Probl\`eme de Cauchy avec des conditions modifi\'ees pour les \'equations d'Euler-Poisson-Darboux

TL;DR

This paper generalizes and unifies solutions to the Euler-Poisson-Darboux equation with modified Cauchy conditions, providing explicit solutions using hypergeometric functions and applying them to wave equations.

Contribution

It introduces a unified approach to solving the Euler-Poisson-Darboux equation with modified conditions, expressing solutions explicitly via hypergeometric functions.

Findings

Explicit solutions in terms of hypergeometric functions.

Applications to classical and radial wave equations.

Generalization of existing results on Euler-Poisson-Darboux equations.

Abstract

Nowadays the Euler-Poisson-Darboux equation is extensively studied in several settings. The main questions on avery spaces are explicit solutions for the classical Cauchy problems with the second data null. In this note we will generalize and unify several results on Euler-Poisson-Darboux equation. In this note we consider the Cauchy problem with modified conditions for the Euler-Poisson-Darboux equation. We give the explicit solutions in terms of Gauss 2F1 and Appel F4 hypergeometric functions with applications to the classical and radial wave equations