Convergence of Fourier method connected with orthogonal splines

Fourier method and Fourier series have wide fields of application. The use of the theory of orthogonal splines, created by the author of this article and developed in the last thirty years, has led to significant progress in a number of numerical and analytical methods of deformable solid mechanics and mathematical physics. In particular, the generalized Fourier method associated with the use of finite Fourier series and orthogonal splines was successfully applied earlier by the author in solving parabolic initial boundary value problems for regions with curved boundaries. Recent article proposes further development and novel full research of the algorithm of this method, designed to solve parabolic initial boundary value problems in non-canonical domains. The method gives approximate analytical solutions in form of finite Fourier series whose structure is similar to that of partial sums of an infinite Fourier series for an exact solution. Full investigation of the method’s convergence presented in this article is based on the theory of finite difference methods. As a number of grid nodes in a region increases, such finite Fourier series approach an exact solution of a parabolic initial boundary value problem. Investigation of convergence shows efficiency of the novel algorithm of the generalized Fourier method in solving parabolic initial boundary value problems for non-canonical regions.