Research Article Compact Finite Difference Scheme Combined with Richardson Extrapolation for Fisher’s Equation Hailu Muleta Chemeda , Ayana Deressa Negassa , and Feyisa Edosa Merga Jimma University Department of Mathematics, Jimma, Ethiopia Correspondence should be addressed to Hailu Muleta Chemeda; hailux2020@gmail.com Received 7 December 2021; Accepted 23 December 2021; Published 27 January 2022 Academic Editor: Francisco Urena Copyright©2022HailuMuletaChemedaetal.isisanopenaccessarticledistributedundertheCreativeCommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this study, the fourth-order compact finite difference scheme combined with Richardson extrapolation for solving the 1D Fisher’s equation is presented. First, the derivative involving the space variable is discretized by the fourth-order compact finite difference method. en, the nonlinear term is linearized by the lagging method, and the derivative involving the temporal variableisdiscretizedbytheCrank–Nicolsonscheme.emethodisfoundtobeunconditionallystableandfourth-orderaccurate inthedirectionofthespacevariableandsecond-orderaccurateinthedirectionofthetemporalvariable.Whencombinedwiththe Richardsonextrapolation,theorderofthemethodisimprovedfromfourthtosixth-orderaccurateinthedirectionofthespace variable. e numerical results displayed in figures and tables show that the proposed method is efficient, accurate, and a good candidate for solving the 1D Fisher’s equation. 1. Introduction Mathematical modeling of most physical systems leads to linear/nonlinear partial differential equations (PDEs) in various fields of science. PDEs have enormous applications comparedtoordinarydifferentialequations(ODEs)suchas in dynamics, electricity, heat transfer, electromagnetic theory, quantum mechanics, and so on [1]. e 1D Fisher’s equation is given by zu zt � β z 2 u zx 2 + αu(1 − u), x ∈ (a, b) ⊆ (−∞, ∞),t ≥ 0. (1) With the initial condition, u(x, 0)� f(x), a ≤ x ≤ b. (2) e boundary conditions are u(a, t)� g 1 (t), u(b, t)� g 2 (t), t ≥ 0, (3) where β( > 0) is the diffusion coefficient, α( > 0) is the re- activefactor, x isthedistance, t isthetime,and u(x, t) isthe population density. f(x),g 1 (t), and g 2 (t) are the smooth functions on the given domain. e 1D Fisher’s equation in equation (1) was first proposed by Fisher [2] as a model for the spatial and temporalpropagationofavirilegeneinaninfinitemedium [3]. It can also be considered as a model equation for the evolutionofaneutronpopulationinanuclearreactor[4,5]. Equation (1) also describes the rate of the advance of a new advantageousgenewithinapopulationofaconstantdensity occupyingaone-dimensionalhabitat[6].Inequation(1),the effect of the linear diffusion is observed along u xx , whereas the nonlinear local multiplication or reaction is observed along u(1 − u) [5, 7, 8]. Some of the application areas of equation (1) include gene propagation [3, 5], tissue engi- neering [9], combustion [10], and neurophysiology [8]. Duetoitswiderapplicationsintherealworldproblems, many researchers have been developing both analytical and numerical methods for solving Fisher’s equation. Gazdag Hindawi Mathematical Problems in Engineering Volume 2022, Article ID 7887076, 7 pages https://doi.org/10.1155/2022/7887076