Higher-Order Numerical Techniques for Solving the Nonlinear Fisher Equation are Based on the Runge-Kutta Method
DOI:
https://doi.org/10.26713/cma.v16i2.2912Keywords:
Fisher’s problems, Method of lines, Finite difference methods, Strong stability preserving Runge-Kutta methodsAbstract
This paper presents higher-order numerical methods for solving nonlinear Fisher equations. These types of equations arise in various fields of sciences and engineering, the main application of this equation has been found in the biomedical sciences. The solution of this equation helps to determine the size of the brain tumor. In this paper explores the utilization of advanced numerical techniques, such as the method of lines and higher-order strong stability preserving schemes of order four and stage seven, to approximate solutions to the Fisher equation with higher-order accuracy. These schemes are explicitly designed and easy to implement, especially for addressing nonlinear problems. Their stability-preserving nature ensures only mild restrictions on time steps. This scheme is then tested on two examples and the results show that it is more efficient methods and requires less computing time. Various test problems are examined to verify the scheme’s performance, including a comparison of l2 and l∞ errors with the exact solution, leading to high accuracy.
Downloads
References
K. M. Agbavon, A. R. Appadu and M. Khumalo, On the numerical solution of Fisher’s equation with coefficient of diffusion term much smaller than coefficient of reaction term, Advances in Difference Equations 2019 (2019), article number 146, DOI: 10.1186/s13662-019-2080-x.
B. S. Alshammari and D. S. Mashat, Numerical study of Fisher’s equation by finite difference schemes, Applied Mathematics 8(8) (2017), 1100 – 1116, DOI: 10.4236/am.2017.88083.
A. J. Ammerman and L. L. Cavalli-Sforza, Measuring the rate of spread of early farming in Europe, Man 6(4) (1971), 674 – 688, DOI: 10.2307/2799190.
M. Bastani and D.K. Salkuyeh, A highly accurate method to solve Fisher’s equation, Pramana 78 (2012), 335 – 346, DOI: 10.1007/s12043-011-0243-8.
M. Bonkile, A. Awasthi and S. Jayaraj, A numerical implementation of higher order time integration algorithm on unsteady Burgers’ equation, in: Proceedings of the International Conference On Mathematical Modeling and Computer Simulation, Indian Institute of Technology Madras, Chennai, India (2014).
M. D. Bramson, Maximal displacement of branching Brownian motion, Communications on Pure and Applied Mathematics 31(5) (1978), 531 – 581, DOI: 10.1002/cpa.3160310502.
J. Canosa, Diffusion in nonlinear multiplicative media, Journal of Mathematical Physics 10(10) (1969), 1862 – 1868, DOI: 10.1063/1.1664771.
V. Chandrakera, A. Awasthib and S. Jayaraja, A numerical treatment of Fisher equation, Procedia Engineering 127 (2015), 1256 – 1262, DOI: 10.1016/j.proeng.2015.11.481.
R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics 7(4) (1937), 355 – 369, DOI: 10.1111/j.1469-1809.1937.tb02153.x.
D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, 2nd enlarged and revised edition, Plenum Press, New York – London, xxvi + 574 pages (1969).
S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Review 43(1) (2001), 89 – 112, DOI: 10.1137/S003614450036757X.
S. Hasnain and M. Saqib, Numerical study of one dimensional Fishers KPP equation with finite difference schemes, American Journal of Computational Mathematics 7(01) (2017), 70 – 83, DOI: 10.4236/ajcm.2017.71006.
A. C. Hindmarsh, Toward a Systematized Collection of ODE Solvers, No. UCRL-87465; CONF-820810-6, Lawrence Livermore National Laboratory, CA (USA), (1982), URL: https://ntrl.ntis.gov/NTRL/dashboard/searchResults/titleDetail/DE82012217.xhtml.
R. Jiwari and R. C. Mittal, A higher order numerical scheme for singularly perturbed Burger-Huxley equation, Journal of Applied Mathematics and Informatics 29(3-4) (2011), 813 – 829, URL: https://koreascience.kr/article/JAKO201121559392442.pdf.
A. Kolmogorov, Étude de l’équation de la diffu`sion avec croissance de la quantité de matière et son application à un problème biologigue, Bulletin de l’Universite d’Etat a Moscou Serie Internationale Section 1 (1937), 12 pages.
J. E. Macías-Díaz, I. E. Medina-Ramírez and A. Puri, Numerical treatment of the spherically symmetric solutions of a generalized Fisher–Kolmogorov–Petrovsky–Piscounov equation, Journal of Computational and Applied Mathematics 231(2) (2009), 851 – 868, DOI: 10.1016/j.cam.2009.05.008.
R. E. Mickens, A best finite-difference scheme for the fisher equation, Numerical Methods for Partial Differential Equations 10(5) (1994), 581 – 585, DOI: 10.1002/num.1690100505.
R. C. Mittal and R. Jiwari, Numerical study of Fisher’s equation by using differential quadrature method, International Journal of Information and System Sciences 5(1) (2009), 143 – 160.
R. C. Mittal and S. Kumar, Numerical study of Fisher’s equation by wavelet Galerkin method, International Journal of Computer Mathematics 83(3) (2006), 287 – 298, DOI: 10.1080/00207160600717758.
O. Oymak and N. Selçuk, Method-of-lines solution of time-dependent two-dimensional Navier-Stokes equations, International Journal for Numerical Methods in Fluids 23(5) (1996), 455 – 466, DOI: 10.1002/(SICI)1097-0363(19960915)23:5%3C455::AID-FLD435%3E3.0.CO;2-J.
R. A. V. Parambu, A. Awasthi, V. Vimal and N. Jha, A numerical implementation of higher-order time integration method for the transient heat conduction equation with a moving boundary based on boundary immobilization technique, AIP Conference Proceedings 2336(1) (2021), 030012, DOI: 10.1063/5.0045874.
Y. Qiu and D. M. Sloan, Numerical solution of Fisher’s equation using a moving mesh method, Journal of Computational Physics 146(2) (1998), 726 – 746, DOI: 10.1006/jcph.1998.6081.
E. Rothe, Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben, Mathematische Annalen 102(1) (1930), 650 – 670, DOI: 10.1007/BF01782368.
N. A. Shah, An analytical view of fractional-order Fisher’s type equations within Caputo operator, Mathematical Problems in Engineering 2021(1) (2021), 5516392, DOI: 10.1155/2021/5516392.
L. F. Shampine, ODE solvers and the method of lines, Numerical Methods for Partial Differential Equations 10(6) (1994), 739 – 755, DOI: 10.1002/num.1690100608.
C.-W. Shu, Total-variation-diminishing time discretizations, SIAM Journal on Scientific and Statistical Computing 9(6) (1988), 1073 – 1084, DOI: 10.1137/0909073.
R. J. Spiteri and S. J. Ruuth, A new class of optimal high-order strong-stability-preserving time discretization methods, SIAM Journal on Numerical Analysis 40(2) (2002), 469 – 491, DOI: 10.1137/S0036142901389025.
R. J. Spiteri and S. J. Ruuth, Non-linear evolution using optimal fourth-order strong-stability preserving Runge-Kutta methods, Mathematics and Computers in Simulation 62(1-2) (2003), 125 – 135, DOI: 10.1016/S0378-4754(02)00179-9.
S. Tang and R. Weber, Numerical study of Fisher’s equation by a Petrov-Galerkin finite element method, The ANZIAM Journal 33(1) (1991), 27 – 38, DOI: 10.1017/S0334270000008602.
J. J. Tyson and P. K. Brazhnik, On traveling wave solutions of Fisher’s equation in two spatial dimensions, SIAM Journal on Applied Mathematics 60(2) (2000), 371 – 391, URL: http://www.jstor.org/stable/118487.
A. Verma, R. Jiwari and M. E. Koksal, Analytic and numerical solutions of nonlinear diffusion equations via symmetry reductions, Advances in Difference Equations 2014 (2014), article number 229, DOI: 10.1186/1687-1847-2014-229.
V. Vimal, R. K. Sinha and L. Pannikkal, Numerical methods for solving nonlinear Fisher equation using backward differentiation formula, Applications and Applied Mathematics 19(4) (2024), Article 4, URL: https://digitalcommons.pvamu.edu/aam/vol19/iss4/4.
V. Vimal, R. K. Sinha and P. Liju, An unconditionally stable numerical scheme for solving nonlinear Fisher equation, Nonlinear Engineering 13(1) (2024), 20240006, DOI: 10.1515/nleng-2024-0006.
V. Vimal, R. Kumari and R. K. Sinha, Higher-order numerical technique based on strong stability preserving method for solving nonlinear Fisher equation, Communications in Mathematics and Applications 15(3) (2024), 997 – 1010, DOI: 10.26713/cma.v15i3.2806.
X. Y. Wang, Exact and explicit solitary wave solutions for the generalised Fisher equation, Physics Letters A 131(4-5) (1988), 277 – 279, DOI: 10.1016/0375-9601(88)90027-8.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
