Generalized Solutions of boundary value problems for the d’Alembert equation with local and associated boundary conditions

Abstract

The initial-boundary value problems for the wave equations with local and non-local linear boundary conditions at the ends of a general segment are considered. To solve them, a generalize functions method has been developed, which translates the original boundary value problems to solving the wave equation with a singular right-hand side containing a singular simple and double layers, the densities of which are determined by the boundary and initial values of 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 boundary functions, it is built in space Fourier transforms in time, a two-leaf resolving system linear algebraic equations, which connects 4 boundary values solution and its derivatives. Together with two boundary conditions of local and non-local type, a resolving system of equations is built for solving the stated initial-boundary value problems. On its basis, given analytical solutions for classical three boundary value problems with conditions Dirichlet, Neumann and mixed at the ends of the segment. The developed method allows solving boundary value problems with different local and nonlocal boundary conditions and must find an application change in solving wave and other equations on graphs of different structures.