Finite Fourier series in hyperbolic initial-boundary value problems for domains with curved boundaries

The Fourier method based on orthogonal splines is used to solve a linear hyperbolic initial-boundary value problem for a domain with a curved boundary. A theoretical analysis and solutions of problems of free vibrations of membranes with curved boundaries show that the sequence of finite Fourier series generated by the algorithm converges at each time to the exact solution of the problem, i.e., to the infinite Fourier series. The structure of these finite Fourier series, each associated with a particular grid in the considered domain, is similar to the structure of the corresponding partial sums of the infinite Fourier series, i.e., the exact solution of the problem. As the number of grid nodes in the domain with a curved boundary increases, the approximate eigenvalues and eigenfunctions of the boundary value problem converge to the exact ones, which determine the convergence of the finite Fourier series to the exact solution of the initial-boundary value problem. For this problem, the Fourier method based on orthogonal splines yields arbitrarily accurate approximate analytical solutions in the form of finite generalized Fourier series similar in structure to the partial sums of the exact solution. The method has an expanded domain of application.