International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

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Volume 2011 |Article ID 682381 | https://doi.org/10.5402/2011/682381

V. B. L. Chaurasia, Ravi Shanker Dubey, "Analytical Solution for the Differential Equation Containing Generalized Fractional Derivative Operators and Mittag-Leffler-Type Function", International Scholarly Research Notices, vol. 2011, Article ID 682381, 9 pages, 2011. https://doi.org/10.5402/2011/682381

Analytical Solution for the Differential Equation Containing Generalized Fractional Derivative Operators and Mittag-Leffler-Type Function

Academic Editor: M. F. El-Sayed
Received26 Mar 2011
Accepted10 May 2011
Published05 Jul 2011

Abstract

We discuss and derive the analytical solution for the fractional partial differential equation with generalized Riemann-Liouville fractional operator 𝐷𝛼,𝛽0,𝑡 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 𝑝𝜓𝑞.

1. Introduction, Definition, and Preliminaries

Applications of fractional calculus require fractional derivatives of different kinds [19]. Differentiation and integration of fractional order are traditionally defined by the right-sided Riemann-Liouville fractional integral operator 𝐼𝑃𝑎+ and the left-sided Riemann-Liouville fractional integral operator 𝐼𝑃𝑎, and the corresponding Riemann-Liouville fractional derivative operators 𝐷𝑃𝑎+ and 𝐷𝑃𝑎, as follows [10, 11]:𝐼𝜇𝑎+𝑓1(𝑥)=Γ(𝜇)𝑥𝑎𝑓(𝑡)(𝑥𝑡)1𝜇𝐼𝑑𝑡(𝑥>𝑎;𝑅(𝜇)>0),(1.1)𝜇𝑎𝑓1(𝑥)=Γ(𝜇)𝑎𝑥𝑓(𝑡)(𝑡𝑥)1𝜇𝐷𝑑𝑡(𝑥<𝑎;𝑅(𝜇)>0),(1.2)𝜇𝑎±𝑓±𝑑(𝑥)=𝑑𝑥𝑛𝐼𝑛𝜇𝑎±𝑓[](𝑥)(𝑅(𝜇)0;𝑛=𝑅(𝜇)+1),(1.3)

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 𝐷𝛼,𝛽𝑎+ and the left-sided fractional derivative 𝐷𝛼,𝛽𝑎 of order 𝛼(0<𝛼<1) and type 𝛽(0𝛽1) with respect to 𝑥 are defined by 𝐷𝛼,𝛽𝑎±𝑓(𝑥)=±𝐼𝛽(1𝛼)𝑎±𝑑𝐼𝑑𝑥(1𝛽)(1𝛼)𝑎±𝑓(𝑥),(1.4) 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 [12] 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 𝐷𝛼,𝛽𝑎± are given [3].
Using the formulas (1.1) and (1.2) in conjunction with (1.3) when 𝑛=1, the fractional derivative operator 𝐷𝛼,𝛽𝑎± can be written in the following form: 𝐷𝛼,𝛽𝑎±𝑓(𝑥)=±𝐼𝛽(1𝛼)𝑎±𝐷𝛼+𝛽𝛼𝛽𝑎±𝑓(𝑥).(1.5) 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] 𝐿𝐷𝛼,𝛽0+𝑓(𝑥)(𝑠)=𝑠𝛼𝐿[]𝑓(𝑥)(𝑠)𝑠𝛽(𝛼1)𝐼(1𝛽)(1𝛼)0+𝑓(0+)(0<𝛼<1),(1.6) where (𝐼(1𝛽)(1𝛼)0+𝑓)(0+) is the Riemann-Liouville fractional integral of order (1𝛽)(1𝛼) evaluated in the limit as 𝑡0+, it being understood (as usual) that [13], 𝐿[]𝑓(𝑥)(𝑠)=0𝑒𝑠𝑥𝑓(𝑥)𝑑𝑥=𝐹(𝑠),(1.7) 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(𝐸𝑧)(𝑧𝐶;𝑅(𝜇)>0),(1.8)𝜇,𝜈(𝑧)=𝑛=0𝑧𝑛Γ(𝜇𝑛+𝜈)(𝑧,𝜈𝐶;𝑅(𝜇)>0),(1.9) respectively. These functions are natural extensions of the exponential, hyperbolic, and trigonometric functions, since 𝐸1(𝑧)=𝑒𝑧,𝐸2𝑧2=cosh𝑧,𝐸2𝑧2𝐸=cos𝑧,1,2𝑒(𝑧)=𝑧1𝑧,𝐸2,2𝑧2=sinh𝑧𝑧.(1.10)
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. [14] and Kilbas et al. [1517]. 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 [20] as follows: 𝐸𝜆𝜇,𝜈(𝑧)=𝑛=0(𝜆)𝑛𝑧Γ(𝜇𝑛+𝜈)𝑛(𝑛!𝑧,𝜈,𝜆𝐶;𝑅(𝜇)>0),(1.11) where (𝜆)𝜈 denotes the familiar Pochhammer symbol, defined (for 𝜆,𝜈𝐶 and in terms of the familiar Gamma function) by (𝜆)𝜈Γ=(𝜆+𝜈)=Γ(𝜆)1(𝜈=0;𝜆𝐶{0})𝜆(𝜆+1)(𝜆+𝑛1)(𝜈=𝑛𝑁;𝜆𝐶).(1.12)
Clearly, we have the following special cases: 𝐸1𝜇,𝜈(𝑧)=𝐸𝜇,𝜈(𝑧),𝐸1𝜇,1(𝑧)=𝐸𝜇(𝑧).(1.13) Indeed, as already observed earlier by Srivastava and Saxena [21], the generalized Mittag-Leffler function 𝐸𝜆𝜇,𝜈(𝑧) itself is actually a very specialized case of a rather extensively investigated function 𝑝Ψ𝑞 as indicated below [17]: 𝐸𝜆𝑢,𝜈1(𝑧)=Γ(𝜆)1Ψ1(𝑧𝜆,1);(𝜈,u);.(1.14) 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 [12]: 𝑝Ψ𝑞=𝑎1,𝐴1𝑎,,𝑝,𝐴𝑝;𝑏1,𝐵1𝑏,,𝑞,𝐵𝑞;𝑧=𝑥=0Γ𝑎1+𝐴1𝑘𝑎Γ𝑝+𝐴𝑝𝑘𝑧𝑘Γ𝑏1+𝐵1𝑘𝑏Γ𝑞+𝐵𝑞𝑘𝑅𝐴𝑘!,(1.15)𝑗𝐵>0(𝑗=1,,𝑝);𝑅𝑗>0(𝑗=1,,𝑞);1+𝑅𝑞𝑗=1𝐵𝑗𝑝𝑗=1𝐴𝑗0,(1.16) in which we assumed in general that 𝑎𝑗,𝐴𝑗𝐶(𝑗=1,,𝑝),𝑏𝑗,𝐵𝑗𝐶(𝑗=1,,𝑞).(1.17)
In application of Mittag-Leffler function, it is useful to have the following Laplace inverse transform formula: 𝐿1𝑆𝛾𝛽(𝑆𝛾+𝐴)𝑘+1=1𝑡𝑘!𝛾𝑘+𝛽1𝐸𝑘𝛾,𝛽(𝐴𝑡𝛾),(1.18) where 𝐸𝑗𝛾,𝛽(𝑧)=(𝑑𝑗/𝑑𝑧𝑗)𝐸𝛾,𝛽(𝑧).

2. Fox’s 𝐻-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 [22]𝐻1,𝑝𝑝,𝑞+1|||||𝑥1𝑎1,𝐴1,,1𝑎𝑝,𝐴𝑝(0,1),1𝑏1,𝐵1,,1𝑏𝑞,𝐵𝑞=𝑘=0Γ𝑎1+𝐴1𝑘𝑎Γ𝑝+𝐴𝑝𝑘𝑏𝑘!Γ1+𝐵1𝑘𝑏Γ𝑝+𝐵𝑞𝑘𝑥𝑘.(2.1)

We need this relation𝐸𝑘𝛼,𝛽(𝑥)=𝑛=𝑘𝑛!𝑥𝑛𝑘=(𝑛𝑘)!Γ(𝛼𝑛+𝛽)𝑗=0Γ(𝑗+𝑘+1)𝑥𝑗𝑗!Γ(𝛼𝑗+𝛼𝑘+𝛽)=𝐻1,11,2||||.𝑥(𝑘,1)(0,1),(1𝛼𝑘𝛽,𝛼)(2.2)

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 [23]𝐻[]=𝑓(𝑟)𝑓𝜆𝑛=𝑎0𝑟𝑓(𝑟)𝐽0𝑟𝜆𝑛𝑑𝑟,(3.1)

where 𝜆𝑛 are the roots of the equation 𝐽0(𝑟)=0. Then at each point of the interval at which 𝑓(𝑟) is continuous:2𝑓(𝑟)=𝑎2𝑛=1𝑓𝜆𝑛𝐽0𝜆𝑛𝑟𝐽21𝜆𝑛𝑎,(3.2)

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 [23]𝐻𝑑2𝑓𝑑𝑟2+1𝑟𝑑𝑓𝑑𝑟=𝜆2𝑛𝑓(𝑟)+𝑎𝜆𝑛𝑓(𝑎)𝐽1𝜆𝑛𝑎.(3.3)

Example 3.1. Solve the differential equation 𝐷2𝛼,𝛽0,𝑡𝑢(𝑟,𝑡)+𝑎𝐷𝛼,𝛽0,𝑡𝜕𝑢(𝑟,𝑡)=𝑑2𝑢(𝑟,𝑡)𝜕𝑟2+1𝑟𝑢(𝑟,𝑡)+𝑓(𝑡),(3.4) where 0<𝛼1/2 and 0𝛽1
with initial condition 𝐼𝑡(1𝛽)(12𝛼)𝑢(𝑟,0)=𝜙1𝐼(𝑟),𝑡(1𝛽)(1𝛼)𝑢(𝑟,0)=𝜙2(𝑟),𝑢(𝑟,𝑡)=0everywherefor𝑡<0,𝑢(𝑟,𝑡)=0for𝑟=1,𝑡>0,𝑢(𝑟,𝑡)=niteat𝑟=0,𝑡>0.(3.5)

Solution 1. Taking Laplace transform of (3.4), we get 𝑠2𝛼̃𝑢(𝑟,𝑠)𝑠𝛽(2𝛼1)𝜙1(𝑟)+𝑎𝑠𝛼̃𝑢(𝑟,𝑠)𝑎𝑠𝛽(𝛼1)𝜙2𝜕(𝑟)=𝑑2̃𝑢(𝑟,𝑠)𝜕𝑟2+1𝑟+̃𝑢(𝑟,𝑠)𝑓(𝑠).(3.6) Taking Hankel transform on both side of the above equation, we get 𝑠2𝛼̃̃𝑢(𝑟,𝑠)𝑠𝛽(2𝛼1)𝜙1(𝑟)+𝑎𝑠𝛼̃̃𝑢(𝑟,𝑠)𝑎𝑠𝛽(𝛼1)𝜙2(𝑟)=𝑑𝜆2𝑛̃+𝐽̃𝑢(𝑟,𝑠)𝑓(𝑠)1𝜆𝑛𝜆𝑛,(3.7) then we get ̃𝑠̃𝑢(𝑟,𝑠)=𝛽(2𝛼1)𝜙1(𝑟)𝑠2𝛼+𝑎𝑠𝛼+𝑑𝜆2𝑛+𝑎𝑠𝛽(𝛼1)𝜙2(𝑟)𝑠2𝛼+𝑎𝑠𝛼+𝑑𝜆2𝑛+𝑓(𝑠)𝑠2𝛼+𝑎𝑠𝛼+𝑑𝜆2𝑛𝐽1𝜆𝑛𝜆𝑛̃𝐺,(3.8)̃𝑢(𝑟,𝑠)=1𝜙1𝐺(𝑟)+𝑎2𝜙2𝐺(𝑟)+3𝐽𝑓(𝑠)1𝜆𝑛𝜆𝑛,(3.9) where 𝐺1=𝑠𝛽(2𝛼1)𝑠2𝛼+𝑎𝑠𝛼+𝑑𝜆2𝑛,𝐺(3.10)2=𝑠𝛽(𝛼1)𝑠2𝛼+𝑎𝑠𝛼+𝑑𝜆2𝑛𝐺,(3.11)3=1𝑠2𝛼+𝑎𝑠𝛼+𝑑𝜆2𝑛.(3.12)
On taking Laplace inverse of (3.10), (3.11), and (3.12), respectively, 𝐿1𝑠𝛽(2𝛼1)𝑠2𝛼+𝑎𝑠𝛼+𝑑𝜆2𝑛=𝑚=0(1)𝑚𝑎𝑚+1𝑡𝛼+𝛽2𝛼𝛽𝑚𝛼1𝐸𝑚!𝑚𝛼,𝛼+𝛽2𝛼𝛽2𝑚𝛼𝑑𝜆2𝑛𝑎𝑡𝛼,𝐿(3.13)1𝑠𝛽(𝛼1)𝑠2𝛼+𝑎𝑠𝛼+𝑑𝜆2𝑛=𝑚=0(1)𝑚𝑎𝑚+1𝑡𝛼+𝛽𝛼𝛽𝑚𝛼1𝐸𝑚!𝑚𝛼,𝛼+𝛽𝛼𝛽2𝑚𝛼𝑑𝜆2𝑛𝑎𝑡𝛼𝐿,(3.14)11𝑠2𝛼+𝑎𝑠𝛼+𝑑𝜆2𝑛=𝑚=0(1)𝑚𝑎𝑚+1𝑡𝛼𝑚𝛼1𝐸𝑚!𝑚𝛼,𝛼2𝑚𝛼𝑑𝜆2𝑛𝑎𝑡𝛼.(3.15) 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+1𝐽0𝜆𝑛𝑟𝑚!𝐽21𝜆𝑛𝜙1(𝑟)𝑡2𝛼𝛽𝑚𝛼+𝛼+𝛽1𝑗=0(𝑗+𝑚+1)!𝑑𝜆2𝑛𝑡𝛼/𝑎𝑗𝑗!Γ(𝑗𝛼+𝛼+𝛽2𝛼𝛽2𝑚𝛼)+2𝑎𝑛=0𝑚=0(1)𝑚𝑎𝑚+1𝐽0𝜆𝑛𝑟𝐽21𝜆𝑛𝜙2(𝑟)𝑡𝛼𝛽𝑚𝛼+𝛼+𝛽1𝑗=0(𝑗+𝑚+1)!𝑑𝜆2𝑛𝑡𝛼/𝑎𝑗𝑗!Γ(𝑗𝛼+𝛼+𝛽𝛼𝛽𝑚𝛼)+2𝑛=0𝑚=0(1)𝑚𝑎𝑚+1𝐽0𝜆𝑛𝑟𝜆𝑛𝐽1𝜆𝑛𝑡0𝑢𝛼𝑚𝛼1𝑗=0(𝑗+𝑚+1)!𝑑𝜆2𝑛𝑢𝛼/𝑎𝑗𝑗!Γ(𝛼𝑗+𝛼𝑚𝛼)𝑓(𝑡𝑢)𝑑𝑢.(3.16)𝑢(𝑟,𝑡)=2𝑛=0𝑚=0(1)𝑚𝑎𝑚+1𝐽0𝜆𝑛𝑟𝐽21𝜆𝑛𝜙1(𝑟)𝑡2𝛼𝛽𝑚𝛼+𝛼+𝛽1𝐻1,11,2𝑑𝜆2𝑛𝑡𝛼𝑎||||(𝑚1,1)(0,1),(1𝛼𝛽+2𝛼𝛽+2𝑚𝛼,𝛼)+2𝑎𝑛=0𝑚=0(1)𝑚𝑎𝑚+1𝐽0𝜆𝑛𝑟𝐽21𝜆𝑛𝜙2(𝑟)𝑡𝛼𝛽𝑚𝛼+𝛼+𝛽1𝐻1,11,2𝑑𝜆2𝑛𝑡𝛼𝑎||||(𝑚1,1)(0,1),(1𝛼𝛽+𝛼𝛽+𝑚𝛼,𝛼)+2𝑛=0𝑚=0(1)𝑛𝐽0𝜆𝑛𝑟𝜆𝑛𝐽1𝜆𝑛𝑡0𝑢𝛼𝑚𝛼1𝐻1,11,2𝑑𝜆2𝑛𝑢𝛼𝑎||||𝑓(𝑚1,1)(0,1),(1𝛼+𝑚𝛼,𝛼)(𝑡𝑢)𝑑𝑢.(3.17)

Example 3.2. Solve the differential equation (3.4) with initial condition 𝐼𝑡(1𝛽)(12𝛼)𝐼𝑢(𝑟,0)=0,𝑡(1𝛽)(1𝛼)𝑢(𝑟,0)=0,𝑢(𝑟,𝑡)=0everywherefor𝑡0,𝑢(𝑟,𝑡)=0for𝑟=1,𝑡>0,𝑢(𝑟,𝑡)=niteat𝑟=0,𝑡>0.(3.18)

Solution 2. Taking Laplace and Hankel transform of (3.4), we get ̃𝐽̃𝑢(𝑟,𝑠)=1𝜆𝑛𝜆𝑛𝑓(𝑠)𝑠2𝛼+𝑎𝑠𝛼+𝑑𝜆2𝑛,(3.19) on taking Inverse Laplace transform of equation (3.19), we get ̃𝑢(𝑟,𝑡)=𝐿1𝐽𝑓(𝑠)1𝜆𝑛𝜆𝑛𝐿11𝑠2𝛼+𝑎𝑠𝛼+𝑑𝜆2𝑛.(3.20)
By using convolution theorem for Laplace transform and taking inverse Hankel transform, we get 𝑢(𝑟,𝑡)=2𝑛=0𝑚=0(1)𝑚𝑎𝑚+1𝐽0𝜆𝑛𝑟𝜆𝑛𝐽1𝜆𝑛𝑡0𝑢𝛼𝑚𝛼1𝐸𝑚𝛼,𝛼2𝑚𝛼𝑑𝜆2𝑛𝑢𝛼𝑎𝑓(𝑡𝑢)𝑑𝑢,(3.21) or 𝑢(𝑟,𝑡)=2𝑛=0𝑚=0(1)𝑚𝑎𝑚+1𝐽0𝜆𝑛𝑟𝜆𝑛𝐽1𝜆𝑛𝑡0𝑢𝛼𝑚𝛼1𝑗=0(𝑗+𝑚+1)!(𝑗)!𝑑𝜆2𝑛𝑢𝛼/𝑎𝑗Γ.(𝛼𝑗+𝛼𝑚𝛼)(3.22) By using the relation (2.2) 𝑢(𝑟,𝑡)=2𝑛=0𝑚=0(1)𝑚𝑎𝑚+1𝐽0𝜆𝑛𝑟𝜆𝑛𝐽1𝜆𝑛𝑡0𝑢𝛼𝑚𝛼1𝐻1,11,2𝑑𝜆2𝑛𝑢𝛼𝑎||||(𝑚1,1)(0,1),(1𝛼+𝑚𝛼,𝛼)𝑓(𝑡𝑢)𝑑𝑢,(3.23) or 𝑢(𝑟,𝑡)=2𝑛=0𝑚=0(1)𝑚𝑎𝑚+1𝐽0𝜆𝑛𝑟𝜆𝑛𝐽1𝜆𝑛𝑡0𝑢𝛼𝑚𝛼11Γ(𝑚)1Ψ1(𝑚,1);(𝛼2𝑚𝛼,𝛼);𝑑𝜆2𝑛𝑢𝛼𝑎,(3.24) which is the required solution.

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Copyright © 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.


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