Skip to main content
SpringerLink
Log in

Identification of a Time Dependent Source Function in a Parabolic Inverse Problem Via Finite Element Approach

  • Published:
Indian Journal of Pure and Applied Mathematics Aims and scope Submit manuscript

Abstract

In the present paper, we study one-dimensional inverse source problem of identifying a time-dependent source function from the over-specification condition. The unknown source term is recovered by solving an operator integral equation of the first kind. Due to the ill-posedness of this operator integral equation, the Tikhonov regularization approach of the 1st order is applied in order to acquire a stable approximation. The stable solution is defined by minimization of the Tikhonov functional. The value of the regularization parameter in this method is obtained as a function of error in input data. The finite element method is applied for numerical simulation based on obtained results. The illustrative example is presented to show the accuracy and applicability of the proposed method.

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

Similar content being viewed by others

References

  1. J. R. Cannon, Determination of an unknown heat source from overspecified boundary data, SIAM J. Numer. Anal., 5(2) (1968), 275–286.

    Article  MathSciNet  Google Scholar 

  2. J. R. Cannon, Y. Lin, and S. Wang, Determination of a control parameter in a parabolic partial differential equation, J. Aust. Math. Soc. Ser. B, 33(2) (1991), 149–163.

    Article  MathSciNet  Google Scholar 

  3. J. R. Cannon, The One-dimensional Heat Equation, Addison-Wesley, Reading, MA, (1984).

    Book  Google Scholar 

  4. M. Dehghan, An inverse problem of finding a source parameter in a semilinear parabolic equation, Appl. Math. Model., 25 (2001), 743–754.

    Article  Google Scholar 

  5. M. Dehghan, Parameter determination in partial differential equation from the overspecified data, Math. Comput. Model., 41(23) (2005), 196–213.

    Article  MathSciNet  Google Scholar 

  6. F. F. Dou, C. L. Fu, and F. Yang, Identifying an unknown source term in a heat equation, Inverse Problem, Sci. Eng., 17(7) (2009), 901–913.

    Article  MathSciNet  Google Scholar 

  7. H. W. Engle, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publisher, Dordrecht, (1996).

    Book  Google Scholar 

  8. A. Farcas and D. Lesnic, The boundary-element method for the determination of a heat source dependent on one variable, J. Eng. Math., 54 (2006), 375–388.

    Article  MathSciNet  Google Scholar 

  9. A. G. Fatullayev and S. Cula, An iterative procedure for determining an unknown spacewise-dependent coefficient in a parabolic equation, Appl. Math. Lett., 22 (2009), 1033–1037.

    Article  MathSciNet  Google Scholar 

  10. S. Foadian, R. Pourgholi, and S. H. Tabasi, Cubic B-spline method for the solution of an inverse parabolic system, Applicable Analysis, 97(3) (2017), 438–465.

    Article  MathSciNet  Google Scholar 

  11. A. Hasanov, Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: weak solution approach, J. Math. Anal. Appl., 330 (2007), 766–779.

    Article  MathSciNet  Google Scholar 

  12. Y. C. Hon and T. Wei, A fundamental solution method for inverse heat conduction problem, Eng. Anal. Bound. Elem., 28 (2004), 489–495.

    Article  Google Scholar 

  13. V. Isakov, Inverse Problems for Partial Differential Equations, Springer, New York, (1998).

    Book  Google Scholar 

  14. B. T. Johansson and D. Lesnic, A procedure for determining a sapcewise dependent heat source and the initial temperature, Appl. Anal., 87 (2008), 265–276.

    Article  MathSciNet  Google Scholar 

  15. B. T. Johansson and D. Lesnic, Determination of a spacewise dependent heat source, J. Comput. Appl. Math., 209(1) (2007), 66–80.

    Article  MathSciNet  Google Scholar 

  16. J. J. Liu, Regularization Methods and Applications for the Ill-posed Problems, Science Press, Beijing, (2005).

    Google Scholar 

  17. J. Liu, B. Wang, and Z. Liu, Determination of a source term in a heat equation, Int. J. Comput. Math., 87(5) (2010), 969–975.

    Article  MathSciNet  Google Scholar 

  18. G. L. Mazzieri, R. D. Spies, and K. G. Temperini, Existence, uniqueness and stability ofminimizers of generalized TikhonovPhillips functionals, Journal of Mathematical Analysis and Applications, 396(1) (2012), 396–411.

    Article  MathSciNet  Google Scholar 

  19. R. Pourgholi, N. Azizi, Y. S. Gasimov, F. Aliev, and H. K. Khalafi, Removal of numerical instability in the solution of an inverse heat conduction problem, Commun. Nonlinear Sci. Numer. Simul., 14(6) (2009), 2664–2669.

    Article  MathSciNet  Google Scholar 

  20. R. Pourgholi, H. Dana, and S. H. Tabasi, Solving an inverse heat conduction problem using genetic algorithm: sequential and multi-core parallelization approach, Appl. Math. Modelling, 38(7) (2014), 1948–1958.

    Article  MathSciNet  Google Scholar 

  21. R. Pourgholi and A. Saeedi, Applications of cubic B-splines collocation method for solving nonlinear inverse parabolic partial differential equations, Numerical Methods for Partial Differential Equations, 34(8) (2016), 1–17.

    MATH  Google Scholar 

  22. A. G. Ramm, An inverse problem for the heat equation, J. Math. Anal. Appl., 264(2) (2001), 691–697.

    Article  MathSciNet  Google Scholar 

  23. A. G. Ramm, Inverse Problems, Springer, New York, (2005).

    MATH  Google Scholar 

  24. B. D. Reddy, Functional analysis and boundary-value problems: an introductory treatment, John Wiley & Sons. Inc., (1986).

  25. A. Shidfar, G. R Karamali, and J. Damirchi, An inverse heat conduction problem with a nonlinear source term, Nonlinear Anal., 65 (2006), 615–621.

    Article  MathSciNet  Google Scholar 

  26. A. N. Tikhonov and Y. V. Arsenin, Solution of Ill-Posed Problems, Winston and Sons, Washington D.C., (1997).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematics, Statistics and Computer Science, Semnan University, Semnan, Iran

    J. Damirchi, T. R. Shamami & A. Janmohammadi

  2. School of Mathematics and Computer Sciences, Damghan University, Damghan, Iran

    R. Pourgholi

  3. Faculty of Engineering, Sabzevar University of New Technology, Sabzevar, Iran

    H. Zeidabadi

Authors
  1. J. Damirchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Pourgholi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. T. R. Shamami
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. H. Zeidabadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. A. Janmohammadi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to J. Damirchi, R. Pourgholi, T. R. Shamami, H. Zeidabadi or A. Janmohammadi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Damirchi, J., Pourgholi, R., Shamami, T.R. et al. Identification of a Time Dependent Source Function in a Parabolic Inverse Problem Via Finite Element Approach. Indian J Pure Appl Math 51, 1587–1602 (2020). https://doi.org/10.1007/s13226-020-0483-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13226-020-0483-8

Key words

2010 Mathematics Subject Classification

Search

Navigation