Some investigations on singular cauchy problems

Abstract

The purpose of our study is to get a solution to the Cauchy problem of (i) The wave equation in n-dimension space Rn which is effectively a good example of regular Cauchy problems (ii) The Euler Poisson Darboux equation which we call singular Cauchy problem by use of Riemann's method. The Riemann-Green function for each case is calculated, which enables us to evaluate any solution at a point by the Cauchy data on a non-characteristic curve. In case (i) the Riemann-Green function is in terms of Legendre polynomial and the solution obtained is shown to solve the wave equation as well. In case (ii) the Riemann-Green function written in terms of the Appell's hyper geometric function of two variables is arrived at, this is of interest and may be a good model for a more general theory. A discussion of the generalized singular Cauchy problem of Euler-Poisson-Darboux equation is included and found to have solution that is continuous and analytic over the interval that contains the singular point.