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

Author

Listed:
  • Ihor Borachok

    (Ivan Franko National University of Lviv)

  • Roman Chapko

    (Ivan Franko National University of Lviv)

  • B. Tomas Johansson

    (Linköping University)

Abstract

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 $$L_2$$ 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.

Suggested Citation

  • Ihor Borachok & Roman Chapko & B. Tomas Johansson, 2022. "A method of fundamental solutions with time-discretisation for wave motion from lateral Cauchy data," Partial Differential Equations and Applications, Springer, vol. 3(3), pages 1-13, June.
    • Handle: RePEc:spr:pardea:v:3:y:2022:i:3:d:10.1007_s42985-022-00177-0
    DOI: 10.1007/s42985-022-00177-0
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