SpringerLink
Log in

High Order Compact Cubic B-spline Collocation Method for the Solution of Fisher’s Equation

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

In this article, we formulate a compact fourth order accurate method using the collocation of cubic B-splines to find the numerical solution of the Fisher’s equation. To achieve fourth order accuracy in space, the proposed method requires only three spatial grid points as compared to the requirement of five grid points in the literature, using the collocation methods based on splines. This helps us in achieving accurate results with less computation time. The method is unconditionally stable. A detailed analysis is done to show fourth order accuracy of the method. A comprehensive analysis of the method has been done to check the precision and effectiveness of the suggested method with the help of some examples.

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
EUR 32.99 /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 (United Kingdom)

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
Fig. 9
Fig. 10
Fig. 11

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.

Availability of data and materials

Not applicable.

References

  1. Yadav, O.P., Jiwari, R.: Finite element analysis and approximation of Burgers’-Fisher equation. Num. Meth. Part. Diff. Eq. 33(5), 1652–1677 (2017)

    Article  MathSciNet  Google Scholar 

  2. Jiwari, R.: A Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation. Comput. Phys. Commun. 183(11), 2413–2423 (2012)

    Article  MathSciNet  Google Scholar 

  3. Jiwari, R.: A hybrid numerical scheme for the numerical solution of the Burgers’ equation. Comput. Phys. Commun. 188, 59–67 (2015)

    Article  MathSciNet  Google Scholar 

  4. Mittal, R.C., Pandit, S.: Sensitivity analysis of shock wave Burgers’ equation via a novel algorithm based on scale-3 Haar wavelets. Int. J. Comput. Math. 95(3), 601–625 (2018)

    Article  MathSciNet  Google Scholar 

  5. Kumar, M., Pandit, S.: A composite numerical scheme for the numerical simulation of coupled Burgers’ equation. Comput. Phys. Commun. 185(3), 809–817 (2014)

    Article  MathSciNet  Google Scholar 

  6. Dhiman, N., Chauhan, A., Tamsir, M., Chauhan, A.: Numerical simulation of Fisher’s type equation via a collocation technique based on re-defined quintic B-splines, Multidiscipline Modeling in Materials and Structures, 16(5), pp.1117-1130 (2020)

  7. Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eug. 7(4), 355–369 (1937)

    Article  Google Scholar 

  8. Gazdag, J., Canosa, J.: Numerical solution of Fisher’s equation, J. Appl. Prob., pp.445-457 (1974)

  9. Aggarwal, S.K.: Some numerical experiments on Fisher equation. Int. Commun. Heat Mass Trans. 12(4), 417–430 (1985)

    Article  Google Scholar 

  10. Wang, X.Y.: Exact and explicit solitary wave solutions for the generalised Fisher equation. Phys. Lett. A 131(4–5), 277–279 (1988)

    Article  MathSciNet  Google Scholar 

  11. Tang, S., Weber, R.O.: Numerical study of Fisher’s equation by a Petrov-Galerkin finite element method. ANZIAM J. 33(1), 27–38 (1991)

    MathSciNet  MATH  Google Scholar 

  12. Mavoungou, T., Cherruault, Y.: Numerical study of Fisher’s equation by Adomian’s method. Math. Comput. Modell. 19(1), 89–95 (1994)

    Article  MathSciNet  Google Scholar 

  13. Carey, G.F., Shen, Y.: Least-squares finite element approximation of Fisher’s reaction-diffusion equation. Num. Meth. Part. Diff. Eq. 11(2), 175–186 (1995)

    Article  MathSciNet  Google Scholar 

  14. Qiu, Y., Sloan, D.: Numerical solution of Fisher’s equation using a moving mesh method. J. Comput. Phys. 146(2), 726–746 (1998)

    Article  MathSciNet  Google Scholar 

  15. Al-Khaled, K.: Numerical study of Fisher’s reaction-diffusion equation by the Sinc collocation method. J. Comput. Appl. Math. 137(2), 245–255 (2001)

    Article  MathSciNet  Google Scholar 

  16. Zhao, S., Wei, G.W.: Comparison of the discrete singular convolution and three other numerical schemes for solving Fisher’s equation. SIAM J. Sci. Comput. 25(1), 127–147 (2003)

    Article  MathSciNet  Google Scholar 

  17. Wazwaz, A.M., Gorguis, A.: An analytic study of Fisher’s equation by using Adomian decomposition method. Appl. Math. Comput. 154(3), 609–620 (2004)

    MathSciNet  MATH  Google Scholar 

  18. Mittal, R.C., Kumar, S.: Numerical study of Fisher’s equation by wavelet Galerkin method. Int. J. Comput. Math. 83(3), 287–298 (2006)

    Article  MathSciNet  Google Scholar 

  19. Olmos, D., Shizgal, B.D.: A pseudospectral method of solution of Fisher’s equation. J. Comput. Appl. Math. 193(1), 219–242 (2006)

    Article  MathSciNet  Google Scholar 

  20. Cattani, C., Kudreyko, A.: Mutiscale analysis of the Fisher equation. In International Conference on Computational Science and Its Applications, Springer, pp. 1171-1180 (2008)

  21. Şahin, A., Dağ, İ., Saka, B.: AB-spline algorithm for the numerical solution of Fisher’s equation, Kybernetes, (2008)

  22. Mittal, R.C., Jiwari, R.: Numerical study of Fisher’s equation by using differential quadrature method. Int. J. Inf. Syst. Sci 5(1), 143–160 (2009)

    MathSciNet  MATH  Google Scholar 

  23. Dağ, İ, Şahin, A., Korkmaz, A.: Numerical investigation of the solution of Fisher’s equation via the B-spline Galerkin method. Num. Meth. Part. Diff. Eq. 26(6), 1483–1503 (2010)

    Article  MathSciNet  Google Scholar 

  24. Mittal, R.C., Arora, G.: Efficient numerical solution of Fisher’s equation by using B-spline method. Int. J. Comput. Math. 87(13), 3039–3051 (2010)

    Article  MathSciNet  Google Scholar 

  25. Shukla, H.S., Tamsir, M.: Extended modified cubic B-spline algorithm for nonlinear Fisher’s reaction-diffusion equation. Alexandria Eng. J. 55(3), 2871–2879 (2016)

    Article  Google Scholar 

  26. Tamsir, M., Dhiman, N., Srivastava, V.K.: Cubic trigonometric B-spline differential quadrature method for numerical treatment of Fisher’s reaction-diffusion equations. Alexandria Eng. J. 57(3), 2019–2026 (2018)

    Article  Google Scholar 

  27. Rohila, R., Mittal, R.: Numerical study of reaction diffusion Fisher’s equation by fourth order cubic B-spline collocation method. Math. Sci. 12(2), 79–89 (2018)

    Article  MathSciNet  Google Scholar 

  28. de Boor, Carl, A practical guide to splines (Revised edition), Applied Mathematical Sciences, Springer-Verlag, New York, 2001,ISBN: 0-387-95366-3

  29. Lucas, Thomas R, Error bounds for interpolating cubic splines under various end conditions, SIAM Journal on Numerical Analysis, 1974, vol. 11, no. 3, pp. 569-584

  30. Dhiman, N., Tamsir, M.: A collocation technique based on modified form of trigonometric cubic B-spline basis functions for Fisher’s reaction-diffusion equation”, Multidis. Model. Mater. Struct., Vol. 14 No. 5, pp. 923-939. https://doi.org/10.1108/MMMS-12-2017-0150

Download references

Acknowledgements

The authors thank the referees for the helpful suggestions which greatly improved the quality of the article.

Funding

Not applicable.

Author information

Authors and Affiliations

  1. Department of Mathematics, Aditi Mahavidyalaya, University of Delhi, Delhi, India

    Suruchi Singh

  2. Department of Mathematics, Sri Venkateswara College, University of Delhi, Delhi, India

    Swarn Singh

  3. Department of Mathematics, University of Delhi, Delhi, India

    Sandeep Bhatt

Authors
  1. Suruchi Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Swarn Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sandeep Bhatt
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

SS did the literature review, SS derived the method and SB did the programming part. All authors read and approve the final manuscript.

Corresponding author

Correspondence to Swarn Singh.

Ethics declarations

Code availability

All the codes were written using MATLAB programming and can be made available on request.

Conflict of interest

On behalf of all the authors the corresponding author states that there is no conflict of interest.

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

Singh, S., Singh, S. & Bhatt, S. High Order Compact Cubic B-spline Collocation Method for the Solution of Fisher’s Equation. Int. J. Appl. Comput. Math 7, 217 (2021). https://doi.org/10.1007/s40819-021-01157-5

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40819-021-01157-5

Keywords

Mathematics Subject Classification

Search

Navigation