APPMATHISRN Applied Mathematics2090-55722090-5564International Scholarly Research Network68238110.5402/2011/682381682381Research ArticleAnalytical Solution for the Differential Equation Containing Generalized Fractional Derivative Operators and Mittag-Leffler-Type FunctionChaurasiaV. B. L.1DubeyRavi Shanker1El-SayedM. F.1Department of MathematicsUniversity of RajasthanJaipur 302004Indiauniraj.ac.in2011572011201126032011100520112011Copyright © 2011 V. B. L. Chaurasia and Ravi Shanker Dubey.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We discuss and derive the analytical solution for the fractional partial differential equation with generalized Riemann-Liouville fractional operator D0,tα,β of order α and β. Here, we derive the solution of the given differential equation with the help of Laplace and Hankel transform in terms of Fox's H-function as well as in terms of Fox-Wright function ψpq.

1. Introduction, Definition, and Preliminaries

Applications of fractional calculus require fractional derivatives of different kinds . Differentiation and integration of fractional order are traditionally defined by the right-sided Riemann-Liouville fractional integral operator Ia+P and the left-sided Riemann-Liouville fractional integral operator Ia-P, and the corresponding Riemann-Liouville fractional derivative operators Da+P and Da-P, as follows [10, 11]:(Ia+μf)(x)=1Γ(μ)axf(t)(x-t)1-μdt(x>a;R(μ)>0),(Ia-μf)(x)=1Γ(μ)xaf(t)(t-x)1-μdt(x<a;R(μ)>0),(Da±μf)(x)=(±ddx)n(Ia±n-μf)(x)(R(μ)0;n=[R(μ)]+1),

where the function f is locally integrable, R(μ) denotes the real part of the complex number μC and [R(μ)] means the greatest integer in R(μ).

Recently, a remarkable large family of generalized Riemann-Liouville fractional derivatives of order α  (0<α<1) and type β  (0β1) was introduced as follows [13, 5, 6, 8].

Definition 1.1.

The right-sided fractional derivative Da+α,β and the left-sided fractional derivative Da-α,β of order α  (0<α<1) and type β  (0β1) with respect to x are defined by (Da±α,βf)(x)=(±Ia±β(1-α)ddx(Ia±(1-β)(1-α)f))(x), whenever the second number of (1.4) exists. This generalization (1.4) yields the classical Riemann-Liouville fractional derivative operator when β=0. Moreover, for β=1, it gives the fractional derivative operator introduced by Liouville  which is often attributed to Caputo now-a-days and which should more appropriately be referred to as the Liouville-Caputo fractional derivative. Several authors [7, 9] called the general operators in (1.4) the Hilfer fractional derivative operators. Applications of Da±α,β are given .

Using the formulas (1.1) and (1.2) in conjunction with (1.3) when n=1, the fractional derivative operator Da±α,β can be written in the following form: (Da±α,βf)(x)=(±Ia±β(1-α)(Da±α+β-αβf))(x). The difference between fractional derivatives of different types becomes apparent from their Laplace transformations. For example, it is found for 0<α<1 that [1, 2, 9] L[(D0+α,βf)(x)](s)=sαL[f(x)](s)-sβ(α-1)(I0+(1-β)(1-α)f)(0+)(0<α<1), where (I0+(1-β)(1-α)f)(0+) is the Riemann-Liouville fractional integral of order (1-β)(1-α) evaluated in the limit as t0+, it being understood (as usual) that , L[f(x)](s)=0e-sxf(x)dx=F(s), provided that the defining integral in (1.7) exists.

The familiar Mittag-Leffler functions Eμ(z) and Eμ,ν(z) are defined by the following series: Eμ(z)=n=0znΓ(μn+1)=Eμ,1(z)(zC;R(μ)>0),Eμ,ν(z)=n=0znΓ(μn+ν)(z,νC;R(μ)>0), respectively. These functions are natural extensions of the exponential, hyperbolic, and trigonometric functions, since E1(z)=ez,E2(z2)=coshz,E2(-z2)=cosz,E1,2(z)=ez-1z,E2,2(z2)=sinhzz.

For a detailed account of the various properties, generalizations, and applications of the Mittag-Leffler functions, the reader may refer to the recent works by, for example, Gorenflo et al.  and Kilbas et al. . The Mittag-Leffler function (1.1) and some of its various generalizations have only recently been calculated numerically in the whole complex plane [18, 19]. By means of the series representation, a generalization of the Mittag-Leffler function Eμ,ν(z) of (1.2) was introduced by Prabhakar  as follows: Eμ,νλ(z)=n=0(λ)nΓ(μn+ν)znn!(z,ν,λC;R(μ)>0), where (λ)ν denotes the familiar Pochhammer symbol, defined (for λ,νC and in terms of the familiar Gamma function) by (λ)ν=Γ(λ+ν)Γ(λ)={1(ν=0;λC{0})λ(λ+1)(λ+n-1)(ν=nN;λC).

Clearly, we have the following special cases: Eμ,ν1(z)=Eμ,ν(z),Eμ,11(z)=Eμ(z). Indeed, as already observed earlier by Srivastava and Saxena , the generalized Mittag-Leffler function Eμ,νλ(z) itself is actually a very specialized case of a rather extensively investigated function pΨq as indicated below : Eu,νλ(z)=1Γ(λ)1Ψ1[(λ,1);(ν,u);z]. Here and in what follows, pΨq denotes the Wright (or more appropriately, the Fox-Wright) generalized of the hypergeometric pFq function, which is defined as follows : pΨq=[(a1,A1),,(ap,Ap);(b1,B1),,(bq,Bq);z]=x=0Γ(a1+A1k)Γ(ap+Apk)zkΓ(b1+B1k)Γ(bq+Bqk)k!,[R(Aj)>0  (j=1,,p);R(Bj)>0  (j=1,,q);1+R(j=1qBj-j=1pAj)0], in which we assumed in general that aj,AjC  (j=1,  ,p),bj,BjC  (j=1,  ,  q).

In application of Mittag-Leffler function, it is useful to have the following Laplace inverse transform formula: L-1{Sγ-β(Sγ+A)k+1}=1k!tγk+β-1Eγ,βk(-Atγ), where Eγ,βj(z)=(dj/dzj)Eγ,β(z).

2. Fox’s <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M59"><mml:mrow><mml:mi>H</mml:mi></mml:mrow></mml:math></inline-formula>-function

The Fox function, also referred as the Fox’s H-function, generalizes the Mellin-Barnes function. The importance of the Fox function lies in the fact that it includes nearly all special functions occurring in applied mathematics and statistics as special cases. Fox H-function is defined as Hp,q+11,p[-x|(1-a1,A1),,(1-ap,Ap)(0,1),(1-b1,B1),,(1-bq,Bq)]=k=0Γ(a1+A1k)Γ(ap+Apk)k!Γ(b1+B1k)Γ(bp+Bqk)xk.

We need this relationEα,βk(x)=n=kn!xn-k(n-k)!Γ(αn+β)=j=0Γ(j+k+1)xjj!Γ(αj+αk+β)=H1,21,1[-x|(-k,1)(0,1),(1-αk-β,α)].

3. Finite Hankel Transform

If f(r) satisfies Dirichlet conditions in closed interval (0,a) and if its finite Hankel transform is defined to be H[f(r)]=f¯(λn)=0arf(r)J0(rλn)dr,

where λn are the roots of the equation J0(r)=0. Then at each point of the interval at which f(r) is continuous:f(r)=2a2n=1f¯(λn)J0(λnr)J12(λna),

where the sum is taken over all positive roots of J0(r)=0, J0 and J1 are Bessel functions of first kind.

In application of the finite Hankel transform to physical problems, it is useful to have the following formula H[d2fdr2+1rdfdr]=-λn2f¯(r)+aλnf(a)J1(λna).

Example 3.1.

Solve the differential equation D0,t2α,βu(r,t)+aD0,tα,βu(r,t)=d(2u(r,t)r2+1ru(r,t))+f(t), where 0<α1/2 and 0β1 with initial condition It(1-β)(1-2α)u(r,0)=ϕ1(r),It(1-β)(1-α)u(r,0)=ϕ2(r),u(r,t)=0everywhere  for  t<0,u(r,t)=0for  r=1,t>0,u(r,t)=finiteat  r=0,t>0.

Solution 1.

Taking Laplace transform of (3.4), we get s2αũ(r,s)-sβ(2α-1)ϕ1(r)+asαũ(r,s)-asβ(α-1)ϕ2(r)=d[2ũ(r,s)r2+1rũ(r,s)]+f̃(s). Taking Hankel transform on both side of the above equation, we get s2αũ̃(r,s)-sβ(2α-1)ϕ̃1(r)+asαũ̃(r,s)-asβ(α-1)ϕ̃2(r)=d[-λn2ũ̃(r,s)]+f̃(s)J1(λn)λn, then we get ũ̃(r,s)=sβ(2α-1)ϕ̃1(r)(s2α+asα+dλn2)+asβ(α-1)ϕ̃2(r)(s2α+asα+dλn2)+f̃(s)(s2α+asα+dλn2)J1(λn)λn,ũ̃(r,s)=G̃̃1ϕ̃1(r)+aG̃̃2ϕ̃2(r)+G̃̃3f̃(s)J1(λn)λn, where G̃̃1=sβ(2α-1)(s2α+asα+dλn2),G̃̃2=sβ(α-1)(s2α+asα+dλn2),G̃̃3=1(s2α+asα+dλn2).

On taking Laplace inverse of (3.10), (3.11), and (3.12), respectively, L-1{sβ(2α-1)(s2α+asα+dλn2)}=m=0(-1)mam+1tα+β-2αβ-mα-1m!Eα,α+β-2αβ-2mαm(-dλn2atα),L-1{sβ(α-1)(s2α+asα+dλn2)}=m=0(-1)mam+1tα+β-αβ-mα-1m!Eα,α+β-αβ-2mαm(-dλn2atα),L-1{1(s2α+asα+dλn2)}=m=0(-1)mam+1tα-mα-1m!Eα,α-2mαm(-dλn2atα). After taking Inverse Laplace and Hankel transform of (3.9) put the value (3.13) through (3.15) in (3.9), we get u(r,t)=2n=0m=0(-1)mam+1J0(λnr)m!J12(λn)ϕ1(r)t-2αβ-mα+α+β-1j=0(j+m+1)!(-dλn2tα/a)j  j!Γ(jα+α+β-2αβ-2mα)+2an=0m=0(-1)mam+1J0(λnr)J12(λn)ϕ2(r)t-αβ-mα+α+β-1j=0(j+m+1)!(-dλn2tα/a)j  j!Γ(jα+α+β-αβ-mα)+2n=0m=0(-1)mam+1J0(λnr)λnJ1(λn)0tuα-mα-1j=0(j+m+1)!(-dλn2uα/a)j  j!Γ(αj+α-mα)f(t-u)du.u(r,t)=2n=0m=0(-1)mam+1J0(λnr)J12(λn)ϕ1(r)t-2αβ-mα+α+β-1H1,21,1[dλn2tαa|(-m-1,1)(0,1),(1-α-β+2αβ+2mα,α)]+2an=0m=0(-1)mam+1J0(λnr)J12(λn)ϕ2(r)t-αβ-mα+α+β-1H1,21,1[dλn2tαa|(-m-1,1)(0,1),(1-α-β+αβ+mα,α)]+2n=0m=0(-1)nJ0(λnr)λnJ1(λn)0tuα-mα-1H1,21,1[dλn2uαa|(-m-1,1)(0,1),(1-α+mα,α)]f(t-u)du.

Example 3.2.

Solve the differential equation (3.4) with initial condition It(1-β)(1-2α)u(r,0)=0,It(1-β)(1-α)u(r,0)=0,u(r,t)=0everywhere  for  t0,u(r,t)=0for  r=1,t>0,u(r,t)=finiteat  r=0,t>0.

Solution 2.

Taking Laplace and Hankel transform of (3.4), we get ũ̃(r,s)=J1(λn)λnf̃(s)(s2α+asα+dλn2), on taking Inverse Laplace transform of equation (3.19), we get ũ(r,t)=L-1{f̃(s)J1(λn)λn}L-1{(1s2α+asα+dλn2)}.

By using convolution theorem for Laplace transform and taking inverse Hankel transform, we get u(r,t)=2n=0m=0(-1)mam+1J0(λnr)λnJ1(λn)0tuα-mα-1Eα,α-2mαm(-dλn2uαa)f(t-u)du, or u(r,t)=2n=0m=0(-1)mam+1J0(λnr)λnJ1(λn)0tuα-mα-1j=0(j+m+1)!(j)!(-dλn2uα/a)jΓ(αj+α-mα). By using the relation (2.2) u(r,t)=2n=0m=0(-1)mam+1J0(λnr)λnJ1(λn)0tuα-mα-1H1,21,1[dλn2uαa|(-m-1,1)(0,1),(1-α+mα,α)]f(t-u)du, or u(r,t)=2n=0m=0(-1)mam+1J0(λnr)λnJ1(λn)0tuα-mα-11Γ(m)1Ψ1[(m,1);(α-2mα,α);-dλn2uαa], which is the required solution.

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