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

Date

2017-07-11

Authors

Shishkina, Elina
Sitnik, Sergei

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Publisher

Texas State University, Department of Mathematics

Abstract

In this article, 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.

Description

Keywords

Bessel operator, Euler-Poisson-Darboux equation, Hankel transform, Transmutation operators

Citation

Shishkina, E. L., & Sitnik, S. M. (2017). General form of the Euler-Poisson-Darboux equation and application of the transmutation method. <i>Electronic Journal of Differential Equations, 2017</i>(177), pp. 1-20.

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Attribution 4.0 International

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