Non-homogeneous wave equation on a cone

TL;DR

This paper establishes the existence and explicit construction of solutions for a non-homogeneous wave equation on a conical domain, utilizing Fourier series of orthogonal polynomials, and addresses boundary value problems in this setting.

Contribution

It provides a unique solution framework for the non-homogeneous wave equation on a cone, with explicit formulas using orthogonal polynomial Fourier series, advancing boundary value problem solutions.

Findings

Unique solutions exist under specified conditions.

Explicit solution formulas using orthogonal polynomial Fourier series.

Applicable to boundary value problems on conical domains.

Abstract

The wave equation $\left(\partial_{tt} - c^2 \Delta_x\right) u(x,t) = e^{-t} f(x,t)$ is shown to have a unique solution if $u$ and its partial derivatives in $x$ are in $L^2(e^{-t})$ on the cone, and the solution can be explicit given in the Fourier series of orthogonal polynomials on the cone. This provides a particular solution for the boundary value problems of the non-homogeneous wave equation on the cone, which can be combined with a solution to the homogeneous wave equation in the cone to obtain the full solution.