On singular and regular cauchy problems

Abstract

Singular Cauchy Problem for the Euler-Poisson-Darboux EquationConsider the partial differential equation Cauchy problem for (1) consists in the determination of a solution of (l), for t > 0, meeting the following initial conditions on the "singular" plane t = 0:where g is a given function. For k any real number, this singular Cauchy problem was first solved by A. Weinstein [7], who employed what he termed the "method of recurrence" and a generalized method of descent. Let, for the time being, the solution of the Cauchy problem (l), (2) be denoted by Z J~. Weinstein shows that for k = -1, -3, * * -the Cauchy problem for uk can be reduced to a Cauchy problem for u k + 2 by means of the following recurrence formulas:

Solutions in the Hyperbolic Region of an Equation, which Approximates Chaplygin's Equation Near the Vacuum Line

“…The singular Cauchy problem has been studied quite extensively by A. Weinstein [10,11], J. B. Diaz [12,16], H. F. Weinberger [12] and E. K. Blum [13,14,15]. Their method consists essentially of a combination of a generalized method of descent applied on the formula of Poisson with two relations for solutions with different k. Also, solutions to the problem were found using an analytical continuation of definite integrals with respect to k.…”