Chaos, Solitons and Fractals 86 (2016) 101–106 Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos The exponential cubic B-spline algorithm for Fisher equation Idiris Dag, Ozlem Ersoy ∗ Eski ¸ sehir Osmangazi University, Department of Mathematics-Computer, Faculty of Science and Art, Eski ¸ sehir, Turkey a r t i c l e i n f o Article history: Received 18 February 2015 Revised 3 January 2016 Accepted 21 February 2016 PACS: 02.70.Jn 47.35.Fg Keywords: Collocation methods Exponential cubic B-spline Fisher’s equation a b s t r a c t In this study, exponential B-spline collocation method is set up for solving Fisher’s equation. Integration of Fisher’s equation is managed by use of the exponential cubic B-spline in space and the Crank–Nicolson method in time. The effect of reaction and diffusion is observed by studying three test problems. A com- parison is performed between the obtained numerical results and some earlier results using L ∞ and rel- ative error norms. © 2016 Elsevier Ltd. All rights reserved. 1. Introduction The reaction–diffusion equations have an important role in modeling some physical phenomena. They are used in many fields such as biology, chemistry and engineering. Because of the complexities in finding their solutions, approximate solutions of reaction–diffusion equations have become a central tool in their study. In one space dimension, the nonlinear reaction–diffusion equations may be written in the following form: U t = λU xx + φ (U ) (1) where U = U (x, t ) is a time dependent real-valued function. λU xx is called diffusivity term where the coefficient λ is a non-negative constant and the function φ(U) describes the reaction term. One of the most popular special case of the Eq. (1) is given by U t = λU xx + βU (1 − U ), −∞ < x < ∞, t > 0 (2) where β is a real parameter. This equation is known as a Fisher’s equation, which was introduced by Fisher [1] who describe it to model the kinetic advancing rate of an advantageous gene. Since then works have been done to extend the model to take into account the other biological, chemical and physical events. Fisher’s equation is also defined to represents the evolution of the population due to the two competing physical processes, flame propagation, nuclear reactor, auto-catalytic chemical reactions, logistic growth models and neurophysiology. ∗ Corresponding author. Tel.: +902222393750. E-mail address: ozersoy@ogu.edu.tr (O. Ersoy). The exponential B-splines are piecewise polynomial functions including a free parameter. It and its properties are introduced by [5]. Although the use of it in the numerical methods is not as wide as the well-known B-splines, there are few papers dealing with the use of the exponential B-splines for finding solutions of the dif- ferential equations: Some variants of the exponential B-spline col- location methods have been set up to numerical solutions of the Singularly perturbed problems in the studies [4,7,8]. Time depen- dent partial differential equations, Convection–Diffusion, Korteweg- de Vries (KdV), generalized Burgers, equal width and Generalized Long Wave equations, have been solved numerically by way of the exponential B-spline collocation method recently [14,16–19]. The numerical solutions of the differential equations have investigated using spline functions. So spline functions are used to build up numerical methods for finding solutions of differential equations. Publications also exist dealing with the numerical solutions of Fisher’s equation by way of spline-related techniques. We want to mention some of them: the Galerkin scheme whose trial function is consist of combination of the cubic B-splines is set up to find numerical solutions of Fisher’s equation over the finite elements [9], the Galerkin method is used with quadratic B-spline base functions to obtain the numerical solutions of Fisher’s equa- tion [10], a cubic B-spline collocation method is given to solve Fisher’s equation [11], the numerical solution of Fisher’s equation is given by using collocation method based on the modified cubic B-spline method [13], cubic B-spline quasi-interpolation is estab- lished to obtain solutions of Fisher’s equation numerically [12], the quintic B-spline collocation method is proposed to get solution of Fisher’s equation in the study [15] http://dx.doi.org/10.1016/j.chaos.2016.02.031 0960-0779/© 2016 Elsevier Ltd. All rights reserved.