Skip to main content
SpringerLink
Log in

Novel Meshfree Scheme For Solving The Inverse Cauchy problem Of Heat Conduction

  • Research Article
  • Published:
Proceedings of the National Academy of Sciences, India Section A: Physical Sciences Aims and scope Submit manuscript

Abstract

The paper presents a meshfree technique based on the method of fundamental solutions (MFS) and the energy equation associated with the model, to optimize the location of source points. Over the years, the simplest way of distributing the source set has been on a fixed contour lying outside of the considered domain. In this study, a method automates the optimal placement of the source points with the help of the energy conservation equation. The implementation of the scheme is carried forward for the inverse Cauchy problem of heat conduction, and numerical results are provided to establish the new scheme as a superior alternative to MFS.

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
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Lesnic D, Elliott L, Ingham DB (1997) An iterative boundary element method for solving numerically the Cauchy problem for the Laplace equation. Eng Anal Bound Elem 20(2):123–133

    Article  Google Scholar 

  2. Mera NS, Elliott L, Ingham DB, Lesnic D (2000) An iterative boundary element method for the solution of a Cauchy steady state heat conduction problem. C Comput Model Eng Sci 1:101–106

    MATH  Google Scholar 

  3. Wang F, Chen W, Qu W, Gu Y (2016) A BEM formulation in conjunction with parametric equation approach for three-dimensional Cauchy problems of steady heat conduction. Eng Anal Bound Elements 63:1–14

    Article  MathSciNet  Google Scholar 

  4. Fu CL, Li HF, Qian Z, Xiong XT (2008) Fourier regularization method for solving a Cauchy problem for the Laplace equation. Inv Probl Sci Eng 16(2):159–169

    Article  MathSciNet  Google Scholar 

  5. Liu CS (2008) A modified collocation Trefftz method for the inverse Cauchy problem of Laplace equation. Eng Anal Bound Elem 32(9):778–785

    Article  Google Scholar 

  6. Liu CS, Wang F, Gu Y (2018) Trefftz energy method for solving the Cauchy problem of the Laplace equation. Appl Math Lett 79:187–195

    Article  MathSciNet  Google Scholar 

  7. Lin J, Chen W, Wang F (2011) A new investigation into regularization techniques for the method of fundamental solutions. Math Comput Simul 81(6):1144–1152

    Article  MathSciNet  Google Scholar 

  8. Kupradze VD, Aleksidze MA (1964) The method of functional equations for the approximate solution of certain boundary value problems. USSR Comput Math Math Phys 4(4):82–126

    Article  MathSciNet  Google Scholar 

  9. Mathon R, Johnston RL (1977) The approximate solution of elliptic boundary-value problems by fundamental solutions. SIAM J Numer Anal 14(4):638–650

    Article  MathSciNet  Google Scholar 

  10. Fairweather G, Karageorghis A (1998) The method of fundamental solutions for elliptic boundary value problems. Adv Comput Math 9(1–2):69

    Article  MathSciNet  Google Scholar 

  11. Golberg MA, Chen CS, Fromme JA (1997) Discrete projection methods for integral equations. Appl Mech Rev 50:B75–B75

    Google Scholar 

  12. Chen CS, Cho HA, Golberg MA (2006) Some comments on the ill-conditioning of the method of fundamental solutions. Eng Anal Bound Elem 30(5):405–410

    Article  Google Scholar 

  13. Tsai CC, Lin YC, Young DL, Atluri SN (2006) Investigations on the accuracy and condition number for the method of fundamental solutions. Comput Model Eng Sci 16(2):103

    Google Scholar 

  14. Alves CJS (2009) On the choice of source points in the method of fundamental solutions. Eng Anal Bound Elem 33(12):1348–1361

    Article  MathSciNet  Google Scholar 

  15. Wang F, Liu CS, Qu W (2018) Optimal sources in the mfs by minimizing a new merit function: energy gap functional. Appl Math Lett 86:229–235

    Article  MathSciNet  Google Scholar 

  16. Lin J, Lamichhane A, Chen CS, Lu J (2018) The adaptive algorithm for the selection of sources of the method of fundamental solutions. Eng Anal Bound Elements 95:154–159

    Article  MathSciNet  Google Scholar 

  17. Johnston RL, Fairweather G (1984) The method of fundamental solutions for problems in potential flow. Appl Math Model 8(4):265–270

    Article  Google Scholar 

  18. Wang J, Ahmed MT, Lavers JD (1990) Nonlinear least squares optimization applied to the method of fundamental solutions for eddy current problems. IEEE Trans Magn 26(5):2385–2387

    Article  ADS  Google Scholar 

  19. Katsurada M, Okamoto H (1996) The collocation points of the fundamental solution method for the potential problem. Comput Math Appl 31(1):123–137

    Article  MathSciNet  Google Scholar 

  20. Chen CS, Karageorghis A, Li Y (2016) On choosing the location of the sources in the MFS. Numer Algorithms 72(1):107–130

    Article  MathSciNet  Google Scholar 

  21. Liu CS (2012) An equilibrated method of fundamental solutions to choose the best source points for the Laplace equation. Eng Anal Bound Elem 36(8):1235–1245

    Article  MathSciNet  Google Scholar 

  22. Liu CS, Wang F (2018) An energy method of fundamental solutions for solving the inverse Cauchy problems of the Laplace equation. Comput Math with Appl 75(12):4405–4413

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Sciences and Engineering, Indian Institute of Technology, Roorkee, Uttarakhand, India

    Surbhi Arora & Jaydev Dabas

Authors
  1. Surbhi Arora
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jaydev Dabas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Surbhi Arora.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Relevance of the work: The new scheme proves to be superior to MFS in terms of accuracy and speed. The simplicity of the work makes it easy to replicate and apply for other potential problems. Further, higher accuracy of the obtained results makes the novel scheme more reliable than others.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arora, S., Dabas, J. Novel Meshfree Scheme For Solving The Inverse Cauchy problem Of Heat Conduction. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 92, 411–418 (2022). https://doi.org/10.1007/s40010-021-00729-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40010-021-00729-w

Keywords

Mathematics Subject Classification

Search

Navigation