General form of the Euler-Poisson-Darboux equation and application of the transmutation method
Date
2017-07-11
Authors
Shishkina, Elina
Sitnik, Sergei
Journal Title
Journal ISSN
Volume Title
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. Electronic Journal of Differential Equations, 2017(177), pp. 1-20.
Rights
Attribution 4.0 International