Integral transform approach to solving Klein-Gordon equation with variable coefficients

TL;DR

This paper introduces a generalized integral transform method for solving Klein-Gordon equations with spatially variable coefficients, extending previous approaches and enabling solutions via related simpler equations.

Contribution

The paper develops a new generalized integral transform that facilitates solving Klein-Gordon equations with variable coefficients, expanding the applicability of previous methods.

Findings

Transform enables solutions of complex Klein-Gordon equations with variable coefficients.

Application to equations in de Sitter and Einstein-de Sitter spacetimes.

Provides a framework for solving PDEs via related simpler equations.

Abstract

In this paper we describe the integral transform that allows to write solutions of one partial differential equation via solution of another one. This transform was suggested by the author in the case when the last equation is a wave equation, and then it was used to investigate several well-known equations such as Tricomi-type equation, the Klein-Gordon equation in the de~Sitter and Einstein-de~Sitter spacetimes. A generalization given in this paper allows us to consider also the Klein-Gordon equations with coefficients depending on the spatial variables.