A direct numerical method for approximate solution of inverse reaction diffusion equation via two-dimensional Legendre hybrid functions | Numerical Algorithms Skip to main content
Springer Nature Link
Log in

A direct numerical method for approximate solution of inverse reaction diffusion equation via two-dimensional Legendre hybrid functions

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

In this paper, we propose an efficient numerical method based on two-dimensional hybrid of block-pulse functions and Legendre polynomials for numerically solving an inverse reaction diffusion equation. The main idea of the present method is based upon some of the important benefits of the hybrid functions such as high accuracy, wide applicability, and adjustability of the orders of the block-pulse functions and Legendre polynomials to achieve highly accurate numerical solutions. By using the spectral method, inverse reaction diffusion equation with initial and boundary conditions would reduce to a system of nonlinear algebraic equations. Due to the ill-posed system of nonlinear algebraic equations, a regularization scheme is employed to obtain a numerical stable solution. Finally, some numerical examples are presented to show the accuracy and effectiveness of this method.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Vries, G., Hillen, T., Lewis, M., Muller, J., Schonfisch, B.: A Course in Mathematical Biology. SIAM, Philadelphia (2006)

    Book  MATH  Google Scholar 

  2. Cannon, J.R., Zackman, D.: Parameter determination in parabolic partial differential equations from over-specified boundary data. Int. J. Eng. Sci. 20, 779–788 (1982)

    Article  Google Scholar 

  3. Dehghan, M.: Identification of a time-dependent coefficient in a partial differential equation subject to an extra measurement. Numer. Methods Partial Differ. Equ. 21 (3), 611–622 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Tinazepe, R., Tatar, S.: Identification of the density dependent coefficient in an inverse reaction-diffusion problem from single boundary data. J. Differ. Equ. 21, 1–44 (2014)

    Google Scholar 

  5. Dehghan, M., Tatari, M.: Solution of a parabolic equation with time-dependent coefficient and an extra measurement using the decomposition procedure of Adomian. Phys. Scr. 72(6), 425–431 (2005)

    Article  MATH  Google Scholar 

  6. Shamsi, M., Dehghan, M.: Recovering a time-dependent coefficient in a parabolic equation from overspecified boundary data using the pseudospectral Legendre method. Numer. Methods Partial Differ. Equ. 23(1), 196–210 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Parzlivand, F., Shahrezaee, A.M.: Numerical solution of an inverse reaction-diffusion problem via collocation method based on radial basis function. Appl. Math. Model. 39, 3733–3744 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lakestani, M., Dehghan, M.: The use of Chebyshev cardinal functions for the solution of a partial differential equation with an unknown time-dependent coefficient subject to an extra measurement. J. Comput. Appl. Math. 235(3), 669–678 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Liao, W.Y., Dehghan, M., Mohebbi, A.: Direct numerical method for an inverse problem of a parabolic partial differential equation. J. Comput. Appl. Math. 232(2), 351–360 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cannon, J.R., Lin, Y.: Aninverse problem of finding a parameter in a semi-linear heat equation. J. Math. Anal. Appl. 145(2), 470–484 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cannon, J.R., Yin, H.M.: Numerical solutions of some parabolic inverse problems. Numer. Methods Partial Differ. Equ. 6(2), 177–191 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cannon, J.R., Yin, H.M.: On a class of nonlinear parabolic equations with nonlinear trace type functionals. Inverse Prob. 7(1), 149–161 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Yin, H.M.: Solvability of a class of parabolic inverse problems. Adv. Differ. Equ. 1(6), 1005–1023 (1996)

    MathSciNet  MATH  Google Scholar 

  14. Cannon, J.R., Yin, H.M.: A class of non-linear non-classical parabolic equations. J. Differ. Equ. 79(2), 266–288 (1989)

    Article  MATH  Google Scholar 

  15. Borukhov, V.T., Kostyukova, O.I.: Identification of time-dependent coefficients of heat transfer by the method of suboptimal stage-by-stage optimization. Int. J. Heat Mass Transf. 59, 286–294 (2013)

    Article  Google Scholar 

  16. Hussein, M.S., Lesnic, D., Ivanchov, M.I.: Simultaneous determination of time-dependent coefficients in the heat equation. Comput. Math. Appl. 67, 1065–1091 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nayroles, B., Touzot, G., Villon, P.: Generalizing the finite element method: diffuse approximation and diffuse elements. Comput. Mech. 10(5), 307–318 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Babuska, I., Melenk, J.: The partition of unity method. Int. J. Numer. Methods Eng. 40(4), 727–758 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Belytschko, T., Lu, Y.Y., Gu, L.: Element-free Galerkin methods. Int. J. Numer. Methods Eng. 37(2), 229–256 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu, W., Jun, S., Zhang, Y.: Reproducing kernel particle methods. Int. J. Numer. Methods Fluids 20(8-9), 1081–1106 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, C.S., Atluri, S.: A double iteration process for solving the nonlinear algebraic equations, especially for ill-posed nonlinear algebraic equations. J. Comput. Model. Eng. Sci. 99(2), 123–149 (2014)

    MathSciNet  Google Scholar 

  22. Almasieh, H., Meleh, J.N.: Hybrid function method based on radial basis function for solving ninlinear fredholm integral equations. J. Math. Ext. 7, 29–38 (2014)

    MATH  Google Scholar 

  23. Almasieh, H., Meleh, J.N.: A meshless method for optimal control problem of Volterra-Fredholm integral equation using multiquadratic radial basis functions. J. New. Res. Math. 2, 86–96 (2016)

    Google Scholar 

  24. Funaro, D.: Polynomial Approximation of Differential Equations. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  25. Carlson, B. C.: Special Function of Applied Mathematics. Academic Press, New York (1977)

    MATH  Google Scholar 

  26. Yuzbasi, S.: A numerical approach for solving the high-order linear singular differential difference equations. J. Comput. Math. 62(5), 2289–2303 (2011)

    MathSciNet  MATH  Google Scholar 

  27. Cannon, J.R., Rundell, W.: Recovering a time dependent coefficient in a parabolic differential equation. J. Math. Anal. Appl. 160(2), 572–582 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  28. Onate, E., Idelsohn, S., Zienkiewicz, O.C., Taylor, R.L.: A finite point method in computational mechanics, application to convective transport and fluid flow. Int. J. Numer. Methods Eng. 39(22), 3839–3866 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, Science Publisher Inc., New York (1978)

    MATH  Google Scholar 

  30. Delves, L. M., Mohamed, J. L.: Computional Methods for Integral Equations. Cambridge University Press, Cambridge (1985)

    Book  Google Scholar 

  31. Irfan, N., Siddiqi, A.H.: Sine-Cosine wavelets approach in numerical evaluation of Hankel transform for seismology. J. Appl. Math. Model. 40(7), 121–129 (2016)

    MathSciNet  MATH  Google Scholar 

  32. Atkinson, K.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  33. Liu, C.S., Atluri, S.: Simple “Residual-Norm” based algorithms, for solution of a large system of nonlinear algebraic equations, which converge faster than the Newton’s method. J. Comput. Model. Eng. Sci. 71(3), 279–304 (2011)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan, Iran

    I. Gholampoor & M. Tavassoli Kajani

Authors
  1. I. Gholampoor
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Tavassoli Kajani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Tavassoli Kajani.

Additional information

Publisher’s note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gholampoor, I., Kajani, M.T. A direct numerical method for approximate solution of inverse reaction diffusion equation via two-dimensional Legendre hybrid functions. Numer Algor 83, 511–528 (2020). https://doi.org/10.1007/s11075-019-00691-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-019-00691-0

Keywords

Search

Navigation