Wave polynomials, transmutations and Cauchy's problem for the Klein-Gordon equation

TL;DR

This paper introduces generalized wave polynomials for the Klein-Gordon equation with variable coefficients, proving their completeness and developing a numerical method for solving the Cauchy problem.

Contribution

It extends the concept of wave polynomials to variable coefficient Klein-Gordon equations using transmutation operators, enabling explicit solution construction.

Findings

Proved completeness of generalized wave polynomials.

Constructed explicit solutions for the Cauchy problem.

Developed and tested a numerical method for the Klein-Gordon equation.

Abstract

We prove a completeness result for a class of polynomial solutions of the wave equation called wave polynomials and construct generalized wave polynomials, solutions of the Klein-Gordon equation with a variable coefficient. Using the transmutation (transformation) operators and their recently discovered mapping properties we prove the completeness of the generalized wave polynomials and use them for an explicit construction of the solution of the Cauchy problem for the Klein-Gordon equation. Based on this result we develop a numerical method for solving the Cauchy problem and test its performance.