An Operator Approach to the Cauchy Problem for the Euler-Poisson-Darboux Equation in Spaces of Constant Curvature

Abstract

We develop an operator approach to the problem named in the title. It is based on operator treatment of some relations between Bessel functions, and hypergeometric functions as well. Our operator version of the DelsartePovzner formulas also plays an essential role in the paper. Introduction.These notes illustrate an operator reading of some formulas of analysis which gives these formulas a new meaning so that they become an heuristic source of non-trivial results.We have in mind some relations concerning Bessel and hypergeometrical functions and the Delsarte-Povzner formulas.Read anew, they make it possible to find, among other things a solution of the singular and regular Cauchy problems for the generalized Euler-Poisson-Darboux (EPD) equation in the spaces S n of constant curvature k = 0.We explain the essence of our approach on the well-investigated [1-6] singular Cauchy problem for the EPD equationThe solution of this problem can be represented symbolically using Bessel functions of the Laplacian. In order to give an explicit meaning to this representation, we notice that for a certain value of the parameter a the solution can be obtained as the average M (t, x; f ) of f over the sphere of radius t centered at x [7,8].On the other hand a Sonin formula gives a connection between Bessel functions with different values of the indices. We introduce an operator version of the Sonin formula which allows one to easily obtain the solution for any a. For a detailed presentation see Sec. 2.

Fractional Klein–Gordon Equations and Related Stochastic Processes

“…The operational solution (8.6) becomes an effective solution when the series converges, and this depends upon the actual form of the initial condition g(x). Operational methods to solve Euler-Poisson-Darboux equations are applied in Olevskii (2004).…”