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.