An effective Legendre wavelet technique for the time-fractional Fisher equation

Document Type : Research Paper

Authors

1 Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey.

2 Department of Mathematics, Izmir Institute of Technology, Izmir, Turkey.

3 1. Department of Mathematics, Izmir Institute of Technology, Izmir, Turkey.\\ 2. Center for Theoretical Physics, Khazar University, 41 Mehseti Street, Baku, AZ1096, Azerbaijan.

4 Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran.

Abstract

This study modifies the time-fractional Fisher equation by adding a source term and generalizing the non-linear power to an arbitrary order. A numerical technique is proposed for the modified time-fractional Fisher equation using Legendre wavelets and the quasilinearization technique. The non-linear term is iteratively linearized using the quasilinearization technique. The convergence analysis and error estimates of the proposed method are studied. Several test problems are solved using the proposed technique, and numerical outcomes are contrasted with those obtained using some other approaches existing in the literature.

Keywords

Main Subjects

References

  • [1] N. Aghazadeh, A.Mohammadi, G. Ahmadnezhad, and S. Rezapour, Solving partial fractional differential equations by using the Laguerre wavelet-Adomian method, Advances in Difference Equations, 2021(1) (2021), 231.
  • [2] N. Aghazadeh, A. Mohammadi, and G. Tanoğlu, Taylor wavelets collocation technique for solving fractional nonlinear singular PDEs, Mathematical Sciences, (2022).
  • [3] G. Ahmadnezhad, N. Aghazadeh, and S. Rezapour, Haar wavelet iteration method for solving time fractional Fisher’s equation, Computational Methods for Differential Equations, 8(3) (2020), 505–522.
  • [4] R. E. Bellman and R. E. Kalaba, Quasilinearization and nonlinear boundary-value problems, RAND Corporation, Santa Monica, CA, (1965).
  • [5] A. H. Bhrawy, A. S. Alofi, and S. S. Ezz-Eldien, A quadrature tau method for fractional differential equations with variable coefficients, Applied Mathematics Letters, 24(12) (2011), 2146–2152.
  • [6] M. D. Bramson, Maximal displacement of branching brownian motion, Communications on Pure and Applied Mathematics, 31(5) (1978), 531–581.
  • [7] A. Das, M. Rabbani, and B. Hazarika, An iterative algorithm to approximate the solution of a weighted fractional integral equation, Asian-European Journal of Mathematics, 17(01) (2024), 2350241.
  • [8] A. Das, M. Rabbani, S. A. Mohiuddine, and B. C. Deuri, Iterative algorithm and theoretical treatment of existence of solution for (k,z)-Riemann–Liouville fractional integral equations, Journal of Pseudo-Differential Operators and Applications, 13(3) (2022), 39.
  • [9] K. Diethelm, The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type, International series of monographs on physics, Springer, (2010).
  • [10] R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7(4) (1937), 355–369.
  • [11] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, (2000).
  • [12] F. Idiz, G. Tanoğlu, and N. Aghazadeh, A numerical method based on Legendre wavelet and quasilinearization technique for fractional Lane-Emden type equations, Numerical Algorithms, (2023), 1–26.
  • [13] W. R. Inc, Mathematica, Version 13.1, Champaign, IL, (2022).
  • [14] B. Jin, Fractional Differential Equations: An Approach Via Fractional Derivatives, Springer Nature, (2022).
  • [15] V. M. Kenkre, Results from variants of the Fisher equation in the study of epidemics and bacteria, Physica A: Statistical Mechanics and its Applications, 342(1) (2004), 242–248.
  • [16] Y. Li and W. Zhao, Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Applied Mathematics and Computation, 216(8) (2010), 2276–2285.
  • [17] K. Maleknejad and A. Hoseingholipour, The impact of Legendre wavelet collocation method on the solutions of nonlinear system of two-dimensional integral equations, International Journal of Computer Mathematics, 97(11) (2020), 2287–2302.
  • [18] S. Mallat, A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way, Academic Press, Inc., USA, 3rd edition, (2008).
  • [19] A. Mohammadi, G. Ahmadnezhad, and N. Aghazadeh, Chebyshev-quasilinearization method for solving fractional singular nonlinear Lane-Emden equations, Communications in Mathematics, (2022).
  • [20] S. T. Mohyud-Din and M. A. Noor, Modified variational iteration method for solving Fisher’s equations, Journal of Applied Mathematics and Computing, 31 (2009), 295–308.
  • [21] Y. R. Molliq, M. S. M. Noorani, and I. Hashim, Variational iteration method for fractional heat- and wave-like equations, Nonlinear Analysis: Real World Applications, 10(3) (2009), 1854–1869.
  • [22] S. Momani and Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Physics Letters A, 365(5) (2007), 345–350.
  • [23] P. Moulin, Chapter 6 - Multiscale Image Decompositions and Wavelets, In A. Bovik, editor, The Essential Guide to Image Processing, Academic Press, Boston, (2009), 123–142.
  • [24] K. B. Oldham, Fractional differential equations in electrochemistry, Advances in Engineering Software, 41(1) (2010), 9–12.
  • [25] Y. Ouedjedi, A. Rougirel, and K. Benmeriem, Galerkin method for time fractional semilinear equations, Fractional Calculus and Applied Analysis, 24(3) (2021), 755–774.
  • [26] R. Rahul, N. K. Mahato, M. Rabbani, and N. Aghazadeh, Existence of the solution via an iterative algorithm for two-dimensional fractional integral equations including an industrial application, Journal of Integral Equations and Applications, 35(4) (2023), 459 – 472.
  • [27] S. S. Ray and R. K. Bera, An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Applied Mathematics and Computation, 167(1) (2005), 561–571.
  • [28] J. Ross, A. F. Villaverde, J. R. Banga, S. Vazquez, and F. Moran, A generalized Fisher equation and its utility in chemical kinetics, Proceedings of the National Academy of Sciences, 107(29) (2010), 12777–12781.
  • [29] F. R. Savadkoohi, M. Rabbani, T. Allahviranloo, and M. R. Malkhalifeh, A fractional multi-wavelet basis in Banach space and solving fractional delay differential equations, Chaos, Solitons & Fractals, 186 (2024), 115313.
  • [30] A. Secer and M. Cinar, A Jacobi wavelet collocation method for fractional fisher’s equation in time, Thermal Science, 24 (2020), 119–129.
  • [31] H. Singh, Jacobi collocation method for the fractional advection-dispersion equation arising in porous media, Numerical Methods for Partial Differential Equations, 38(3) (2022), 636–653.
  • [32] P. J. Torvik and R. L. Bagley, On the Appearance of the Fractional Derivative in the Behavior of Real Materials, Journal of Applied Mechanics, 51(2) (1984), 294–298.
  • [33] L. Wang, Y. Ma, and Z. Meng, Haar wavelet method for solving fractional partial differential equations numerically, Applied Mathematics and Computation, 227 (2014), 66–76.
  • [34] A. M. Wazwaz and A. Gorguis, An analytic study of Fisher’s equation by using Adomian decomposition method, Applied Mathematics and Computation, 154(3) (2004), 609–620.
  • [35] S. Yan, G. Ni, and T. Zeng, Image restoration based on fractional-order model with decomposition: texture and cartoon, Computational and Applied Mathematics, 40(304) (2021).
Computational Methods for Differential Equations
Volume 14, Issue 1
January 2026
Pages 145-164

Files

History

  • Receive Date: 27 September 2024
  • Revise Date: 18 January 2025
  • Accept Date: 27 January 2025

Share

How to cite

Statistics