Table of Contents
ISRN Applied Mathematics
Volume 2012, Article ID 608712, 10 pages
http://dx.doi.org/10.5402/2012/608712
Research Article

New Inversion Techniques for Some Integral Transforms via the Generalized Product Theorem of the Mellin Transform

Department of Mathematics, Faculty of Sciences, Shahrekord University, P.O. Box 115, Shahrekord, Iran

Received 21 December 2011; Accepted 17 January 2012

Academic Editors: C. Chen, F. Ding, and H. Homeier

Copyright © 2012 Alireza Ansari and Mohammadreza Ahmadi Darani. 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.

Abstract

We introduce the generalized product theorem for the Mellin transform, and we solve certain classes of singular integral equations with kernels coincided with conditions of this theorem. Also, new inversion techniques for the Wright, Mittag-Leffler, Stieltjes, and Widder potential transforms are obtained.

1. Introduction and Preliminaries

One of the classical integral transforms is the Mellin transform {𝑓(𝑥);𝑝}=𝐹(𝑝)=0𝑥𝑝1𝑓(𝑥)𝑑𝑥,𝑐1<𝑝<𝑐2,(1.1) and its inversion formula is written in terms of the Bromwich’s integral in the following form:1𝑓(𝑥)=2𝜋𝑖𝑐+𝑖𝑐𝑖𝐹(𝑝)𝑥𝑝𝑑𝑝,𝑐1<𝑐<𝑐2.(1.2)

This transform is used for expressing many problems in applied sciences. An application of this transform that may occur in problems led to following singular integral equation:0𝑘(𝑥,𝑦)𝑔(𝑦)𝑑𝑦=𝑓(𝑥),𝑥>0.(1.3)

In operational calculus of the Mellin transform, in this paper, we state the generalized product theorem for the Mellin transform and consider a certain class of singular integral equation (1.3) which its kernel is coincided with the conditions of the generalized product theorem.

This technique enables us to get formal solution of singular integral equation in terms of an improper integral in the following form:𝑔(𝑦)=0(𝑥,𝑦)𝑓(𝑥)𝑑𝑥.(1.4) For this purpose, at first step by writing some main properties of the Mellin transform, we solve some singular integral equations with kernels in terms of elementary functions in Section 2. In the next section, we introduce new approaches for finding inversion formulas for some integral transforms. The Wright, Mittag-Leffler, Stieltjes, and Widder potential transforms [4, 7, 8] are the selected integral transforms that we find new inversion techniques for them. Finally, an Appendix has been drawn for transformed functions used in paper.

First, we recall some fundamental properties of the Mellin transform which can be easily written with respect to definition (1.1). For more details and properties of this transform, see [1].(i) The convolution theorem for the Mellin transform is𝐹(𝑝)𝐺(𝑝)={𝑓𝑔}=0𝑥𝑔(𝑢)𝑓𝑢𝑑𝑢𝑢.(1.5)(ii)The Mellin transform of 𝛿𝑥-derivatives is𝛿𝑛𝑥𝑓(𝑥)=(𝑝)𝑛𝐹(𝑝),𝛿𝑥𝑑=𝑥𝑑𝑥.(1.6)(iii)Change of scale property of the Mellin transform is{𝑓(𝑥𝑎1)}=𝑎𝐹𝑝𝑎,𝑎>0.(1.7)(iv)Translation property of the Mellin transform is{𝑥𝑎𝑓(𝑥)}=𝐹(𝑝+𝑎),𝑎>0.(1.8)

Now, we state the generalized product theorem for the Mellin transform.

Theorem 1.1 (generalized product theorem). Let {𝑔(𝑥);𝑝}=𝐺(𝑝) and assume that Ψ1(𝑝), Ψ2(𝑝) are analytic functions such that, {𝑘(𝑥,𝑦);𝑝}=Ψ1(𝑝)𝑦Ψ2(𝑝)1, then the following relation holds true for continuous function 𝑘(𝑥,𝑦) on the rectangular region 𝑎𝑥𝑏, 𝑐𝑦𝑑, [𝑎,𝑏]×[𝑐,𝑑][0,)×[0,), 0𝑘(𝑥,𝑦)𝑔(𝑦)𝑑𝑦=Ψ1Ψ(𝑝)𝐺2(𝑝).(1.9)

Proof. Using the definition of the Mellin transform and considering the condition of continuous function 𝑘(𝑥,𝑦) in order to change the order of integration, we get 0=𝑘(𝑥,𝑦)𝑔(𝑦)𝑑𝑦0𝑥𝑝10=𝑘(𝑥,𝑦)𝑔(𝑦)𝑑𝑦𝑑𝑥0𝑔(𝑦)0𝑥𝑝1𝑘(𝑥,𝑦)𝑑𝑥𝑑𝑦=Ψ1(𝑝)0𝑦Ψ2(𝑝)1𝑔(𝑦)𝑑𝑦=Ψ1Ψ(𝑝)𝐺2.(𝑝)(1.10)

With considering the above theorem, in next section, we find formal solutions of some singular integral equations.

2. Singular Integral Equations with Kernels of Elementary Functions

Problem 1. Solve the singular integral equation with the following logarithmic kernel [2]: 0||||ln𝑥𝑦𝑔(𝑦)𝑑𝑦=𝑓(𝑥),𝑥>0.(2.1)

By showing the above equation in the following form:0||||ln𝑥𝑦𝑦||||𝑔(𝑦)𝑑𝑦=𝑓(𝑥)𝑓(0),𝑥>0(2.2) and applying the Mellin transform on both sides of equation, we get0||||ln𝑥𝑦𝑦||||𝑔(𝑦)𝑑𝑦;𝑝={𝜙(𝑥),𝑝},(2.3)

where 𝜙(𝑥) is defined as 𝜙(𝑥)=𝑓(𝑥)𝑓(0).

Now, by using the generalized product theorem (1.9) and considering the relation (A.1) for the Mellin transform of ln(|(𝑥/𝑦)1|), we rewrite the above equation in the form𝜋𝑝cot(𝜋𝑝)0𝑦𝑝𝑔(𝑦)𝑑𝑦=Φ(𝑝).(2.4)

The above equation can be rewritten as the Mellin transform of function 𝑔(𝑥) as𝐺𝑝(𝑝+1)=𝜋tan(𝜋𝑝)𝜙(𝑝).(2.5) By implementation of the inverse of the Mellin transform and considering the translation property (1.8) and the convolution property (1.5), simultaneously, we obtain1𝑦𝑔(𝑦)=𝜋01𝑦𝑝tan(𝜋𝑝),𝑢𝜙(𝑢)𝑑𝑦𝑢.(2.6) At this point, by reconsidering the following relation for 1{𝑝tan(𝜋𝑝)} in view of the relation (A.7) in the appendix:1{𝑝tan(𝜋𝑝);𝑦}=1𝑝2tan(𝜋𝑝)𝑝=𝑦𝑑;𝑦𝑑𝑦21tan(𝜋𝑝)𝑝=1;𝑦𝜋𝑦𝑑𝑑𝑦2ln1+𝑦1𝑦(2.7) and substituting in the relation (2.5), the solution of the singular integral equation (2.1) is written as1𝑔(𝑦)=𝜋2𝑑𝑦𝑑𝑑𝑦𝑑𝑦0|||||ln𝑦+𝑢𝑦𝑢|||||𝜙(𝑢)𝑑𝑢𝑢.(2.8)

Problem 2. Solve the singular integral equation with the following inverse trigonometric kernel: 0sin1𝑦𝑦2+𝑥2𝑔(𝑦)𝑑𝑦=𝑓(𝑥),𝑥>0.(2.9)

By applying the Mellin transform on both sides of equation0sin1𝑦𝑦2+𝑥2𝑔(𝑦)𝑑𝑦;𝑝={𝑓(𝑥),𝑝}(2.10)

and using the relation (A.2) in the appendix for the Mellin transform of kernel sin1(𝑦𝑦2+𝑥2), we get1𝑝sin𝜋𝑝20𝑦𝑝𝑔(𝑦)𝑑𝑦=𝐹(𝑝),(2.11)

which implies that𝐺(𝑝+1)=𝑝csc𝜋𝑝2𝐹(𝑝).(2.12) By applying the inverse Mellin transform and using the translation and convolution properties, we obtain𝑦𝑔(𝑦)=01𝑝csc𝜋𝑝2,𝑦𝑢𝑓(𝑢)𝑑𝑢𝑢.(2.13) According to the relation (A.8) in the appendix and the Mellin transform of delta derivatives (1.6), we finally get the solution of (2.9) as follows:2𝑔(𝑦)=𝜋𝑑𝑑𝑦01𝑢𝑦+𝑢𝑓(𝑢)𝑑𝑢.(2.14)

Problem 3. Solve the singular integral equation with the following exponential kernel: 0𝑒(1/𝑦)𝑥𝛼𝑔(𝑦)𝑑𝑦=𝑓(𝑥),𝛼>0.(2.15)

In the same procedure to previous problem, after applying the Mellin transform on equation and using the relation (A.3) in the appendix, we getΓ𝑝𝛼𝐺𝑝𝛼+1=𝛼𝐹(𝑝),(2.16) or equivalentlyΓ(𝑝)𝐺(𝑝+1)=𝛼𝐹(𝛼𝑝).(2.17) Also, by applying the inverse of Mellin transform and using the convolution property, we obtain0𝑒𝑦/𝑢𝑦𝑔(𝑢)𝑑𝑢=𝑓1/𝛼.(2.18) The above equation implies that the function 𝑔 can be obtained in terms of the Laplace transform inversion as [3]1𝑔(𝑦)=𝑦21𝑓𝑢1/𝛼;1𝑦.(2.19)

3. Inversion Techniques for Some Integral Transforms

Similar to the procedures in the previous section for solving singular integral equations, in this section, we find new inversion formulas for some integral transforms. First, we consider the Wright transform and Mittag-Leffler transforms. These integral transforms have been recently arisen in fractional calculus [4], and it is necessary to have inversion techniques for them.

3.1. The Wright Transform

For the following Wright transform [4]:0𝑊1𝛾,0;𝑦𝑥𝛾𝑔(𝑦)𝑑𝑦=𝑓(𝑥),𝑥>0,0<𝛾<1,(3.1) where the Wright function is presented by the following relation:𝑊(𝛼,𝛽;𝑧)=𝑘=0𝑧𝑘𝑘!Γ(𝛼𝑘+𝛽),𝛼>1,𝛽,𝑧,(3.2) we show an inversion formula for function 𝑔(𝑦). At first, by applying the Laplace transform on both sides of equation with respect to 𝑥,0𝑊1𝛾,0;𝑦𝑥𝛾𝑔(𝑦)𝑑𝑦;𝑠={𝑓(𝑥);𝑠},𝑥>0,(3.3) and using the fact that [5]𝑒(1/𝑦)𝑠𝛾=0𝑒𝑠𝑥1𝑥𝑊1𝛾,0;𝑦𝑥𝛾𝑑𝑥,(3.4) we get0𝑒(1/𝑦)𝑠𝛾𝑔(𝑦)𝑑𝑦=Φ(𝑠),(3.5) where Φ is the Laplace transform of function 𝜙(𝑥)=𝑓(𝑥)/𝑥. Now, by considering the singular integral equation with exponential kernel (2.15) and its solution, we obtain the following relation for the inverse function 𝑔:1𝑔(𝑦)=𝑦21Φ𝑠1/𝛾;1𝑦=1𝑦21𝑓(𝑥)𝑥;𝑠1/𝛾;1𝑦.(3.6)

Example 3.1. Find the inverse function for the following Wright transform: 0𝑊1𝛾,0;𝑦𝑥𝛾𝑔(𝑦)𝑑𝑦=𝑥𝐸𝛼(𝑥𝛼),𝑥>0,0<𝛾<1,(3.7) where the Mittag-Leffler function, 𝐸𝛼(𝑧) is given by 𝐸𝛼(𝑧)=𝑘=0𝑧𝑘Γ(𝛼𝑘+1),𝛼>0,𝑧.(3.8)

With considering the relation (3.6) and the Laplace transform of the Mittag-Leffler function,𝐸𝛼(𝑥𝛼=𝑠);𝑠𝛼1𝑠𝛼+1,(3.9) and the function 𝑔(𝑦) can be obtained as1𝑔(𝑦)=𝑦21𝑠(𝛼/𝛾)1𝑠𝛼/𝛾;1+1𝑦=1𝑦2𝐸𝛼/𝛾1𝑦𝛼/𝛾.(3.10)

3.2. The Mittag-Leffler Transform

If we consider the Mittag-Leffler transform [4]0𝐸𝛼(𝑦𝑠𝛼)𝑔(𝑦)𝑑𝑦=𝑓(𝑠),0<𝛼<1,(3.11) where the Mittag-Leffler function is given by the relation (3.8), then by applying the inverse Laplace transform with respect to 𝑠 and using 1𝐸𝛼(𝑦𝑠𝛼=1);𝑟𝜋𝑦𝑟𝛼1sin(𝛼𝜋)𝑦2+2𝑦𝑟𝛼cos(𝛼𝜋)+𝑟2𝛼,(3.13) we get a transformed equation in the following form:1𝜋0𝑦𝑟𝛼1sin(𝛼𝜋)𝑦2+2𝑦𝑟𝛼cos(𝛼𝜋)+𝑟2𝛼𝑔(𝑦)𝑑𝑦=𝜙(𝑟),0<𝛼<1,(3.14) where 𝜙(𝑟) is the inverse Laplace transform of 𝑓(𝑠). The above relation can be obtained in view of the Titchmarsh theorem 1𝑓(𝑟)=𝜋0𝑒𝑠𝑟𝐹𝑠𝑒𝑖𝜋𝑑𝑠(3.12) for inverses of the Laplace transform of functions which have branch cut on the real negative semiaxis, see [6].

According to the relation (A.4) in the appendix, the integral equation (3.14) is coincided with the generalized product theorem; therefore, by applying the Mellin transform on the above equation, we get1𝛼𝐺𝑝𝛼𝜋𝑝+1csc𝛼sin(𝜋𝑝)=𝜙(𝑟)𝑟;𝑝,(3.15)

or equivalently𝐺(𝑝+1)csc(𝜋𝑝)sin(𝛼𝜋𝑝)=𝛼𝜙(𝑟)𝑟;𝛼𝑝.(3.16)

At this point, by using the inverse Mellin transform and using convolution and change of scale properties, we get the function 𝑔 as follows:𝑔1(𝑦)=𝑦0𝜓𝑦𝑢𝜙𝑢1/𝛼𝑢1/𝛼+1𝑑𝑢,(3.17) provided that the above integral converges absolutely. Also, the function 𝜓 involved in the above integral is given by𝜓(𝑥)=1sin(𝜋𝑝)=sin(𝛼𝜋𝑝);𝑥𝑛=1(1)𝑛𝑥𝑛/𝛼𝜋𝑛sin𝛼.(3.18)

Example 3.2. Find the inverse function for the following Mittag-Leffler transform: 0𝐸𝛼(𝑦𝑠𝛼1)𝑔(𝑦)𝑑𝑦=𝑠,0<𝛼<1.(3.19)

With considering the relation (3.17) and the inverse Laplace transform of 𝑓(𝑠)=1/𝑠,11𝑠;𝑢=𝜙(𝑢)=1,(3.20) the function 𝑔(𝑦) can be written in the following form:1𝑔(𝑦)=𝑦0𝜓𝑦𝑢𝑑𝑢𝑢1/𝛼+1.(3.21)

3.3. The Stieltjes and Widder Potential Transforms

At the same procedures to previous problems by using the generalized product theorem and implementation of relations (A.5), (A.6), we get a new inversion formula for the Stieltjes transform [7] as follows:011𝑦+𝑥𝑔(𝑦)𝑑𝑦=𝑓(𝑥),𝑔(𝑦)=2𝜋2𝑖𝑐+𝑖𝑐𝑖𝐹(𝑝)sin(𝜋𝑝)𝑦𝑝𝑑𝑝,(3.22) swhere 𝐹(𝑝) is the Mellin transform of function 𝑓(𝑥). Also, a new inversion formula for the the Widder potential transform [8] is obtained in the following form:0𝑦𝑦2+𝑥2𝑦𝑔(𝑦)𝑑𝑦=𝑓(𝑥),𝑔(𝑦)=2𝜋2𝑖𝑐+𝑖𝑐𝑖𝐹(𝑝)csc𝜋𝑝2𝑦𝑝𝑑𝑝.(3.23)

4. Conclusions

This paper provides new results in operational calculus for the Mellin transform. These results are derived form the generalized product theorem. New inversion formulas for the Wright and Mittag-Leffler transforms were obtained. These formulas may be considered as promising approaches in expressing the Wright and Mittag-Leffler functions in fractional calculus especially fractional differential equations theory. Also, new inversion techniques for the Stieltjes and Widder potential transform (arising in potential theory) were written. These techniques have been presented in terms of Bromwich’s integral, and inclusion of complex analysis in evaluating these transforms can be considered as an advantage of these techniques. Numerous calculations in Bromwich’s integral with respect to the residue theorem may be also considered as a disadvantage of them.

Appendix

The Mellin transform of functions coincided with the generalized product theorem [9]||||𝑥ln𝑦||||=𝜋1;𝑝𝑝𝑦𝑝cot(𝜋𝑝),1<𝑝<0,(A.1)sin1𝑦𝑦2+𝑥2=1;𝑝𝑝𝑦𝑝sin𝜋𝑝2𝑒,0<𝑝<1,(A.2)(1/𝑦)𝑥𝛼=1;𝑝𝛼𝑦𝑝/𝛼Γ𝑝𝛼,𝑝>0,(A.3)𝑦𝑥sin(𝛼𝜋)𝑦2+2𝑦𝑥cos(𝛼𝜋)+𝑥2;𝑝=𝜋𝑦𝑝1csc(𝜋𝑝)sin(𝛼𝜋𝑝),1<𝑝<1,(A.4)𝑦+𝑥;𝑝=𝜋𝑦𝑝1𝑦csc(𝜋𝑝),0<𝑝<1,(A.5)𝑦2+𝑥2=𝜋;𝑝2𝑦𝑝1csc𝜋𝑝2,0<𝑝<2.(A.6)

The Mellin transform of other functions used in this paper|||||ln1+𝑥1𝑥|||||;𝑝=𝜋tan(𝜋𝑝)𝑝2,1<𝑝<1,(A.7)𝜋11+𝑥;𝑝=csc𝜋𝑝2,0<𝑝<1.(A.8)

Acknowledgment

The authors would like to thank the referees for valuable comments.

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