Обобщенные решения краевых задач для уравнения Даламбера с локальнымии связанными граничными условиями

Abstract

The initial-boundary value problems for the wave equations with local and non-local linear boundaryconditions at the ends of a general segment are considered. To solve them, a generalize functions method has beendeveloped, which translates the original boundary value problems to solving the wave equation with a singular right-handside containing a singular simple and double layers, the densities of which are determined by the boundary and initial valuesof the desired function and its derivatives. Received integral representation of the solution in terms of boundary functions,which are a generalization of Green’s formula for solutions of the wave equation. To determine the unknown boundaryfunctions, it is built in space Fourier transforms in time, a two-leaf resolving system linear algebraic equations, whichconnects 4 boundary values solution and its derivatives. Together with two boundary conditions of local and non-localtype, a resolving system of equations is built for solving the stated initial-boundary value problems. On its basis, givenanalytical solutions for classical three boundary value problems with conditions Dirichlet, Neumann and mixed at the endsof the segment. The developed method allows solving boundary value problems with different local and nonlocal boundaryconditions and must find an application change in solving wave and other equations on graphs of different structures.