We would like to consider an application of generalized Laplace transform in partial differential equations(PDEs) by using the n-th partial derivatives. The tool of this research is the induction, and the proposed idea gives an easy solution to engineering problems by freely selecting the integer α in the definition.

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.

In this paper, we define the diamond Klein-Gordon Kernel T α and the diamond Klein-Gordon operator of order α on the function f bywhere α ∈ C, the symbol * designates the convolution, and f ∈ S, S is the Schwartz space of functions. In this paper, we aim to study the convolution of T α and obtain the operator

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