International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 682381 | 9 pages | 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 ğ‘ğœ“ğ‘ž.

1. Introduction, Definition, and Preliminaries

Applications of fractional calculus require fractional derivatives of different kinds [1–9]. Differentiation and integration of fractional order are traditionally defined by the right-sided Riemann-Liouville fractional integral operator ğ¼ğ‘ƒğ‘Ž+ and the left-sided Riemann-Liouville fractional integral operator ğ¼ğ‘ƒğ‘Žâˆ’, and the corresponding Riemann-Liouville fractional derivative operators ğ·ğ‘ƒğ‘Ž+ and ğ·ğ‘ƒğ‘Žâˆ’, as follows [10, 11]:î€·ğ¼ğœ‡ğ‘Ž+𝑓1(𝑥)=Γ(𝜇)ğ‘¥ğ‘Žğ‘“(𝑡)(𝑥−𝑡)1−𝜇𝐼𝑑𝑡(𝑥>ğ‘Ž;𝑅(𝜇)>0),(1.1)ğœ‡ğ‘Žâˆ’ğ‘“î€¸1(𝑥)=Γ(𝜇)ğ‘Žğ‘¥ğ‘“(𝑡)(𝑡−𝑥)1−𝜇𝐷𝑑𝑡(𝑥<ğ‘Ž;𝑅(𝜇)>0),(1.2)ğœ‡ğ‘ŽÂ±ğ‘“î€¸î‚€Â±ğ‘‘(𝑥)=î‚ğ‘‘ğ‘¥ğ‘›î€·ğ¼ğ‘›âˆ’ğœ‡ğ‘ŽÂ±ğ‘“î€¸[](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 𝜇∈𝐶 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(0≤𝛽≤1) was introduced as follows [1–3, 5, 6, 8].

Definition 1.1. The right-sided fractional derivative 𝐷𝛼,ğ›½ğ‘Ž+ and the left-sided fractional derivative 𝐷𝛼,ğ›½ğ‘Žâˆ’ of order 𝛼(0<𝛼<1) and type 𝛽(0≤𝛽≤1) with respect to 𝑥 are defined by 𝐷𝛼,ğ›½ğ‘ŽÂ±ğ‘“î‚î‚€(𝑥)=±𝐼𝛽(1−𝛼)ğ‘ŽÂ±ğ‘‘î‚€ğ¼ğ‘‘ğ‘¥(1−𝛽)(1−𝛼)ğ‘ŽÂ±ğ‘“î‚î‚(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,ğ›½ğ‘ŽÂ± are given [3].
Using the formulas (1.1) and (1.2) in conjunction with (1.3) when 𝑛=1, the fractional derivative operator 𝐷𝛼,ğ›½ğ‘ŽÂ± can be written in the following form: 𝐷𝛼,ğ›½ğ‘ŽÂ±ğ‘“î‚î‚€(𝑥)=±𝐼𝛽(1−𝛼)ğ‘ŽÂ±î‚€ğ·ğ›¼+ğ›½âˆ’ğ›¼ğ›½ğ‘ŽÂ±ğ‘“î‚î‚(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] 𝐿𝐷𝛼,𝛽0+𝑓(𝑥)(𝑠)=𝑠𝛼𝐿[]𝑓(𝑥)(𝑠)−𝑠𝛽(𝛼−1)𝐼(1−𝛽)(1−𝛼)0+𝑓(0+)(0<𝛼<1),(1.6) where (𝐼(1−𝛽)(1−𝛼)0+𝑓)(0+) is the Riemann-Liouville fractional integral of order (1−𝛽)(1−𝛼) evaluated in the limit as 𝑡→0+, it being understood (as usual) that [13], 𝐿[]𝑓(𝑥)(𝑠)∶=∞0𝑒−𝑠𝑥𝑓(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)∶=âˆžî“ğ‘›=0𝑧𝑛Γ(𝜇𝑛+1)∶=𝐸𝜇,1(𝐸𝑧)(𝑧∈𝐶;𝑅(𝜇)>0),(1.8)𝜇,𝜈(𝑧)∶=âˆžî“ğ‘›=0𝑧𝑛Γ(𝜇𝑛+𝜈)(𝑧,𝜈∈𝐶;𝑅(𝜇)>0),(1.9) respectively. These functions are natural extensions of the exponential, hyperbolic, and trigonometric functions, since 𝐸1(𝑧)=𝑒𝑧,𝐸2𝑧2=cosh𝑧,𝐸2−𝑧2𝐸=cos𝑧,1,2𝑒(𝑧)=𝑧−1𝑧,𝐸2,2𝑧2=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. [15–17]. 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)=âˆžî“ğ‘›=0(𝜆)𝑛𝑧Γ(𝜇𝑛+𝜈)𝑛(𝑛!𝑧,𝜈,𝜆∈𝐶;𝑅(𝜇)>0),(1.11) where (𝜆)𝜈 denotes the familiar Pochhammer symbol, defined (for 𝜆,𝜈∈𝐶 and in terms of the familiar Gamma function) by (𝜆)𝜈Γ∶=(𝜆+𝜈)=Γ(𝜆)1(𝜈=0;𝜆∈𝐶⧵{0})𝜆(𝜆+1)⋯(𝜆+𝑛−1)(𝜈=𝑛∈𝑁;𝜆∈𝐶).(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 ğ‘Î¨ğ‘ž as indicated below [17]: 𝐸𝜆𝑢,𝜈1(𝑧)=Γ(𝜆)1Ψ1(𝑧𝜆,1);(𝜈,u);.(1.14) Here and in what follows, ğ‘Î¨ğ‘ž denotes the Wright (or more appropriately, the Fox-Wright) generalized of the hypergeometric 𝑝Fğ‘ž function, which is defined as follows [12]: ğ‘Î¨ğ‘ž=îƒ¬î€·ğ‘Ž1,𝐴1î€¸î€·ğ‘Ž,…,𝑝,𝐴𝑝;𝑏1,𝐵1𝑏,…,ğ‘ž,ğµğ‘žî€¸;𝑧=âˆžî“ğ‘¥=0Î“î€·ğ‘Ž1+𝐴1ğ‘˜î€¸î€·ğ‘Žâ‹¯Î“ğ‘+𝐴𝑝𝑘𝑧𝑘Γ𝑏1+𝐵1ğ‘˜î€¸î€·ğ‘â‹¯Î“ğ‘ž+ğµğ‘žğ‘˜î€¸îƒ¬ğ‘…î€·ğ´ğ‘˜!,(1.15)𝑗𝐵>0(𝑗=1,…,𝑝);𝑅𝑗>0(𝑗=1,…,ğ‘ž);1+ğ‘…ğ‘žâˆ‘ğ‘—=1𝐵𝑗−𝑝∑𝑗=1𝐴𝑗≥0,(1.16) in which we assumed in general that ğ‘Žğ‘—,𝐴𝑗∈𝐶(𝑗=1,…,𝑝),𝑏𝑗,𝐵𝑗∈𝐶(𝑗=1,…,ğ‘ž).(1.17)
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=1𝑡𝑘!𝛾𝑘+𝛽−1𝐸𝑘𝛾,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,ğ‘ž+1⎡⎢⎢⎣|||||−𝑥1âˆ’ğ‘Ž1,𝐴1,…,1âˆ’ğ‘Žğ‘,𝐴𝑝(0,1),1−𝑏1,𝐵1,…,1âˆ’ğ‘ğ‘ž,ğµğ‘žî€¸âŽ¤âŽ¥âŽ¥âŽ¦=âˆžî“ğ‘˜=0Î“î€·ğ‘Ž1+𝐴1ğ‘˜î€¸î€·ğ‘Žâ‹¯Î“ğ‘+𝐴𝑝𝑘𝑏𝑘!Γ1+𝐵1𝑘𝑏⋯Γ𝑝+ğµğ‘žğ‘˜î€¸ğ‘¥ğ‘˜.(2.1)

We need this relation𝐸𝑘𝛼,𝛽(𝑥)=âˆžî“ğ‘›=𝑘𝑛!𝑥𝑛−𝑘=(𝑛−𝑘)!Γ(𝛼𝑛+𝛽)âˆžî“ğ‘—=0Γ(𝑗+𝑘+1)𝑥𝑗𝑗!Γ(𝛼𝑗+𝛼𝑘+𝛽)=𝐻1,11,2||||.−𝑥(−𝑘,1)(0,1),(1−𝛼𝑘−𝛽,𝛼)(2.2)

3. Finite Hankel Transform

If 𝑓(𝑟) satisfies Dirichlet conditions in closed interval (0,ğ‘Ž) and if its finite Hankel transform is defined to be [23]𝐻[]=𝑓(𝑟)𝑓𝜆𝑛=î€œğ‘Ž0𝑟𝑓(𝑟)𝐽0𝑟𝜆𝑛𝑑𝑟,(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)=ğ‘Ž2âˆžî“ğ‘›=1𝑓𝜆𝑛𝐽0𝜆𝑛𝑟𝐽21î€·ğœ†ğ‘›ğ‘Žî€¸,(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]𝐻𝑑2𝑓𝑑𝑟2+1𝑟𝑑𝑓𝑑𝑟=−𝜆2𝑛𝑓(𝑟)+ğ‘Žğœ†ğ‘›ğ‘“(ğ‘Ž)𝐽1î€·ğœ†ğ‘›ğ‘Žî€¸.(3.3)

Example 3.1. Solve the differential equation 𝐷2𝛼,𝛽0,𝑡𝑢(𝑟,𝑡)+ğ‘Žğ·ğ›¼,𝛽0,𝑡𝜕𝑢(𝑟,𝑡)=𝑑2𝑢(𝑟,𝑡)𝜕𝑟2+1𝑟𝑢(𝑟,𝑡)+𝑓(𝑡),(3.4) where 0<𝛼≤1/2 and 0≤𝛽≤1
with initial condition 𝐼𝑡(1−𝛽)(1−2𝛼)𝑢(𝑟,0)=𝜙1𝐼(𝑟),𝑡(1−𝛽)(1−𝛼)𝑢(𝑟,0)=𝜙2(𝑟),𝑢(𝑟,𝑡)=0everywherefor𝑡<0,𝑢(𝑟,𝑡)=0for𝑟=1,𝑡>0,𝑢(𝑟,𝑡)=finiteat𝑟=0,𝑡>0.(3.5)

Solution 1. Taking Laplace transform of (3.4), we get 𝑠2𝛼̃𝑢(𝑟,𝑠)−𝑠𝛽(2𝛼−1)𝜙1(𝑟)+ğ‘Žğ‘ ğ›¼Ìƒğ‘¢(𝑟,𝑠)âˆ’ğ‘Žğ‘ ğ›½(𝛼−1)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(2𝛼−1)𝜙1(𝑟)+ğ‘Žğ‘ ğ›¼ÌƒÌƒğ‘¢(𝑟,𝑠)âˆ’ğ‘Žğ‘ ğ›½(𝛼−1)𝜙2(𝑟)=𝑑−𝜆2𝑛̃+𝐽̃𝑢(𝑟,𝑠)𝑓(𝑠)1𝜆𝑛𝜆𝑛,(3.7) then we get ̃𝑠̃𝑢(𝑟,𝑠)=𝛽(2𝛼−1)𝜙1(𝑟)𝑠2𝛼+ğ‘Žğ‘ ğ›¼+𝑑𝜆2𝑛+ğ‘Žğ‘ ğ›½(𝛼−1)𝜙2(𝑟)𝑠2𝛼+ğ‘Žğ‘ ğ›¼+𝑑𝜆2𝑛+𝑓(𝑠)𝑠2𝛼+ğ‘Žğ‘ ğ›¼+𝑑𝜆2𝑛𝐽1𝜆𝑛𝜆𝑛̃𝐺,(3.8)̃𝑢(𝑟,𝑠)=1𝜙1𝐺(𝑟)+ğ‘Ž2𝜙2𝐺(𝑟)+3𝐽𝑓(𝑠)1𝜆𝑛𝜆𝑛,(3.9) where 𝐺1=𝑠𝛽(2𝛼−1)𝑠2𝛼+ğ‘Žğ‘ ğ›¼+𝑑𝜆2𝑛,𝐺(3.10)2=𝑠𝛽(𝛼−1)𝑠2𝛼+ğ‘Žğ‘ ğ›¼+𝑑𝜆2𝑛𝐺,(3.11)3=1𝑠2𝛼+ğ‘Žğ‘ ğ›¼+𝑑𝜆2𝑛.(3.12)
On taking Laplace inverse of (3.10), (3.11), and (3.12), respectively, 𝐿−1𝑠𝛽(2𝛼−1)𝑠2𝛼+ğ‘Žğ‘ ğ›¼+𝑑𝜆2𝑛=âˆžî“ğ‘š=0(−1)ğ‘šğ‘Žğ‘š+1𝑡𝛼+𝛽−2𝛼𝛽−𝑚𝛼−1𝐸𝑚!𝑚𝛼,𝛼+𝛽−2𝛼𝛽−2𝑚𝛼−𝑑𝜆2ğ‘›ğ‘Žğ‘¡ğ›¼î‚¶,𝐿(3.13)−1𝑠𝛽(𝛼−1)𝑠2𝛼+ğ‘Žğ‘ ğ›¼+𝑑𝜆2𝑛=âˆžî“ğ‘š=0(−1)ğ‘šğ‘Žğ‘š+1𝑡𝛼+𝛽−𝛼𝛽−𝑚𝛼−1𝐸𝑚!𝑚𝛼,𝛼+𝛽−𝛼𝛽−2𝑚𝛼−𝑑𝜆2ğ‘›ğ‘Žğ‘¡ğ›¼î‚¶ğ¿,(3.14)−11𝑠2𝛼+ğ‘Žğ‘ ğ›¼+𝑑𝜆2𝑛=âˆžî“ğ‘š=0(−1)ğ‘šğ‘Žğ‘š+1𝑡𝛼−𝑚𝛼−1𝐸𝑚!𝑚𝛼,𝛼−2𝑚𝛼−𝑑𝜆2ğ‘›ğ‘Žğ‘¡ğ›¼î‚¶.(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)=2âˆžî“âˆžğ‘›=0𝑚=0(−1)ğ‘šğ‘Žğ‘š+1𝐽0𝜆𝑛𝑟𝑚!𝐽21𝜆𝑛𝜙1(𝑟)ğ‘¡âˆžâˆ’2𝛼𝛽−𝑚𝛼+𝛼+𝛽−1𝑗=0(𝑗+𝑚+1)!−𝑑𝜆2𝑛𝑡𝛼/ğ‘Žğ‘—ğ‘—!Γ(𝑗𝛼+𝛼+𝛽−2𝛼𝛽−2𝑚𝛼)+2ğ‘Žâˆžî“âˆžğ‘›=0𝑚=0(−1)ğ‘šğ‘Žğ‘š+1𝐽0𝜆𝑛𝑟𝐽21𝜆𝑛𝜙2(𝑟)ğ‘¡âˆžâˆ’ğ›¼ğ›½âˆ’ğ‘šğ›¼+𝛼+𝛽−1𝑗=0(𝑗+𝑚+1)!−𝑑𝜆2𝑛𝑡𝛼/ğ‘Žğ‘—ğ‘—!Γ(𝑗𝛼+𝛼+𝛽−𝛼𝛽−𝑚𝛼)+2âˆžî“âˆžğ‘›=0𝑚=0(−1)ğ‘šğ‘Žğ‘š+1𝐽0𝜆𝑛𝑟𝜆𝑛𝐽1𝜆𝑛𝑡0ğ‘¢âˆžğ›¼âˆ’ğ‘šğ›¼âˆ’1𝑗=0(𝑗+𝑚+1)!−𝑑𝜆2𝑛𝑢𝛼/ğ‘Žğ‘—ğ‘—!Γ(𝛼𝑗+𝛼−𝑚𝛼)𝑓(𝑡−𝑢)𝑑𝑢.(3.16)𝑢(𝑟,𝑡)=2âˆžî“âˆžğ‘›=0𝑚=0(−1)ğ‘šğ‘Žğ‘š+1𝐽0𝜆𝑛𝑟𝐽21𝜆𝑛𝜙1(𝑟)𝑡−2𝛼𝛽−𝑚𝛼+𝛼+𝛽−1⋅𝐻1,11,2𝑑𝜆2ğ‘›ğ‘¡ğ›¼ğ‘Ž||||(−𝑚−1,1)(0,1),(1−𝛼−𝛽+2𝛼𝛽+2𝑚𝛼,𝛼)+2ğ‘Žâˆžî“âˆžğ‘›=0𝑚=0(−1)ğ‘šğ‘Žğ‘š+1𝐽0𝜆𝑛𝑟𝐽21𝜆𝑛𝜙2(𝑟)𝑡−𝛼𝛽−𝑚𝛼+𝛼+𝛽−1⋅𝐻1,11,2𝑑𝜆2ğ‘›ğ‘¡ğ›¼ğ‘Ž||||(−𝑚−1,1)(0,1),(1−𝛼−𝛽+𝛼𝛽+𝑚𝛼,𝛼)+2âˆžî“âˆžğ‘›=0𝑚=0(−1)𝑛𝐽0𝜆𝑛𝑟𝜆𝑛𝐽1𝜆𝑛𝑡0𝑢𝛼−𝑚𝛼−1𝐻1,11,2𝑑𝜆2ğ‘›ğ‘¢ğ›¼ğ‘Ž||||𝑓(−𝑚−1,1)(0,1),(1−𝛼+𝑚𝛼,𝛼)(𝑡−𝑢)𝑑𝑢.(3.17)

Example 3.2. Solve the differential equation (3.4) with initial condition 𝐼𝑡(1−𝛽)(1−2𝛼)𝐼𝑢(𝑟,0)=0,𝑡(1−𝛽)(1−𝛼)𝑢(𝑟,0)=0,𝑢(𝑟,𝑡)=0everywherefor𝑡≤0,𝑢(𝑟,𝑡)=0for𝑟=1,𝑡>0,𝑢(𝑟,𝑡)=finiteat𝑟=0,𝑡>0.(3.18)

Solution 2. Taking Laplace and Hankel transform of (3.4), we get ̃𝐽̃𝑢(𝑟,𝑠)=1𝜆𝑛𝜆𝑛𝑓(𝑠)𝑠2𝛼+ğ‘Žğ‘ ğ›¼+𝑑𝜆2𝑛,(3.19) on taking Inverse Laplace transform of equation (3.19), we get ̃𝑢(𝑟,𝑡)=𝐿−1𝐽𝑓(𝑠)1𝜆𝑛𝜆𝑛𝐿−11𝑠2𝛼+ğ‘Žğ‘ ğ›¼+𝑑𝜆2𝑛.(3.20)
By using convolution theorem for Laplace transform and taking inverse Hankel transform, we get 𝑢(𝑟,𝑡)=2âˆžî“âˆžğ‘›=0𝑚=0(−1)ğ‘šğ‘Žğ‘š+1𝐽0𝜆𝑛𝑟𝜆𝑛𝐽1𝜆𝑛𝑡0𝑢𝛼−𝑚𝛼−1𝐸𝑚𝛼,𝛼−2𝑚𝛼−𝑑𝜆2ğ‘›ğ‘¢ğ›¼ğ‘Žî‚¶ğ‘“(𝑡−𝑢)𝑑𝑢,(3.21) or 𝑢(𝑟,𝑡)=2âˆžî“âˆžğ‘›=0𝑚=0(−1)ğ‘šğ‘Žğ‘š+1𝐽0𝜆𝑛𝑟𝜆𝑛𝐽1𝜆𝑛𝑡0ğ‘¢âˆžğ›¼âˆ’ğ‘šğ›¼âˆ’1𝑗=0(𝑗+𝑚+1)!(𝑗)!−𝑑𝜆2𝑛𝑢𝛼/ğ‘Žğ‘—Î“.(𝛼𝑗+𝛼−𝑚𝛼)(3.22) By using the relation (2.2) 𝑢(𝑟,𝑡)=2âˆžî“âˆžğ‘›=0𝑚=0(−1)ğ‘šğ‘Žğ‘š+1𝐽0𝜆𝑛𝑟𝜆𝑛𝐽1𝜆𝑛𝑡0𝑢𝛼−𝑚𝛼−1𝐻1,11,2𝑑𝜆2ğ‘›ğ‘¢ğ›¼ğ‘Ž||||(−𝑚−1,1)(0,1),(1−𝛼+𝑚𝛼,𝛼)𝑓(𝑡−𝑢)𝑑𝑢,(3.23) or 𝑢(𝑟,𝑡)=2âˆžî“âˆžğ‘›=0𝑚=0(−1)ğ‘šğ‘Žğ‘š+1𝐽0𝜆𝑛𝑟𝜆𝑛𝐽1𝜆𝑛𝑡0𝑢𝛼−𝑚𝛼−11Γ(𝑚)1Ψ1(−𝑚,1);(𝛼−2𝑚𝛼,𝛼);𝑑𝜆2ğ‘›ğ‘¢ğ›¼ğ‘Žîƒ­,(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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