Fourier method related with orthogonal splines in parabolic initial boundary value problem for domain with curvilinear boundary

The Fourier method allows one to find solutions to boundary value problems and initial boundary value problems for partial differential equations admitting the separation of variables. The application of the method for problems of many types faces significant difficulties. One of the directions on extending the domain of applicability of the Fourier method is to overcome the mathematical problems related with this method, for instance, ones related with a nature of boundary conditions. Another direction concerns the usage of special functions for the domains of classical forms defined by coordinate lines and surfaces of orthogonal curvilinear coordinates. But in the general case of domains with curvilinear boundaries such approach is ineffective. The directions of developing the Fourier method for solving problems in domains with curvilinear boundary are related also, first, with developing and applying variation grid and projection grid method and second, with a modification of the Fourier method itself. The present paper belongs to the second direction and is aimed on extending the applicability domain of the Fourier method, which is determined by constructing a sequence of finite generalized Fourier series related with orthogonal splines and giving analytic solutions to a parabolic initial boundary value problem in the domain with a curved boundary. For such problem, we propose and study an algorithm of the Fourier method related with the application of orthogonal splines. A sequence of finite generalized Fourier series generated by this algorithm converges to the exact solution given by an infinite Fourier series at each time moment. While increasing the number of the nodes in the grid in the considered domain with a curvilinear boundary, the structure of the finite Fourier series approaches the structure of an infinite Fourier series being an exact solution of initial boundary value problem. The method provides approximate analytic solutions with an arbitrary accuracy in the form of orthogonal series, which are generalized Fourier series, and this gives new opportunities of the classical Fourier method.