A method of fundamental solutions with time-discretisation for wave motion from lateral Cauchy data

• Published in: SN partial differential equations and applications 2022-06, Vol.3 (3), Article 37
• Main Authors: Borachok, Ihor, Chapko, Roman, Johansson, B. Tomas
• Format: Article
• Language: English
• Subjects: 2. Computational Approaches; Analysis; Mathematical and Computational Biology; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Numerical Analysis; Original Paper; Partial Differential Equations; Theoretical
• ISSN: 2662-2963; 2662-2971
• DOI: 10.1007/s42985-022-00177-0

A method of fundamental solutions (MFS) is proposed and analyzed for the ill-posed problem of finding the wave motion from given lateral Cauchy data in annular domains. A finite difference scheme, known as the Houbolt method, is applied for the time-discretisation rendering a sequence of elliptic systems corresponding to the number of time steps. The solution of the elliptic problems is sought as a linear combination of elements in what is known as a fundamental sequence with source points placed outside of the solution domain. Collocating on the boundary part where Cauchy data is given, a sequence of linear equations is obtained for finding the coefficients in the MFS approximation. Tikhonov regularization is employed to generate a stable solution to the obtained systems of linear equations. It is outlined that the elements in the fundamental sequence constitute a linearly independent and dense set on the boundary of the solution domain in the L 2 -sense. Numerical results both in two and three-dimensional domains confirm the applicability of the proposed strategy for the considered lateral Cauchy problem for the wave equation both for exact and noisy data.