Convergence of Fourier Method connected with Orthogonal Splines

Title	Convergence of Fourier Method connected with Orthogonal Splines
Authors	V. L. Leontiev¹ ¹World-Class Research Center for Advanced Digital Technologies at Peter the Great St. Petersburg Polytechnic University
Annotation	Fourier series, Fourier method and spline approximations have wide scopes. The generalized Fourier method associated with the use of finite Fourier series and orthogonal splines was applied earlier for solving the parabolic initial boundary value problems for regions with curvilinear boundaries. Here another analogous generalized Fourier method is applied for solving parabolic initial boundary value problems in noncanonical regions and the investigation of convergence of this method is proposed. The study of convergence is based on the theory of finite difference methods. This method gives solutions in form of finite Fourier series which structure is similar to that of partial sums of an infinite Fourier series of an exact solution. As a number of grid nodes increases in a region, a finite Fourier series approach an exact solution of a parabolic initial boundary value problem. The investigation of convergence shows efficiency of the algorithm of the generalized Fourier method in solving parabolic initial boundary value problems for noncanonical regions. This method yields the approximate analytical solutions in the form of the sequence of finite Fourier series. The use of orthogonal splines brings together numerical and analytical methods – finite difference methods and the Fourier method, expanding the scope of their applications.
Keywords	parabolic initial boundary value problems, noncanonical regions, curvilinear boundary, the method of separation of variables, finite Fourier series, orthogonal splines.
Citation	Leontiev V. L. ''Convergence of Fourier Method connected with Orthogonal Splines'' [Electronic resource]. Proceedings of the International Scientific Youth School-Seminar "Mathematical Modeling, Numerical Methods and Software complexes" named after E.V. Voskresensky (Saransk, July 26-28, 2024). Saransk: SVMO Publ, 2024. - pp. 226-228. Available at: https://conf.svmo.ru/files/2024/papers/paper44.pdf. - Date of access: 16.10.2024.

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Convergence of Fourier Method connected with Orthogonal Splines

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