Classical solution of a problem with an integral condition for the one-dimensional wave equation

Abstract

We find a closed-form classical solution of the homogeneous wave equation with Cauchy conditions, a boundary condition on the lateral boundary, and a nonlocal integral condition involving the values of the solution at interior points of the domain. A classical solution is understood as a function that is defined everywhere in the closure of the domain and has all classical derivatives occurring in the equation and conditions of the problem. The derivatives are defined via the limit values of finite-difference ratios of the function and corresponding increments of the arguments.