Electron. J. Differential Equations, Vol. 2017 (2017), No. 177, pp. 1-20.

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

Elina L. Shishkina, Sergei M. Sitnik

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.

Submitted May 22, 2017. Published July 11, 2017.
Math Subject Classifications: 26A33, 44A15.
Key Words: Bessel operator; Euler-Poisson-Darboux equation; Hankel transform; transmutation operators.

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Elina L. Shishkina
Voronezh State University
Faculty of Applied Mathematics, Informatics and Mechanics
Universitetskaya square, 1
Voronezh 394006, Russia
email: ilina_dico@mail.ru
Sergei M. Sitnik
Belgorod State National Research University
Belgorod, Russia.
RUDN University, 6 Miklukho-Maklaya st
Moscow, Russia
email: pochtasms@gmail.com

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