Exact solution of Singular Cauchy problem for Euler-Poisson-Darboux equation

Abstract

In this work we studied Lie group symmetry of non-linear ordinary differential equations and partial differential equations. And we tried solve a singular Cauchy problem for Euler-Poisson-Darboux equation by use this method because we want to found exact solution. Lie group symmetry takes exact solution if and only if any equation is invariant. When we applied lie group symmetry on a singular Cauchy problem for Euler-Poisson-Darboux equation we have got a polynomial of equations and we equal this polynomial for zero, after that we got many equations equal to zero, when we solve those equations we can’t found infinitesimal transformation of a singular Cauchy problem for Euler-Poisson-Darboux equation. And for this reason, we raised the order of the derivative to a higher level, we also derived a new generator corresponding to the order of the new equation. When we solve new equation we can find infinitesimal transformation of these equations by many method of Mathematics, and by used algorithm of Lie group Symmetries, we got general solution of singular Cauchy problem for Euler-Poisson-Darboux equation.