The Cauchy problem for partial differential equations of the second order and the method of ascent

Abstract

A method of “ascent” is used to solve the Cauchy problem for linear partial differential equations of the second order in p space variables with constant coefficients (i.e., the pure wave equation, the damped wave equation, and the heat equation). This method consists of inferring the solution of the problem referred to from the well known solution of the same problem for one space variable. The commutability of repeated pf ∝, the solution deduced by the method of singularities for the Cauchy problem for the damped wave equation, and the solution of singular integral equations of the Volterra type are also considered.