Electron. J. Diff. Equ., Vol. 2015 (2015), No. 09, pp. 1-10.

Particular solutions of generalized Euler-Poisson-Darboux equation

Rakhila B. Seilkhanova, Anvar H. Hasanov

Abstract:
In this article we consider the generalized Euler-Poisson-Darboux equation
$$ {u}_{tt}+\frac{2\gamma }{t}{{u}_{t}}={u}_{xx}+{u}_{yy} +\frac{2\alpha }{x}{{u}_{x}}+\frac{2\beta }{y}{{u}_y},\quad x>0,\;y>0,\;t>0. $$
We construct particular solutions in an explicit form expressed by the Lauricella hypergeometric function of three variables. Properties of each constructed solutions have been investigated in sections of surfaces of the characteristic cone. Precisely, we prove that found solutions have singularity $1/r$ at $r\to 0$, where ${{r}^2}={{( x-{{x}_0})}^2}+{{( y-{{y}_0})}^2}-{{( t-{{t}_0})}^2}$.

Submitted October 21, 2014. Published January 5, 2015.
Math Subject Classifications: 35Q05, 35L80, 35C65.
Key Words: Generalized Euler-Poisson-Darboux equation; hyperbolic equation; Lauricelli hypergeometric functions.

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Rakhila B. Seilkhanova
S. Baishev University Aktobe
Department for Information Systems and Applied Mathematics
030000 Aktobe, Zhubanov str. 302, Kazakhstan
email: srahila@inbox.ru
Anvar H. Hasanov
S. Baishev University Aktobe
Department for Information Systems and Applied Mathematics
030000 Aktobe, Zhubanov str. 302, Kazakhstan
email: anvarhasanov@yahoo.com

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