Abstract
This study presents a numerical solution of Fisher’s equation. For time integration, Crank–Nicolson and fourth-order one-step implicit schemes are used and for space discretization, quintic B-spline Collocation and quintic B-spline Galerkin methods are employed. The truncation error is analyzed and the stability of the suggested methods is discussed matrix stability analysis. Three examples are studied to compare the present results with existing numerical results by computing error norm \(L_{\infty }\) and the order of convergence. The obtained results show that the proposed methods are satisfactorily efficient in terms of accuracy.
Similar content being viewed by others
References
Ablowitz M, Zepetella A (1979) Explicit solution of Fisher’s equation for a special wave speed. Bull. Math. Biol. 41:835–840
Aghamohamadi M, Rashidinia J, Ezzati R (2014) Tension spline method for solution of non-linear Fisher equation. Appl. Comput. 249:399–407
Al-Khaled K (2001) Numerical study of Fisher’s reaction-diffusion equation by the sinc collocation method. J. Comput. Appl. Math. 137:245–255
Brawson MD (1978) Maximal displacement of branching Brownian motion. Commun. Pure Appl. Math. 31:531–581
Canosa J (1969) Diffusion in nonlinear multiplicative media. J. Math. Phys. 10:1862–1868
Dag I, Ersoy O (2016) The exponential B-spline algorithm for Fisher equation. Chaos Solit. Frac. 86:101–106
Dag I, Sahin A, Korkmaz A (2010) Numerical investigation of the solution of Fisher’s equation via the B-spline Galerkin method. Numer. Methods Partial Differ. Equ. 26(6):1458–1503
Fife PC, McLeod JB (1977) The aproach of solutions of nonlinear diffusion equations to travelling front solution. Arch. Rational Mech. Anal. 65:335–361
Fisher RA (1937) The wave of advance of advantgeous genes. Ann. Eugen 7:355–369
Gazdag J, Canosa J (1974) Numerical solutions of Fisher’s equation. J. Appl. Probab. 11:445–457
Kapoor M, Jashi U (2020) Solution of non-linear Fisher’s reaction–diffusion equation by using Hyperbolic B-spline based differential quadrature method. J. Phys.: Conf. Ser. 1531:012064
Kırlı E, Irk D (2021) A fourth-order one step method for numerical solution of good Boussinesq equation. Turk. J. Math. 45:2154–2170
Mittal RC, Arora G (2010) Efficient numerical solution of Fisher’s eqution by using B-spline method. Int. J. Comput. Math. 87(13):3039–3051
Mittal RC, Jain RK (2013) Numerical solutions of nonlinear Fisher’s equation with modified cubic B-spline collocation method. Math. Sci. 7(1):1–10
Mittal RC, Kumar S (2006) Numerical study of Fisher’s equation by wavelet Galerkin method. Int. J. Comput. Math. 83:287–298
Rohila R, Mittal RC (2018) Numerical study of reaction diffusion Fisher’s equation by fourth-order cubic B-spline collocation method. Math. Sci. 12:79–89
Sahin A, Dag I, Saka B (2008) A B-spline algorithm for the numerical solution of Fisher’s equation. Kybernetes 37:326–342
Sahin A, Ozmen O (2014) Usage of higher order B-splines in numerical solution of Fisher’s equation. Int. J. Nonlinear Sci. 17(3):241–253
Shallu Kukreja VK (2021) An improvised collocation algorithm with specific end conditions for solving modified Burgers equation. Numer. Methods Partial Differ. Equ. 37:874–896
Singh BK, Arora A (2014) A numerical scheme to solve Fisher-type reaction–diffusion equations. MESA 5(2):153–164
Singh S, Singh S, Bhatt S (2021) High order compact cubic B-spline collocation method for the solution of Fisher’s equation. Int. J. Appl. Comput. Math. 217:2
Tamsir M, Dhiman N, Srivastava VK (2017) Cubic trigonometric B-spline differential quadrature method for numerical treatment of Fisher’s reaction–diffusion equations. Alex. Eng. J. 57(3):2019–2026
Tamsir M, Huntul MJ (2021) A numerical approach for solving Fisher’s reaction–diffusion equation via a new kind of spline functions. Ain Shams Eng. J. 12(3):3157–3165
Tang S, Weber RO (1991) Numerical study of Fisher’s equation by a Petrov–Galerkin finite element method. J. Aust. Math. Soc. Sci. B 33:27–38
Zeldovich JB, Frank-Kamenetzk DA (1938) A theory of thermal propagation of flame. Acta Physiochim. U.R.S.S 7(2):341–350
Zorsahin-Gorgulu M, Dag I (2017) Exponential B-splines Galerkin method for the numerical solution of the Fisher’s equation. Iran J. Sci. Tech. Trans. Sci. 42:2189–2198
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Frederic Valentin.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kırlı, E., Irk, D. Efficient techniques for numerical solutions of Fisher’s equation using B-spline finite element methods. Comp. Appl. Math. 42, 151 (2023). https://doi.org/10.1007/s40314-023-02292-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-023-02292-z