Studying a system of non-local condition hyperbolic equations

Abstract

Local boundary value problems for hyperbolic differential equations have been studied in considerable detail. However, the mathematical modeling of a number of real-world processes leads to nonlocal boundary value problems involving nonlinear hyperbolic differential equations, which remain poorly understood. In this paper, we consider a system of hyperbolic equations defined by both point and integral boundary conditions in a rectangular domain. To the best of our knowledge, such a problem is studied here for the first time. We note that this formulation is quite general and encompasses several special cases. The classical Goursat-Darboux problem-a problem with integral boundary conditions in which some boundary conditions are specified as point conditions and others as integral conditions-is derived from this formulation as a particular case. Under natural conditions on the initial data, the necessary conditions for the solvability of a nonlocal boundary value problem are established. A corresponding Green‘s function for the boundary value problem is constructed and the problem is reduced to an equivalent integral equation. Using the principle of contracting Banach maps, conditions for the existence and uniqueness of a solution to the boundary value problem are established. An example is given illustrating the validity of the obtained results.