On the Solution of n-Dimensional Regular Cauchy Problem of Euler-Poisson-Darboux Equation (EPD)

Abstract

A general exact solution to the n-dimensional regular Cauchy prob- lem of Euler-Poisson-Darboux (EPD) equation has been studied. Firstly, the general exact solution for the one dimensional regular Cauchy prob- lem of EPD has been worked out. The EPD which is a second order Partial Di erential Equation (PDE) is converted into an Ordinary Dif- ferential Equation (ODE) by method of separation of variables. On solving the ODE, the rst complementary function (cf) is obtained di- rectly. The second cf is obtained when the rst derivative is eliminated from the ODE and then the ODE solved. When the expression for eliminating the rst derivative is solved, a third term is obtained. The general solution for the one dimensional regular Cauchy EPD is there- fore the product of the three terms. The procedure has been repeated for the two dimensional and n-dimensional cases. The general solutions for these cases are products of four terms and n+2 terms respectively. Finally, the general exact solution for n-dimensional regular Cauchy wave equation when k = 0, has also been obtained. Mathematics Subject Classi cation: 35Q05