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 problem of Euler-Poisson-Darboux (EPD) equation has been studied. Firstly, the general exact solution for the one dimensional regular Cauchy problem of EPD has been worked out. The EPD which is a second order Partial Differential Equation (PDE) is converted into an Ordinary Differential Equation (ODE) by method of separation of variables. On solving the ODE, the first complementary function (cf) is obtained directly. The second cf is obtained when the first derivative is eliminated from the ODE and then the ODE solved. When the expression for eliminating the first derivative is solved, a third term is obtained. The general solution for the one dimensional regular Cauchy EPD is therefore 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.