Inverse problem for a system of mixed-type second-order partial differential equations

Abstract

This paper investigates an inverse problem for a system of mixed-type second-order differential equations with a variable time direction. The problem involves finding the unknown right-hand side functions from the solution of the system under given boundary and initial conditions. The method of separation of variables is applied to reduce the problem to a spectral problem, in which eigenvalues and eigenfunctions are determined. The obtained spectral problem is shown to coincide with previously established spectral problems. The orthogonality of the eigenfunctions is used to derive explicit formulas for the coefficients. The existence and uniqueness of the solution are established using the properties of the eigenfunction system and the Hilbert-Schmidt theorem.