Weak Solution of the Singular Cauchy Problem of Euler-Poisson-Darboux Equation for n= 4

Abstract

We consider the singular Cauchy problem of Euler-Poisson-Darboux equation (EPD) of the form utt+ k t ut=∇ 2u u (x, 0)= f (x), ut (x, 0)= 0 where∇ 2 is the Laplacian operator in R n, n will refer to dimension and k a real parameter. The EPD equation finds applications in geometry, applied mathematics, physics etc. Various authors have solved this problem for different values of n and k using various techniques since the time of Euler [1]. In this paper, we shall take the Fourier transform of the EPD equation with respect to the space coordinate. The equation so obtained is transformed into a Bessel differential equation. We solve this equation and obtain the inverse Fourier transform of its solution. Finally on using the convolution theorem, we obtain the weak solution of the EPD equation for n= 4. The case for n= 1 is an interesting one for this problem as the solution will be that of a 1− D wave equation. We therefore deduce the weak solution for the 1− D wave equation as well.