Singular Cauchy problem for generalized homogeneous Euler-Poisson-Darboux equation

Abstract

In this paper, we solve singular Cauchy problem for a generalised form of an homogeneous Euler–Poisson–Darboux equation with constant potential, where Bessel operator acts instead of the each second derivative. In the classical formulation, the Cauchy problem for this equation is not correct. However, S. A. Tersenov observed that, considering the form of a general solution of the classical Euler–Poisson–Darboux equation, the derivative in the second initial condition must be multiplied by a power function whose degree is equal to the index of the Bessel operator acting on the time variable. The first initial condition remains in the usual formulation. With the chosen form of the initial conditions, the considering equation has a solution. Obtained solution is represented as the sum of two terms. The first tern is an integral containing the normalized Bessel function and the weighted spherical mean. The second term is expressed in terms of the derivative of the square of the time variable from the integral, which is similar in structure to the first term.