[12,13] were also proposed.

On the other hand, in 2003, Nonlaopon and Kananthai [14] proposed and studied ann -dimensional generalization (but not in the fractional calculus context) of the heat equation in which the Laplacian operator was replaced by the ultra-hyperbolic operator. Following

CONTACT Gustavo A. Dorrego gusad82@gmail.com © 2016 Informa UK Limited, trading as Taylor & Francis Group

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the same idea, Sansanit and Kananthai [15] proposed a generalization of the wave equation. Motivated by works mentioned in the previous paragraphs the main objective of this paper is to propose and to solve a Cauchy problem, that contains, as particular case, the equations studied in [14,15]. For this purpose, we replace the ordinary time derivative by a fractional derivative of orderα(1< α <2) and typer(0≤r≤1) in the Hilfer sense and maintain the ultra-hyperbolic operator in the space variable. The objective of using the Hilfer deriva-tive is to include two Cauchy problems that could arise separately and where the ordinary time derivative is replaced by Riemann–Liouville and by Caputo fractional derivatives. To solve the equation, we follow the technique developed by Yakubovich and Luchko , [16] which consists in applying Laplace and Fourier transform together with the Mellin trans-form. This technique has proven a powerful tool for solving fractional partial differential equations, as seen for example in [17–20].

The paper is organized as follows. Section2provides the definitions of differential and integral operators of non-integer order, Mittag–Leffler function, the Laplace transform, the Fourier transform and Mellin transform. In Section3the diffusion-wave equation is solved and inverse Fourier transform of a Mittag–Leffler-type function which contains in its argument a positive-definite quadratic form is calculated. Then several special cases are analysed.

2. Preliminary results

2.1. Mittag–Leffler function

It is known the distinguished role played by the Mittag–Leffler function in solving differen-tial equations of non-integer order. This function is that a generalization of the exponendifferen-tial function was introduced by the Swedish Mathematician G. Mittag–Leffler in 1903 and is given by

Eα(z)= ∞

n=0

zn

(αn+1), α,z∈C, (α) >0. (2.1)

In 1905, Wiman studied the following generalization to two parameters of (2.1) :

Eα,β(z)=

n=0

zn

(αn+β), α,β,z∈C, (α) >0, (β) >0. (2.2)

The Mittag–Leffler function has been studied by many authors who have proposed and studied various generalizations and applications. A very interesting work that meets many results about this function is due to Haubold et al .[21]

2.2. Fox’s H-function.

The Fox’s H-function is another of the so-called special functions of the fractional calculus and contains as particular case the Mittag–Leffler function. The H-function was intro-duced by Fox [22] as generalizations of the Meijer function. Here we adopt the definition and properties mentioned in [23] with minimal modifications regarding notation.

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The H-function is defined by means of a Mellin–Barnes-type integral in the following manner :

Hpm,q,n

z

(a1,A1),. . .,(ap,Ap)

(b1,B1),. . .,(bq,Bq)

=Hmp,q,n

z((ap,Ap) bq,Bq)

= 1

2πi

C(s)z

−sds, (2.3)

where the integrand is

(s)=

m

j=1(bj+Bjs)nk=1(1−akAks)

p

k=n+1(ak+Aks)

q

j=m+1(1−bjBjs)

, (2.4)

wherei=(−1)1/2,z=0, andzs=exp[−s{ln|z| +i argz}], where ln|z|represents the natural logarithm of|z|and argzis not necessarily the principal argument.

In (2.4), an empty product is always interpreted as unity; m,n,p,q∈N0, with 0≤np, 1≤mq,Ak,Bj∈R+,ak,bj ∈RorC,k=1,. . .,p;j=1,. . .,q.

The contourCstarting at the pointp−i∞and going top+i∞wherep∈Rsuch that all the poles of(bj+Bjs)(j=1,. . .,m), are separated from those of(1−akAks)

(k=1,. . .,n).

The integral is convergent in any of the following cases :

(1) C>0,|argz|< 12πCandz=0;

(2) C=0,pD+ (E) <−1, argz=0 andz=0

where

C=

n

j=1

Ajp

j=n+1

Aj+ m

j=1

Bjq

j=m+1

Bj

D=

q

j=1

Bjp

j=1

Aj

and

E=

q

j=1

bjp

j=1

aj+ pq

2 .

A more detailed study about the H-function can be seen at [23]. We only mentioned some properties that will be used in this paper.

2.2.1. Properties of H-function.

Hpm,q,n

z((ap,Ap) bq,Bq)

=Hnq,,pm

1

z

(1−bq,Bq)

(1−ap,Ap)

, (2.5)

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Hpm,q,n

z((ap,Ap) bq,Bq)

=cHpm,q,n

zc((ap,cAp) bq,cBq)

, c∈R+, (2.6)

zρHpm,q,n

z((ap,Ap) bq,Bq)

=Hmp,q,n

z((ap+ρAp,Ap) bq+ρBq,Bq)

, ρ∈C. (2.7)

2.3. Brief review of fractional calculus

In this section, we make a brief review of definitions about fractional calculus and integral transforms to be used in this work.

Definition 2.1 (Riemann–Liouville integral): Let fL1loc[a,b] where

−∞ ≤a<t<b≤ ∞.The Riemann–Liouville integral of orderνis defined as

Iνf(t):= 1

(ν)

t

a (tτ)

ν−1f(τ)dτ =(f j

ν)(t), ν >0. (2.8)

where jν(t)=tν/(ν), t>0. Whenν=0, is defined

I0f(t)=f(t)

Definition 2.2 (Riemann–Liouville derivative): Let ν∈R such that n−1< νn,

n∈N;fL1[a,b]and fjn−νWn,1[a,b];where−∞ ≤a<t<b≤ ∞and

Wn,1[a,b]= fL1[a,b] : d

n

dtnL

1[a,b]

.

The Riemann–Liouville derivative of orderνis defined by

R−LDνf(t)= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

dn

dtnIn−νf(t), if n−1< ν <n;

dn

dtnf(t), ifν =n.

(2.9)

Definition 2.3 (Caputo derivative): Let ν∈R such that n−1< νn, n∈N, and fACn[a,b].The Caputo derivative of orderνis given by

CDνf(t)= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Inνd n

dtnf(t), if n−1< ν <n;

dn

dtnf(t), ifν =n.

(2.10)

Where ACn[a,b]= {f : [a,b]→R: ddxnn−−11f(x)AC[a,b]}, and AC[a,b] is the space of absolutely continuous functions.

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Definition 2.4 (Hilfer derivative): Letα∈Rsuch that n−1< αn,n∈N,andβ∈R such that 0≤β≤1; fL1[a,b] and fj(1−β)(n−α)AC1[a,b] where

−∞ ≤a<t<b≤ ∞.The Hilfer derivative of orderαand typeβof fis given by

,βf(t)=

Iβ(n−α)d n

dtn(I(

1−β)(n−α)f)

(t). (2.11)

Remark 1: In this paper we takea=0 for lower limit at the Riemann–Liouville fractional integral (2.8).

Remark 2: We note that Definition 2.4 contains as particular cases Definitions 2.2 and 2.3. Indeed:

• Ifβ =0, then

,0f(t)=

dn

dtn(In−αf)

(t)=R−LDαf(t). (2.12)

• Ifβ =1, then

,1f(t)=

In−αd n

dtnf

(t)=CDαf(t). (2.13)

• Ifα=n, then

Dn,βf(t)= d n

dtnf(t). (2.14)

Definition 2.5 (Laplace transform): Let f :R+ →Ran exponential order and piecewise continuous function, then the Laplace transform of fis

L{f(t)}(s):= ˜f(s)=

0

e−stf(t)dt. (2.15)

The integral exist forRe(s) >0.

An important property that we use in this work is the Laplace transform of the Hilfer fractional derivative, which is given by the following lemma.

Lemma 2.6 (cf. [24], Laplace transform Hilfer’s derivative): Let n∈N and α∈R, such that n−1< αn and0≤β≤1;then the Laplace transform of Hilfer derivative is given by

L{,βf(t)}(s)=sαL{f(t)}(s)n−1

k=0

sn−k−1−β(n−α)d k

dtkI(

1−β)(n−α)f(0). (2.16)

Definition 2.7 (cf.[25], Mellin transform): Let fLloc1 (0,∞)The Mellin transform of f(t) is defined by

M{f(t)}(p)=

0

tp−1f(t)dt (p∈C). (2.17)

The domain of definition of the Mellin transform turns out to be an open strip of complex number p=σ +it withσa,b. The largest open strip a,b, where the integral converges, is called fundamental strip.

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Definition 2.8 (cf.[25], Inverse Mellin transform): Let fL1(0,∞) with fundamental strip a,b. If γ is such that a< γ <b,andM{f(t)} +it)integrable, then the equality

(M−1M(p))(x)= 1 2πi

γ+i

γ−i∞

xpM{f(t)}(p)dp (2.18)

holds almost everywhere. Moreover, if f is continuous, the equality holds everywhere on (0,∞).

Between the Laplace transform and Mellin transform the following equality is true.

M{f(x)}(p)= 1

(1−p)M{L{f(x)}(s)}(1−p). (2.19)

Definition 2.9 (Fourier transform): Let fL1(Rn),the Fourier transform of f is defined by

F{f(x)}(ξ)=f(ξ)= 1 (2π)n/2

Rne

−i,x)f(x)dx, (2.20)

whereξ =1,ξ2,. . .,ξn),x=(x1,x2,. . .xn)∈Rn,(ξ,x)=ξ1x1+ · · · +ξnxnand dx= dx1dx2· · ·dxnand the inverse Fourier transform is given by

F−1{f(ξ)}(ξ)= 1

(2π)n/2

Rne

i,x)f(ξ)dξ. (2.21)

3. Ultra-hyperbolic time-fractional diffusion-wave equation

In this section we solve a Cauchy problem associated with the time-fractional diffusion-wave equation, which results from replacing the Laplacian operator by ultra-hyperbolic operator.

Let us consider the following Cauchy problem :

t,ru(x,t)c2u(x,t)=0, t>0;xD,

I(2−α)(1−r)u(x,t)|t=0=f(x),

∂tI

(2−α)(1−r)u(x,t)|

t=0=g(x).

(3.1)

where t,r is the Hilfer derivative (2.11) of order 1< α≤2, and type 0≤r≤1, the operatoris defined by

=

2

∂x21 + · · · + 2

∂x2

μ2

∂xμ2+1 − · · · − 2

∂x2μ+ν

, (3.2)

μ+ν=nis the dimension of the Euclidean spaceRn,f,gL1(Rn)andDis defined by

D=(x1,. . .xμ,. . .xμ+ν)∈Rn:μ+ν =n,x12

+ x22+ · · · +xμ2 ≥xμ2+1+ · · · +x2μ+ν. (3.3)

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To solve the Cauchy problem (3.1), first we apply Fourier transform with respect to the space variable

t,ruˆ,t)+c212+ · · · +ξμ2−ξμ2+1− · · · −ξμ2+ν)uˆ,t)=0. (3.4)

Now, applying the Laplace transform in the time variable to both sides of the above equation and putting

ξ2

μ,ν =ξ12+ · · · +ξμ2−ξμ2+1− · · · −ξμ2+ν, (3.5)

sαu˜ˆ,s)s1−r(2−α)I(1−r)(2−α)uˆ, 0)sr(2−α)∂ ∂tI

(1−r)(2−α)uˆ(x, 0)

+c2ξ2μ,νu˜ˆ,s)=0 (3.6)

using the initial conditions

sαu˜ˆ,s)s1−r(2−α)fˆ(ξ)sr(2−α)gˆ(x)+c2ξμ2,νu˜ˆ,s)=0, (3.7)

˜ˆ

u(ξ,s)= s

1−r(2−α)

+c2ξ2

μ,ν

ˆ

f(ξ)+ s −r(2−α)

+c2ξ2

μ,ν

ˆ

g(ξ). (3.8)

Taking inverse Laplace transform is that

ˆ

u(ξ,t)=t(1−r+2r−2,(1−r+2r−1(c2ξ2μ,νtα)fˆ(ξ) (3.9)

+t(1−r)α+2r−1Eα,(1−r)α+2r(c2ξ2μ,νtα)gˆ(ξ). (3.10)

Then, by inverse Fourier transform

u(x,t)=F−1{t(1−r+2r−2,(1−r+2r−1(c2ξ2μ,νtα)} ∗f(x)

+F−1{t(1−r+2r−1E

α,(1−r+2r(c2ξ2μ,νtα)} ∗g(x). (3.11)

To calculate F−1{t(1−r+2r−2,(1−r+2r−1(c2ξ2μ,νtα)} y F−1{t(1−r+2r−1

Eα,(1−r)α+2r(c2ξ2μ,νtα)}use the procedure followed by Yakubovich and Luchko, [16]. To do this, consider

E(t,ξ;α,β)=. −1,β(±2μ,νtα), (3.12) whereDis a positive constant andξ2μ,νis given by (3.5).

It is known that the Laplace transform of the functionE(t,ξ;α,β)with respect the time variabletis given by

L{E(t,ξ;α,β)}(s)= s αβ

2

μ,ν

. (3.13)

Now we calculate the Mellin transform of E(t,ξ;α,β) with respect to t using the relation (2.19). Then

M{E(t,ξ;α,β)}(p)= 1 (1−p)

0

βp 2

μ,ν

ds. (3.14)

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Making the change of variabless=(2μ,ν)1/2θ, result

M{E(t,ξ;α,β)}(p)= [∓

2

μ,ν](1−βp)/α (1−p)

0

θαβp

θα+1 dθ. (3.15)

Takingθ =t1, we have by definition of Beta function,

0

θαβ−p θα+1 dθ =

1

αB

β+p−1

α ,

1−βp

α +1

. (3.16)

Finally, from (3.14)–(3.16) , we have that

M{E(t,ξ;α,β)}(p)= [∓

2

μ,ν](1−βp)/α

(1−p)α B

β+p−1

α ,

1−βp

α +1

. (3.17)

3.1. Inverse Fourier transform of a Mittag–Leffler-type function containing in its argument a positive-definite quadratic form

Now let us calculate the inverse Fourier transform with respect to the variableξ of the

E(t,ξ;α,β)function.

GivenE(t,x;α,β)=F−1{E(t,ξ;α,β)}(x), then, taking the Mellin transform both sides with respect totresult

M{E(t,x;α,β)}(p)=F−1{M{E(t,ξ;α,β)}(p)}(x)

= (D)(1−β−p)/α α(1−p)

β+p−1

α

1−βp

α +1

×F−1{(ξ2

μ,ν)(1−β−p)/α}(x). (3.18)

To calculateF−1{2μ,ν)(1−β−p)/α}(x)we use a result due to Aguirre (cf.[26], f 2.3)

F{(x2μ,ν}(ξ)= eνπ

i/222λ+nπn/2+n/2)

(λ) 2μ,ν)λ−n/2. (3.19)

This formula is valid if x21+x22+ · · · +x2μx2μ+1+ · · · +x2μ+ν and ξ12+ξ22+ · · · +

ξ2

μξμ2+1+ · · · +ξμ2+ν.

Then, forλ=(p+β−1)/αn/2, we have

F−1{(ξ2

μ,ν)(1−β−p)/α}(x)=

iν(n/2−((p+β−1)/α))

22((p+β−1)/α)πn/2((p+β1)/α)(x

2

μ,ν)(p+β−1)/α−n/2 (3.20) then, returning to (3.18)

M{E(t,x;α,β)}(p)= (D)(

1−βp)/α α(1−p)

1−βp

α +1

(3.21)

×iν(n/2−((p+β−1)/α))

22((p+β−1)/α)πn/2 (x

2

μ,ν)(p+β−1)/α−n/2. (3.22)

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Taking inverse Mellin transform results

E(t,x;α,β)= i

ν(x2

μ,ν)(β−1)/αn/2(D)(1−β)/α α4−1)/απn/2

× 1

2πi p+i

p−i∞

((1−βp)/α+1)((1−βp)/α+n/2)

(1−p)

×

⎡ ⎣

x2μ,ν

4(D)tα 1

pdp= iq(x2μ,ν)(β−1)/αn/2(D)(1−β)/α α4−1)/απn/2

× 1

2πi p+i∞

p−i∞

((1−βp)/α+1)((1−βp)/α+n/2)

(1−p)

×

⎡ ⎣

4(D)tα

x2

μ,ν

1

⎦ −p

dp (3.23)

= iν(x2μ,ν)(β−1)/αn/2(D)(1−β)/α α4−1)/απn/2

× 1

2πi p+i

p−i∞

(1−((β−1)/α)(1/α)p)(1−((β+α−1)/αn/2)(1/α)p) (1−p)

×

⎡ ⎣

4(D)tα

x2

μ,ν

1

⎦ −p

dp. (3.24)

From the definition of the H-function, for m=0, n=2, p=2, q=1, and a1=

α)/α,A1=1,a2=α−1)/αn/2,A2=1,b1=0,B1=1

arg

4(D)tα

x2

μ,ν

1

< C2π

withC=(2+α)/α >0, we have

E(t,x;α,β)= i

ν(x2

μ,ν)(β−1)/α−n/2(D)(1−β)/α

α4−1)/απn/2

×H0,22,1 ⎛ ⎝

4(D)tα

x2

μ,ν

1

β−1

α , 1 α ,

β+α−1

αn 2, 1 α

(0, 1)

⎞ ⎠

= iνtβ−1

(4π(D)tα)n/2

x2μ,ν

4(D)tα

1)/αn/2

× 1 αH 0,2 2,1 ⎛ ⎝

4(D)tα

x2

μ,ν

1

β−1

α ,1α

,

β+α−1

αn 2, 1 α

(0, 1)

⎞ ⎠

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from (2.5) and (2.6) result

E(t,x;α,β)= i

νtβ−1 (4π(D)tα)n/2

x2μ,ν

4(D)tα

1)/α−n/2

×H2,01,2 ⎛ ⎝ x2μ,ν

4(D)tα

(1,α)

1−

β−1

α , 1 , n 2−

β−1

α

, 1

.

Now, using (2.7) we have

E(t,x;α,β)= i νtβ−1 (4π(D)tα)n/2

×H1,22,0

⎛ ⎜ ⎜ ⎝ x

2

μ,ν

4(D)tα 1+

β−1

α

n

2

α,α

1−

β−1

α

+β−1

α

n

2, 1 , n 2−

β−1

α

+β−1

α

n

2, 1 ⎞ ⎟ ⎟ ⎠, (3.25) i.e.

F−1{E(t,ξ;α,β)}(x)= iνtβ−1

(4π(D)tα)n/2

×H1,22,0 ⎛ ⎜ ⎝ x

2

μ,ν

4(D)tα

β

2 ,α

1− n 2, 1

,(0, 1)

⎞ ⎟

⎠. (3.26)

Returning to (3.11) and using (3.26) for cases

β =(1−r)α+2r−1

and

β=(1−r)α+2r,

we have that the solution of the Cauchy problem (3.1) is

u(x,t)= i

νt(1−r)α+2r−2 (4πc2tα)n/2

×H1,22,0 ⎛ ⎜ ⎝x

2

μ,ν 4c2tα

(1−r)α+2r−1− 2 ,α

1−n 2, 1

,(0, 1)

⎞ ⎟ ⎠∗f(x)

+iνt(1−r)α+2r−1 (4πc2tα)n/2

×H1,22,0 ⎛ ⎜ ⎝x

2

μ,ν 4c2tα

(1−r)α+2r

2 ,α

1− n 2, 1

,(0, 1)

⎞ ⎟

⎠∗g(x) (3.27)

provided thatx12+x22+ · · · +x2μx2μ+1+ · · · +x2μ+ν.

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3.2. Particular cases

(1) Note that ifr=1, the Cauchy problem (3.1) it is expressed in terms of the Caputo fractional derivative

cDα

tu(x,t)c2u(x,t)=0; u(x, 0)=f(x),

∂tu(x,t)|t=0=g(x)

(3.28)

and then, from (3.27) the solution is

u(x,t)= iν (4πc2tα)n/2

×H1,22,0 ⎛ ⎜ ⎝x

2

μ,ν

4c2tα

1− 2 ,α

1− n 2, 1

,(0, 1)

⎞ ⎟ ⎠∗f(x)

+ iνt

(4πc2tα)n/2

×H1,22,0 ⎛ ⎝x2μ,ν

4c2tα

2− 2 ,α

1−n2, 1!,(0, 1)

g(x). (3.29)

Forx21+x22+ · · · +x2μx2μ+1+ · · · +x2μ+ν.

(2) Ifr=0, the Cauchy problem (3.1) is expressed in terms of the Riemann–Liouville fractional derivative

R−LDα

tu(x,t)c2u(x,t)=0; I2−αu(x, 0)=f(x),

∂tI

2−αu(x,t)|

t=0=g(x)

(3.30)

and then, from (3.27) the solution is

u(x,t)= i

νtα−2 (4πc2tα)n/2

×H1,22,0 ⎛ ⎜ ⎝x

2

μ,ν 4c2tα

α−1− 2 ,α

1− n 2, 1

,(0, 1)

⎞ ⎟ ⎠∗f(x)

+ iνtα−1 (4πc2tα)n/2

×H1,22,0 ⎛ ⎝x2μ,ν

4c2tα

α

2 ,α

1−n2, 1!,(0, 1)

g(x). (3.31)

Provided thatx21+x22+ · · · +x2μxμ2+1+ · · · +x2μ+ν.

(13)

(3) Now, if α↓1 and r=1, the problem (3.1) is reduced to the Cauchy problem (3.1)–(3.2) of Nonlaopon and Kananthai,[14] and the solution is given by

u(x,t)= iν (4πc2t)n/2

×H1,22,0 ⎛ ⎝x2μ,ν

4c2t

1− n 2, 1

1− n2, 1!,(0, 1)

f(x) (3.32)

but, as

H1,22,0 ⎛ ⎜ ⎝x

2

μ,ν

4c2t

1− n 2, 1

1− n 2, 1

,(0, 1)

⎞ ⎟ ⎠=∞

p=0 Res

(p)

x2μ,ν

4c2t

p

=∞

p=0

(−1)p

p!

x2μ,ν

4c2t

p

=exp

x2μ,ν 4c2t

(3.33)

finally results that (3.32) coincides with (3.3) of Nonlaopon and Kananthai,[14] provided thatx21+x22+ · · · +x2μx2μ+1+ · · · +x2μ+ν.

(4) On the other hand, ifα=2 andrtake any value, the problem (3.1) is reduced to the Cauchy problem (8)-(9) whenk=1 of Sansanit and Kananthai [15] and the explicit solution is given by

u(x,t)= i

ν (4πc2t2)n/2

×H2,01,2 ⎛ ⎝x2μ,ν

4c2t2

(1−n, 2)

1− n 2, 1

,(0, 1)

⎞ ⎠f(x)

+ iνt

(4πc2t2)n/2

×H2,01,2 ⎛ ⎝x2μ,ν

4c2t2

(2−n, 2)

1− n 2, 1

,(0, 1)

g(x). (3.34)

Forx21+x22+ · · · +x2μx2μ+1+ · · · +x2μ+ν.

Remark 3: An important question about this solution is that it was not given explicitly in [15].

Disclosure statement

No potential conflict of interest was reported by the author.

(14)

References

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[3] Wyss W. The fractional diffusion equation. J Math Phys. 1986;27:2782–2785.

[4] Schneider WR, Wyss W. Fractional diffusion and wave equations. J Math Phys. 1989;30: 134–144.

[5] Mainardi F. The fundamental solutions for the fractional diffusion-wave equation. Appl Math Lett. 1996;9(6):23–28.

[6] Hilfer R. Fractional diffusion based on Riemann–Liouville fractional derivatives. J Phys Chem B. 2000;104(16):3914–3917.

[7] Huang F, Liu F. The time fractional diffusion and wave equations in an n-dimensional half space with mixed boundary conditions. Pacific J App Math. 2008;1(4):67–77.

[8] Sandev T, Metzler R, Tomovski Ž. Fractional diffusion equation with a generalized Rie-mann–Liouville time fractional derivative. J Phys A: Math Theor. 2011;44(25):255203. [9] Tomovski Ž, Sandev T, Metzler R, Dubbeldam R. Generalized space-time fractional

dif-fusion equation with composite fractional time derivative. Phys A: Statist Mech Appl. 2012;391(8):2527–2542.

[10] Gorenflo R, Mainardi F. Approximation of Lévy–Feller diffusion by random walk. J Anal Appl. 1999;18(2):231–246.

[11] Yu R, Zhang H. New function of Mittag–Leffler type and its application in the fractional diffusion-wave equation. Chaos Solitios Fractals. 2006;30(4):946–955.

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[16] Yakubovich SB, Luchko Y, The hypergeometric approach to integral transforms and convo-lutions, Mathematics and its applications, Vol. 287. Dordrecht: Kluwer Academic Publishers; 1994.

[17] Rodrigues MM, Vieira N, Yakubovich S. Operational calculus for Bessel’s fractional equation. In: Almeida A, Castro L, Speck FO, editors. Advances in harmonic analysis and operator theory. Vol 229, Operator theory: advances and applications. Basel: Springer AG; 2013. p. 357–370. [18] Yakubovich S, Rodrigues MM. Fundamental solutions of the fractional two-parameter

tele-graph equation. Integral Transforms Spec Funct. 2012;23(7):509–519.

[19] Dalla Riva M, Yakubovich S. On a Riemann–Liouville fractional analog of the Laplace operator with positive energy. Integral Transforms Spec Funct. 2012;23(4):277–295.

[20] Yakubovich S. Eigenfunctions and fundamental solutions of the fractional two-parameter Laplacian. Int J Math Math Sci. 2010;2010:18 p.

[21] Haubold HJ, Mathai AM, Saxena RK. Mittag–Leffler functions and their applications. J App Math. 2011;2011:51 p.

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