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
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.
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
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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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