Skip to main content
SpringerLink
Log in

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

  • Original Paper
  • Published:
Partial Differential Equations and Applications Aims and scope Submit manuscript

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 \(L_2\)-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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Data availability

All data generated or analysed during this study are included in this published article.

References

  1. Alves, C.J.S.: On the choice of source points in the method of fundamental solutions. Eng. Anal. Bound. Elem. 33, 1348–1361 (2009)

    Article  MathSciNet  Google Scholar 

  2. Amirov, A., Yamamoto, M.: A timelike Cauchy problem and an inverse problem for general hyperbolic equations. Appl. Math. Lett. 21, 885–891 (2008)

    Article  MathSciNet  Google Scholar 

  3. Bécache, E., Bourgeois, L., Franceschini, L., Dardé, J.: Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: the 1D case, Inverse Probl. Imaging 9, 971–1002 (2015)

    MathSciNet  MATH  Google Scholar 

  4. Bourgeois, L., Ponomarev, D., Dardé, J.: An inverse obstacle problem for the wave equation in a finite time domain. Inverse Probl. Imaging 13, 377–400 (2019)

    Article  MathSciNet  Google Scholar 

  5. Bogomolny, A.: Fundamental solutions method for elliptic boundary value problems. SIAM J. Numer. Anal. 22, 644–669 (1985)

    Article  MathSciNet  Google Scholar 

  6. Borachok, I., Chapko, R., Johansson, B.T.: A method of fundamental solutions for heat and wave propagation from lateral Cauchy data. Numer. Algorithms (2021). https://doi.org/10.1007/s11075-021-01120-x

    Article  MATH  Google Scholar 

  7. Cao, Y.H., Kuo, L.H.: Hybrid method of space-time and Houbolt methods for solving linear time-dependent problems. Eng. Anal. Bound. Elem. 128, 58–65 (2021)

    Article  MathSciNet  Google Scholar 

  8. Chapko, R., Johansson, B.T.: A boundary integral equation method for numerical solution of parabolic and hyperbolic Cauchy problems. Appl. Numer. Math. 129, 104–119 (2018)

    Article  MathSciNet  Google Scholar 

  9. Chapko, R., Johansson, B.T.: Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations. J. Eng. Math. 103, 23–37 (2017)

    Article  MathSciNet  Google Scholar 

  10. Chapko, R., Johansson, B.T., Muzychuk, Y., Hlova, A.: Wave propagation from lateral Cauchy data using a boundary element method. Wave Motion 91, 102385 (2019)

    Article  MathSciNet  Google Scholar 

  11. Cheng, A.H.D., Hong, Y.: An overview of the method of fundamental solutions–solvability, uniqueness, convergence, and stability. Eng. Anal. Bound. Elem. 120, 118–152 (2020)

    Article  MathSciNet  Google Scholar 

  12. Fairweather, G., Karageorghis, A.: The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math. 9, 69–95 (1998)

    Article  MathSciNet  Google Scholar 

  13. Gladwell, I., Thomas, R.M.: Stability properties of the Newmark, Houbolt and Wilson \(\theta \) methods. Int. j. Numer. Anal. Methods Geomech. 4, 143–158 (1980)

    Article  Google Scholar 

  14. Golberg, M.A., Chen, C.S.: The method of fundamental solutions for potential, Helmholtz and diffusion problems. In: Golberg, M.A. (ed.) Boundary Integral Methods: Numerical and Mathematical Aspects, pp. 103–176. WIT Press, Boston (1999)

    Google Scholar 

  15. Gu, M.H., Young, D.L., Fan, C.M.: The method of fundamental solutions for the multi-dimensional wave equations. J. Mar. Sci. Technol. 19, 586–595 (2011)

    Article  Google Scholar 

  16. Houbolt, J.C.: A recurrence matrix solution for the dynamic response of elastic aircraft. J. Aeronaut. Sci. 17, 540–550 (1950)

    Article  MathSciNet  Google Scholar 

  17. Hughes, T.J.R.: The Finite Element Method. Prentice Hall Inc, Englewood Cliffs (1987)

    MATH  Google Scholar 

  18. Isakov, V.: Inverse Problems for Partial Differential Equations, 3rd edn. Springer, Cham (2017)

    Book  Google Scholar 

  19. Johnson, D.E.: A proof of the stability of the Houbolt method. AIAA J. 8, 1450–1451 (1966)

    Article  Google Scholar 

  20. Jovanović, B.: On the estimates of the convergence rate of the finite difference schemes for the approximation of solutions of hyperbolic problems. Publ. Inst. Math. (Beograd) (N.S.) 52(66), 127–135 (1992)

    MathSciNet  MATH  Google Scholar 

  21. Jovanović, B.S., Süli, E.: Analysis of Finite Difference Schemes. Springer, London (2014)

    Book  Google Scholar 

  22. Karageorghis, A., Lesnic, D., Marin, L.: A survey of applications of the MFS to inverse problems. Inverse Prob. Sci. Eng. 19, 309–336 (2011)

    Article  MathSciNet  Google Scholar 

  23. Klibanov, M., Rakesh: Numerical solution of a time-like Cauchy problem for the wave equation. Math. Methods Appl. Sci 15, 559–570 (1992)

    Article  MathSciNet  Google Scholar 

  24. Le, T.T., Nguyen, L.H., Nguyen, T.-P., Powell, W.: The Quasi-reversibility method to numerically solve an inverse source problem for hyperbolic equations. J. Sci. Comput. (2021). https://doi.org/10.1007/s10915-021-01501-3

    Article  MathSciNet  MATH  Google Scholar 

  25. Lin, J., Chen, W., Chen, C.S.: A new scheme for the solution of reaction diffusion and wave propagation problems. Appl. Math. Model. 38, 5651–5664 (2014)

    Article  MathSciNet  Google Scholar 

  26. Lin, J., Chen, C.S., Liu, C.-S., Lu, J.: Fast simulation of multi-dimensional wave problems by the sparse scheme of the method of fundamental solutions. Comput. Math. Appl. 72, 555–567 (2016)

    Article  MathSciNet  Google Scholar 

  27. Wood, W.L.: Practical Time-Stepping Schemes. Oxford University Press, New York (1990)

    MATH  Google Scholar 

  28. Young, D.L., Gu, M.H., Fan, C.M.: The time-marching method of fundamental solutions for wave equations. Eng. Anal. Bound. Elem. 33, 1411–1425 (2009)

    Article  MathSciNet  Google Scholar 

  29. Zhou, Y., Qu, W., Gu, Y., Gao, H.: A hybrid meshless method for the solution of the second order hyperbolic telegraph equation in two space dimensions. Eng. Anal. Bound. Elem. 115, 21–27 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Applied Mathematics and Informatics, Ivan Franko National University of Lviv, Lviv, 79000, Ukraine

    Ihor Borachok & Roman Chapko

  2. ITN, Campus Norrköping, Linköping University, Linköping, Sweden

    B. Tomas Johansson

Authors
  1. Ihor Borachok
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roman Chapko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Tomas Johansson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ihor Borachok.

Ethics declarations

Conflicts of interest

The authors have no relevant financial or non-financial interests to disclose.

Additional information

This article is part of the section “Computational Approaches” edited by Siddhartha Mishra.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Borachok, I., Chapko, R. & Johansson, B.T. A method of fundamental solutions with time-discretisation for wave motion from lateral Cauchy data. Partial Differ. Equ. Appl. 3, 37 (2022). https://doi.org/10.1007/s42985-022-00177-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s42985-022-00177-0

Keywords

Mathematics Subject Classification (2020)

Search

Navigation