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arXiv · 1707.04733

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

Elina L. Shishkina·Sergei M. Sitnik
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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.

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Elina L. Shishkina, Sergei M. Sitnik. 2017-07-15. The general form of the Euler--Poisson--Darboux equation and application of transmutation method. https://arxiv.org/abs/1707.04733

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