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.
Highlights
Introduction and PreliminariesOne of the classical integral transforms is the Mellin transform ∞M f x ;p F p xp−1f x dx, c1 < p < c2, 1.1 and its inversion formula is written in terms of the Bromwich’s integral in the following form: fx 1 2π i c i∞ F p x−pdp, c−i∞ c1 < c < c2.This transform is used for expressing many problems in applied sciences
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
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
Summary
M f x ;p F p xp−1f x dx, c1 < p < c2, 1.1 and its inversion formula is written in terms of the Bromwich’s integral in the following form: fx. 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: gy h x, y f x dx. With considering the above theorem, in section, we find formal solutions of some singular integral equations
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