The general form of the Euler--Poisson--Darboux equation and application of transmutation method

TL;DR

This paper derives integral solution representations for the Euler--Poisson--Darboux equation with Bessel operators, including exceptional cases, using a Hankel transform and transmutation methods, highlighting their importance in differential equations.

Contribution

It provides a unified approach to solving the Euler--Poisson--Darboux equation with Bessel operators for all parameter values, including previously unstudied negative odd values, using transmutation methods.

Findings

Derived integral solution representations for all parameter values.

Proved a transmutation property for generalized spherical mean.

Established conditions under which distributional solutions are classical.

Abstract

In the paper we find solution representations in the compact integral form to the Cauchy problem for a general form of the Euler--Poisson--Darboux equation with Bessel operators via generalized translation and spherical mean operators for all values of the parameter $k$, including also not studying before exceptional odd negative values. We use a Hankel transform method to prove results in a unified way. Under additional conditions we prove that a distributional solution is a classical one too. A transmutation property for connected generalized spherical mean is proved and importance of applying transmutation methods for differential equations with Bessel operators is emphasized. The paper also contains a short historical introduction on differential equations with Bessel operators and a rather detailed reference list of monographs and papers on mathematical theory and applications of this class of differential equations.