Wavelet based solutions to the Poisson and the Helmholtz equations on the $n$-dimensional unit sphere

TL;DR

This paper introduces a wavelet-based approach for solving Poisson and Helmholtz equations on the n-dimensional sphere, providing explicit solutions and Green functions for certain cases.

Contribution

It develops a novel wavelet transform method for PDEs on spheres, yielding explicit solutions and Green functions for these equations.

Findings

Explicit analytical solutions for Poisson and Helmholtz equations.

Closed-form Green functions for specific parameter values.

Method applicable to n-dimensional spheres.

Abstract

We present a method of solving partial differential equations on the $n$-dimensional unit sphere using methods based on the continuous wavelet transform derived from approximate identities. We give an explicit analytical solution to the Poisson equation and to the Helmholtz equations. For the first one and for some special values of the parameter in the latter one, we derive a closed formula for the generalized Green function.