Classical Solution of a Problem with Integral Conditions of the Second Kind for the One-Dimensional Wave Equation

Abstract

for the one-dimensional wave equation given in a half-strip, we consider a boundary value problem with the Cauchy conditions and two nonlocal integral conditions. Each integral condition is a linear combination of a linear Fredholm integral operator along the lateral side of the half-strip applied to the solution and the values of the solution and of a given function on the corresponding base of the half-strip. Under the assumption of appropriate smoothness of the right-hand side of the equation and the initial data, we obtain a necessary and sufficient condition for the existence and uniqueness of a classical solution of this problem and propose a method for finding it in analytical form. A classical solution is understood as a function that is defined everywhere in the closure of the domain where the equation is considered and has all classical derivatives occurring in the equation and in the conditions of the problem.