Electron. J. Differential Equations,
Vol. 2017 (2017), No. 177, pp. 120.
General form of the EulerPoissonDarboux 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 EulerPoissonDarboux
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; EulerPoissonDarboux 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 MiklukhoMaklaya st
Moscow, Russia
email: pochtasms@gmail.com

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