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Efficient computational method for singularly perturbed Burger-Huxley equations

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Abstract

This paper focuses on an efficient computational method for solving the singularly perturbed Burger-Huxley equations. The difficulties encountered in solving this problem come from the nonlinearity term. The quasilinearization technique linearizes the nonlinear term in the differential equation. The finite difference approximation is formulated to approximate the derivatives in the differential equations and then accelerate its rate of convergence to improve the accuracy of the solution. The stability and consistency analysis were investigated to guarantee the convergence analysis of the formulated method. Numerical examples are considered for numerical illustrations. Numerical experiments were conducted to sustain the theoretical results and to show that the proposed method produces a more correct solution than some surviving methods in the literature.

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References

  1. L. Li-Bin, L. Ying, Z. Jian, B. **aobing, A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive 28(4), 1439–1457 (2020)

    Article  Google Scholar 

  2. M.J. Kabeto, G.F. Duressa, Accelerated nonstandard finite difference method for singularly perturbed Burger-Huxley equations. BMC. Res. Notes 14(1), 446 (2021)

    Article  PubMed  PubMed Central  Google Scholar 

  3. K.M. Jima, D.G. File, Implicit finite difference scheme for singularly perturbed Burger-Huxley equations. J. Partial Differ. Equ. 35(1), 87–100 (2022)

    Article  Google Scholar 

  4. M.J. Kabeto, G.F. Duressa, Second-order robust finite difference method for singularly perturbed Burgers’ equation. Heliyon 8(6), e09579 (2022)

    Article  PubMed  PubMed Central  Google Scholar 

  5. I.T. Daba, G.F. Duressa, A fitted numerical method for singularly perturbed Burger-Huxley equation. Bound. Value Probl. 2022(1), 102 (2022)

    Article  Google Scholar 

  6. M.J. Kabeto, G.F. Duressa, A robust numerical method for singularly perturbed semilinear parabolic differential-difference equations. Math. Comput. Simul 188, 537–547 (2021)

    Article  Google Scholar 

  7. T.A. Bullo, G.R. Kusi, Fitted mesh scheme for singularly perturbed parabolic convection–diffusion problem exhibiting twin boundary layers. React. Kinet. Mech. Catal. 137(7), 1–14 (2023)

    Google Scholar 

  8. B.T. Reda, T.A. Bullo, G.F. Duressa, Fourth-order fitted mesh scheme for semilinear singularly perturbed reaction–diffusion problems. BMC. Res. Notes 16(1), 354 (2023)

    Article  PubMed  PubMed Central  Google Scholar 

  9. G.R. Kusi, A.H. Habte, T.A. Bullo, Layer resolving numerical scheme for a singularly perturbed parabolic convection-diffusion problem with an interior layer. MethodsX 10, 101953 (2023)

    Article  PubMed  Google Scholar 

  10. M.M. Woldaregay, T.W. Hunde, V.N. Mishra, Fitted exact difference method for solutions of a singularly perturbed time delay parabolic PDE. Partial Differ. Equ. Appl. Math. 8, 100556 (2023)

    Article  Google Scholar 

  11. T.A. Bullo, Accelerated fitted mesh scheme for singularly perturbed turning point boundary value problems. J. Math. 2022, 3767246 (2022)

    Article  Google Scholar 

  12. T.A. Bullo, G.A. Degla, G.F. Duressa, Fitted mesh method for singularly perturbed parabolic problems with an interior layer. Math. Comput. Simul 193, 371–384 (2022)

    Article  Google Scholar 

  13. T.A. Bullo, G.F. Duressa, G. Degla, Accelerated fitted operator finite difference method for singularly perturbed parabolic reaction-diffusion problems. Comput. Methods Differ. Equ. 9(3), 886–898 (2021)

    Google Scholar 

  14. T.A. Bullo, G.A. Degla, G.F. Duressa, Uniformly convergent higher-order finite difference scheme for singularly perturbed parabolic problems with non-smooth data. J. Appl. Math. Comput. Mech. 20(1), 5–16 (2021)

    Article  Google Scholar 

  15. T.A. Bullo, G.A. Degla, G.F. Duressa, Parameter-uniform finite difference method for a singularly perturbed parabolic problem with two small parameters. Int. J. Comput. Methods Eng. Sci. Mech. 23(3), 210–218 (2022)

    Article  Google Scholar 

  16. A.H. Ejere, T.G. Dinka, M.M. Woldaregay, G.F. Duressa, A tension spline fitted numerical scheme for singularly perturbed reaction-diffusion problem with negative shift. BMC. Res. Notes 16(1), 1–16 (2023)

    Article  Google Scholar 

  17. W.T. Aniley, G.F. Duressa, A uniformly convergent numerical method for time-fractional convection–diffusion equation with variable coefficients. Partial Differential Equations in Applied Mathematics 8, 100592 (2023)

    Article  Google Scholar 

  18. V. Gupta, M.K. Kadalbajoo, A singular perturbation approach to solve Burgers-Huxley equation via monotone finite difference scheme on layer-adaptive mesh. Commun. Nonlinear Sci. Numer. Simul. 16(4), 1825–1844 (2011)

    Article  Google Scholar 

  19. Ö. Oruç, A non-uniform Haar wavelet method for numerically solving two-dimensional convection-dominated equations and two-dimensional near singular elliptic equations. Comput. Math. Appl. 77(7), 1799–1820 (2019)

    Article  Google Scholar 

  20. Ö. Oruç, Two meshless methods based on pseudo spectral delta-shaped basis functions and barycentric rational interpolation for numerical solution of modified Burgers equation. Int. J. Comput. Math. 98(3), 461–479 (2021)

    Article  Google Scholar 

  21. A.L.A.A.T.T.İN. Esen, F.A.T.İH. Bulut, Ö. Oruç, A unified approach for the numerical solution of time fractional Burgers’ type equations. Eur. Phys. J. Plus 131(4), 116 (2016)

    Article  Google Scholar 

  22. Ö. Oruç, F.A.T.İH. Bulut, A.L.A.A.T.T.İN. Esen, A Haar wavelet-finite difference hybrid method for the numerical solution of the modified Burgers’ equation. J. Math. Chem. 53, 1592–1607 (2015)

    Article  Google Scholar 

  23. J. Lu, Y. Sun, Numerical approaches to time fractional boussinesq–burgers equations. Fractals 29(08), 2150244 (2021)

    Article  Google Scholar 

  24. K.L. Wang, A novel approach for fractal Burgers–BBM equation and its variational principle. Fractals 29(03), 2150059 (2021)

    Article  Google Scholar 

  25. B. Chen, L. Chen, Z.Z. **a, He-laplace method for time fractional burgers-type equations. Therm. Sci. 27(3 Part A), 1947–1955 (2023)

    Article  Google Scholar 

  26. J. Lu, Application of variational principle and fractal complex transformation to (3+ 1)-dimensional fractal potential-YTSF equation. Fractals 32(01), 2450027 (2024)

    Article  Google Scholar 

  27. J. Lu, Variational approach for (3+ 1)-dimensional shallow water wave equation. Res. Phys. 56, 107290 (2024)

    Google Scholar 

  28. G. Liu, Z. Zhang, Y. Cao, X. Wang, H. Liu, B. Li, W. Guan, An Analogical Method On Fractal Dimension For Three-Dimensional Fracture Tortuosity In Coal Based On Ct Scanning. Fractals 31(07), 2350072 (2023)

    Article  Google Scholar 

  29. R. Jiwrai, R.C. Mittal, A higher order numerical scheme for singularly perturbed Burger-Huxley equation. J. Appl. Math. Inform. 29(3), 813–829 (2011)

    Google Scholar 

  30. R.C. Mittal, R. Jiwari, Numerical study of Burger-Huxley equation by differential quadrature method. Int. J. Appl. Math. Mech 5, 1–9 (2009)

    Google Scholar 

  31. R. Jiwari, Local radial basis function-finite difference based algorithms for singularly perturbed Burgers’ model. Math. Comput. Simul 198, 106–126 (2022)

    Article  Google Scholar 

  32. O.P. Yadav, R. Jiwari, Finite element analysis and approximation of Burgers’-Fisher equation. Numer. Methods Partial Differ. Equ. 33(5), 1652–1677 (2017)

    Article  Google Scholar 

  33. J. Singh, N. Kumar, R. Jiwari, A robust weak Galerkin finite element method for two parameter singularly perturbed parabolic problems on nonuniform meshes. J. Comput. Sci. 77, 102241 (2024)

    Article  Google Scholar 

  34. N. Kumar, Ş Toprakseven, R. Jiwari, A numerical method for singularly perturbed convection–diffusion–reaction equations on polygonal meshes. Comput. Appl. Math. 43(1), 44 (2024)

    Article  Google Scholar 

  35. S. Pandit, Local radial basis functions and scale-3 Haar wavelets operational matrices based numerical algorithms for generalized regularized long wave model. Wave Motion 109, 102846 (2022)

    Article  Google Scholar 

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Acknowledgements

The authors thank Jimma University for their material support.

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Authors and Affiliations

  1. Department of Mathematics, College of Natural Science, Jimma University, Jimma, Ethiopia

    Masho Jima Kabeto, Tesfaye Aga Bullo, Habtamu Garoma Debela, Gemadi Roba Kusi & Sisay Dibaba Robi

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  1. Masho Jima Kabeto
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  2. Tesfaye Aga Bullo
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  3. Habtamu Garoma Debela
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  4. Gemadi Roba Kusi
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  5. Sisay Dibaba Robi
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Contributions

MJK is ongoing and has organized the plans for this research work. TAB and HGD formulated the numerical scheme and examined the numerical analysis of the study. GRK and SDR revised the study's techniques, analysis, and results. All authors have equal assistance to the paper and decide on the submitted version. All authors read and approved the final manuscript.

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Correspondence to Tesfaye Aga Bullo.

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Kabeto, M.J., Bullo, T.A., Debela, H.G. et al. Efficient computational method for singularly perturbed Burger-Huxley equations. J Math Chem (2024). https://doi.org/10.1007/s10910-024-01627-3

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