Two Problems for One Hyperbolic Equation of the Third Order in Three-Dimensional Space

TL;DR

This paper addresses solving a modified Cauchy problem for a third-order hyperbolic equation in three-dimensional space using Riemann's method, introducing a special class for simplified solutions and analyzing the problem's parameter space.

Contribution

It introduces a special class for the third-order hyperbolic equation that simplifies solutions and extends the parameter range for which solutions are found.

Findings

Unique solutions for the integral equations at various parameter values

Expanded parameter range for the special class solutions

Reduction of mixed problem to two-dimensional Volterra integral equations

Abstract

In the present article, a modified Cauchy problem (problem C) for the hyperbolic equation of the third order with the data on the equation's coefficients singularity plane is solved by Riemann method. The special class in which the solution of the problem C has more simple appearance is introduced and the area of values of the parameter p entering into the equation is considerably expanded. In the special class the mixed problem, which decision was been reduced to the two-dimensional Volterra's integral equations of the first order with uncurtailed operators, is considered. Authors found the unique solution of these equations at various values of the parameter p.