Solution of nonlocal boundary value problems for the heat equation with discontinuous coefficients, in the case of two discontinuity points

Abstract

In this paper, the solution of the initial-boundary value problem for the heat equation with a discontinuous coefficient under periodic or antiperiodic boundary conditions in the case of two discontinuity points is substantiated using the method of separation of variables. Using the replacement, the problem under consideration is reduced to a self-adjoint problem. By means of the Fourier method, this problem is reduced to the corresponding spectral problem. Then, the eigenvalues and eigenfunctions of this self-adjoint spectral problem are found. In conclusion, the main theorem on the existence and uniqueness of the classical solution to the problem under consideration is proved. The peculiarity of the problem under consideration is the non-local boundary conditions and the presence of two discontinuity points, which have not been considered before. The authors were able to find eigenvalues explicitly and construct eigenfunctions. This technique is also applicable in the case of more than two discontinuity points. The solution obtained in explicit form can be further used for numerical calculations.