General analytical solution for the one dimensional regular Cauchy problem of Euler-Poisson-Darboux equation

Abstract

A general analytical solution to the one dimensional regular Cauchy problem of Euler-Poisson-Darboux (EPD) equation has been investigated. The one dimensional EPD which is a Partial Differential Equation (PDE) with initial conditions is transformed into Ordinary Differential Equation (ODE) using Similarity Transformation. The first derivative of the ODE is eliminated by substitution technique. The coefficient of the first derivative is equated to zero and then solved. The general solution is a product of two terms. The first term is the one obtained when the first derivative is eliminated from the ODE and the second term is the complementary function (cf) obtained from the remaining part of ODE. The arbitrary constants of the cf are obtained in terms of x and t when the initial conditions are substituted in the general solution. The general solution is a solution for one dimensional regular Cauchy EPD’s and degenerate EPD’s, which by coincident are one dimensional wave equations.