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

Ihor Borachok (), Roman Chapko () and B. Tomas Johansson ()
Additional contact information
Ihor Borachok: Ivan Franko National University of Lviv
Roman Chapko: Ivan Franko National University of Lviv
B. Tomas Johansson: Linköping University

Partial Differential Equations and Applications, 2022, vol. 3, issue 3, 1-13

Abstract: 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.

Keywords: Cauchy problem; Heat equation; Houbolt method; Inverse problem; L-curve rule; Method of fundamental solutions; Tikhonov regularization; Wave equation; 65M32; 65M60; 65M80 (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s42985-022-00177-0 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:pardea:v:3:y:2022:i:3:d:10.1007_s42985-022-00177-0

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/42985/

DOI: 10.1007/s42985-022-00177-0

Access Statistics for this article

Partial Differential Equations and Applications is currently edited by Zhitao Zhang

More articles in Partial Differential Equations and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().