Particular solutions of generalized Euler-Poisson-Darboux equation

Seilkhanova, Rakhila B. ; Hasanov, Anvar H.

Electronic Journal of Differential Equations, Tome 2015 (2015).

Résumé

Summary: 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 ${{r}^2}={{( x-{{x}_0})}^2}+{{( y-{{y}_0})}^2}-{{( t-{{t}_0})}^2}$.

Détail

Classification : 35Q05, 35L80, 35C65
Keywords: generalized Euler-Poisson-Darboux equation, hyperbolic equation, lauricelli hypergeometric functions