Skip to main content
SpringerLink
Log in

Error analysis and numerical solution of Burgers–Huxley equation using 3-scale Haar wavelets

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

In this article, we develop an efficient and accurate numerical scheme based on the Crank–Nicolson finite difference method and Haar wavelet analysis to evaluate the numerical solution of the Burgers–Huxley equation. The present method is extended form of Haar wavelet 2D scaling which shows that it is reliable for solving nonlinear partial differential equations. The numerical results are more accurate than other existing methods available in the literature and very close to the exact solution. The Haar basis function is generated from multi-resolution analysis and used to evaluate fast and accurate approximate solutions on the collocation points. The convergence of the proposed method is demonstrated by its error analysis. We compared numerical solutions with the exact solutions and solutions available in the literature. The proposed method is found to be straight forward, accurate with small computational cost and can be easily implemented in mathematical software MATLAB.

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. Çelik İ (2012) Haar wavelet method for solving generalized burgers-huxley equation. Arab J Math Sci 18(1):25–37

    MathSciNet  MATH  Google Scholar 

  2. Mohammadi R (2013) B-spline collocation algorithm for numerical solution of the generalized burger’s-huxley equation. Numer Methods Partial Differ Equ 29(4):1173–1191

    Article  MathSciNet  Google Scholar 

  3. Lu BQ, Xiu BZ, Pang ZL, Jiang XF (1993) Exact traveling wave solution of one class of nonlinear diffusion equations. Phys Lett A 175(2):113–115

    Article  MathSciNet  Google Scholar 

  4. Wazwaz A-M (2008) Analytic study on burgers, fisher, huxley equations and combined forms of these equations. Appl Math Comput 195(2):754–761

    MathSciNet  MATH  Google Scholar 

  5. Molabahrami A, Khani F (2009) The homotopy analysis method to solve the burgers-huxley equation. Nonlinear Anal Real World Appl 10(2):589–600

    Article  MathSciNet  Google Scholar 

  6. Batiha B, Noorani MSM, Hashim I (2008) Application of variational iteration method to the generalized burgers-huxley equation. Chaos Solitons Fractals 36(3):660–663

    Article  Google Scholar 

  7. Arora G, Singh BK (2013) Numerical solution of burgers’ equation with modified cubic b-spline differential quadrature method. Appl Math Comput 224:166–177

    MathSciNet  MATH  Google Scholar 

  8. Rashidi MM, Erfani E (2009) New analytical method for solving burgers’ and nonlinear heat transfer equations and comparison with ham. Comput Phys Commun 180(9):1539–1544

    Article  MathSciNet  Google Scholar 

  9. Ismail HNA, Raslan K, Rabboh AAA (2004) Adomian decomposition method for burger’s-huxley and burger’s-fisher equations. Appl Math Comput 159(1):291–30

    MathSciNet  MATH  Google Scholar 

  10. El-Hawary HM, Abdel-Rahman EO (2005) Numerical solution of the generalized burger’s equation via spectral/spline methods. Appl Math Comput 170(1):267–279

    MathSciNet  MATH  Google Scholar 

  11. Çiçek Y, Tanoǧlu G (2016) Strang splitting method for burgers-huxley equation. Appl Math Comput 276:454–467

    MathSciNet  MATH  Google Scholar 

  12. Seydaoğlu M, Erdoğan U, Öziş T (2016) Numerical solution of burgers’ equation with high order splitting methods. J Comput Appl Math 291:410–421

    Article  MathSciNet  Google Scholar 

  13. Ashpazzadeh E, Han B, Lakestani M (2017) Biorthogonal multiwavelets on the interval for numerical solutions of burgers’ equation. J Comput Appl Math 317:510–534

    Article  MathSciNet  Google Scholar 

  14. Oruç Ö, Esen A, Bulut F (2019) A haar wavelet approximation for two-dimensional time fractional reaction-subdiffusion equation. Eng Comput 35(1):75–86

    Article  Google Scholar 

  15. Saeedi A, Foadian S, Pourgholi R (2019) Applications of two numerical methods for solving inverse benjamin-bona-mahony-burgers equation. Eng Comput 96:1–14

    Google Scholar 

  16. Tamsir M, Dhiman N, Srivastava VK (2016) Extended modified cubic b-spline algorithm for nonlinear burgers’ equation. Beni-Suef Univ J Basic Appl Sci 5(3):244–254

    Google Scholar 

  17. Nourazar SS, Soori M, Nazari-Golshan A (2015) On the exact solution of burgers-huxley equation using the homotopy perturbation method. J Appl Math Phys 3:285–294

    Article  Google Scholar 

  18. Wasim I, Abbas M, Amin M (2018) Hybrid b-spline collocation method for solving the generalized burgers-fisher and burgers-huxley equations. Math Probl Eng 2018:18

    Article  MathSciNet  Google Scholar 

  19. Sari M, Gürarslan G, Zeytinoğlu A (2011) High-order finite difference schemes for numerical solutions of the generalized burgers-huxley equation. Numer Methods Partial Differ Equ 27(5):1313–1326

    Article  MathSciNet  Google Scholar 

  20. Soori Mohsen (2016) A comparison between the variational iteration method and the homotopy perturbation method for the burgers-huxley equation. bepress. http://works.bepress.com/mohsensoori/15/

  21. Nadeema M, Yaob S-W (2019) Solving system of partial differential equations using varia-tional iteration method with he’s polynomials. J Math Comput Sci 19:203–211

    Article  Google Scholar 

  22. Hashim I, Noorani MSM, Al-Hadidi MRS (2006) Solving the generalized burgers-huxley equation using the adomian decomposition method. Math Comput Model 43(11–12):1404–1411

    Article  MathSciNet  Google Scholar 

  23. Wang X (1985) Nerve propagation and wall in liquid crystals. Phys Lett A 112(8):402–406

    Article  Google Scholar 

  24. Mittal RC, Pandit S (2018) 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

    Article  MathSciNet  Google Scholar 

  25. Mittal RC, Pandit S (2017) Numerical simulation of unsteady squeezing nanofluid and heat flow between two parallel plates using wavelets. Int J Therm Sci 118:410–422

    Article  Google Scholar 

  26. Babolian E, Saeidian J (2009) Analytic approximate solutions to burgers, fisher, huxley equations and two combined forms of these equations. Commun Nonlinear Sci Numer Simul 14(5):1984–1992

    Article  MathSciNet  Google Scholar 

  27. Ray SS (2015) A new coupled fractional reduced differential transform method for the numerical solution of fractional predator-prey system. Comput Model Eng Sci 105(3):231–249

    Google Scholar 

  28. Ray SS (2019) Nonlinear differential equations in physics. Springer. https://doi.org/10.1007/978-981-15-1656-6

  29. Srivastava VK, Mishra N, Kumar S, Singh BK, Awasthi MK (2014) Reduced differential transform method for solving (1+ n)-dimensional burgers’ equation. Egypt J Basic Appl Sci 1(2):115–119

    Article  Google Scholar 

  30. Odibat Z, Momani S (2008) A generalized differential transform method for linear partial differential equations of fractional order. Appl Math Lett 21(2):194–199

    Article  MathSciNet  Google Scholar 

  31. Yang X-J, Machado JAT, Srivastava HM (2016) A new numerical technique for solving the local fractional diffusion equation: two-dimensional extended differential transform approach. Appl Math Comput 274:143–151

    MathSciNet  MATH  Google Scholar 

  32. Tamsir M, Srivastava VK, Jiwari R (2016) An algorithm based on exponential modified cubic b-spline differential quadrature method for nonlinear burgers’ equation. Appl Math Comput 290(111–124):2016

    MathSciNet  MATH  Google Scholar 

  33. Oruç Ö, Bulut F, Esen A (2016) Numerical solutions of regularized long wave equation by haar wavelet method. Mediter J Math 13(5):3235–3253

    Article  MathSciNet  Google Scholar 

  34. Oruç Ö, Bulut F, Esen A (2019) Chebyshev wavelet method for numerical solutions of coupled burgers’ equation. Hacettepe J Math Stat 48(1):1–16

    MathSciNet  MATH  Google Scholar 

  35. Oruç Ö, Esen A, Bulut F (2019) A unified finite difference chebyshev wavelet method for numerically solving time fractional burgers’ equation. Discrete Continuous Dyn Syst S 12(3):533

    Article  MathSciNet  Google Scholar 

  36. Oruç Ö, Bulut F, Esen A (2015) A haar wavelet-finite difference hybrid method for the numerical solution of the modified burgers’ equation. J Math Chem 53(7):1592–1607

    Article  MathSciNet  Google Scholar 

  37. Pandit S, Kumar M, Tiwari S (2015) Numerical simulation of second-order hyperbolic telegraph type equations with variable coefficients. Comput Phys Commun 187:83–90

    Article  MathSciNet  Google Scholar 

  38. Mittal RC, Pandit S (2019) New scale-3 haar wavelets algorithm for numerical simulation of second order ordinary differential equations. Proc Natl Acad Sci India Sect A Phys Sci 89:799–808

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Authors express heartedly thanks to editor in chief, editor and reviewers for their valuable suggestions to revise this manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Prayagraj, UP, 211004, India

    Shitesh Shukla & Manoj Kumar

Authors
  1. Shitesh Shukla
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manoj Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shitesh Shukla.

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

Shukla, S., Kumar, M. Error analysis and numerical solution of Burgers–Huxley equation using 3-scale Haar wavelets. Engineering with Computers 38, 3–11 (2022). https://doi.org/10.1007/s00366-020-01037-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-020-01037-4

Keywords

Search

Navigation