Certain Solutions of Abel’s Integral Equations on Distribution Spaces via Distributional Gα-Transform

Abstract

Abel’s integral equation is an efficient singular integral equation that plays an important role in diverse fields of science. This paper aims to investigate Abel’s integral equation and its solution using Gα-transform, which is a symmetric relation between Laplace and Sumudu transforms. Gα-transform, as defined via distribution space, is employed to establish a solution to Abel’s integral equation, interpreted in the sense of distributions. As an application to the given theory, certain examples are given to demonstrate the efficiency and suitability of using the Gα-transform method in solving integral equations.

Solution of linear PDEs obtained using Laplace transform

“…The structure and properties of the generalized Laplace transform were studied in [2]. Laplace transforms of generalized hypergeometric functions [3] were studied, and as an application of the generalized Laplace transform, certain solutions of Abel's integral equations on distribution spaces via the distributional G α -transform [4] were also studied. Of course, the method used here can be used for Laplace-typed transforms [2][3][4].…”

“…Laplace transforms of generalized hypergeometric functions [3] were studied, and as an application of the generalized Laplace transform, certain solutions of Abel's integral equations on distribution spaces via the distributional G α -transform [4] were also studied. Of course, the method used here can be used for Laplace-typed transforms [2][3][4]. As a related application, brand-new exact solutions to the Sawada-Kotera equation were investigated [5].…”

Solution of Fractional Third-Order Dispersive Partial Differential Equations and Symmetric KdV via Sumudu–Generalized Laplace Transform Decomposition

“…The researchers in [11] examined the applicability range of the G α -transform in solving ordinary differential equations with variable coefficients. The study conducted in [12] delves into the solutions of Abel's integral equations on distribution spaces using the distributional G α -transform. Furthermore, the author of [13] analyzed the analytic solution of third-order dispersive partial differential equations.…”