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
| Sergei M. Sitnik |
Belgorod State National Research University
RUDN University, 6 Miklukho-Maklaya st
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