The Goursat Problem for a Higher-Order Hyperbolic Equation with a Spectral Parameter

Abstract

In this article, the Goursat problem for higher-order hyperbolic equations with a spectral parameter has been formulated and studied. The main objective of the article is to find an exact analytical solution to this problem based on the Riemann function. The study considers a higher-order hyperbolic equation defined in a domain bounded by characteristic lines. The conditions of the Goursat problem imply that the function and its higher-order normal derivatives are given on these characteristic lines. To find the solution, the method involves transitioning to characteristic coordinates and determining the corresponding Riemann function. This function is expressed using a special function of the Mittag-Leffler. By applying Green's formula and taking the boundary conditions into account, the solution to the Goursat problem is found in an explicit form at any point and then transformed back to the original variables. The article proves that if the boundary conditions are given by sufficiently smooth functions, the problem has a unique solution. The main novelty lies in the fact that this type of problem is solved for the first time using the Riemann method, and the solution is explicitly expressed.