Geeta Arora and Gurpreet Singh Bhatia* A Meshfree Numerical Technique Based on Radial Basis Function Pseudospectral Method for Fisher’s Equation https://doi.org/10.1515/ijnsns-2018-0091 Received April 08, 2018; accepted September 10, 2019 Abstract: This paper concerns with the implementation of radial basis function pseudospectral (RBF-PS) method for solving Fisher’s equation. Pseudospectral methods are well known for being highly accurate but are limited in terms of geometric flexibility. Radial basis function (RBF) in combi- nation with the pseudospectral method is capable to over- come this limitation. Using RBF, Fisher’s equation is approximated by transforming it into a system of ordinary differential equations (ODEs). An ODE solver is used to solve the resultant ODEs. In this approach, the optimal value of the shape parameter is discussed with the help of leave-one out cross validation strategy which plays an important role in the accuracy of the result. Several exam- ples are given to demonstrate the accuracy and efficiency of the method. RBF-PS method is applied using different types of basis functions and a comparison is done based upon the numerical results. A two-dimensional problem that general- izes the Fisher’s equation is also solved numerically. The obtained numerical results and comparisons confirm that the use of RBF in pseudospectral mode is in good agreement with already known results in the literature. Keywords: Fisher’s equation, radial basis function, pseu- dospectral method, meshfree, shape parameter MSC ® (2010). 65M70, 65N35, 65M99 1 Introduction Most of the physical phenomena are modeled with the help of partial differential equations (PDEs). Reaction diffusion equation is one such PDE which plays a significant role in understanding various physical and chemical phenomena. The reaction diffusion equation is a model equation used in various fields of science like ecological invasions, epidemic, pattern formation, oscillating chemical reactions, popula- tion biology, etc. Recently, the application of reaction dif- fusion equation was found in wound healing and tumor growth. Fisher’s equation is one of the most important and simplest reaction diffusion equations. Fisher [1] introduced an equation to describe the growth of advantageous gene due to mutation known as Fisher’s equation which has numerous applications in the field of science and engineer- ing such as flame propagation, tissue engineering, growth models, heat and mass transfer. The nonlinear Fisher’s reaction diffusion equation is given by: ∂u ∂t = λ ∂ 2 u ∂x 2 + βuð1 - uÞ, a ≤ x ≤ b, t >0 [1] where λ is the diffusion constant and β >0 is a parameter. Due to its application in various fields, numerous efforts have been made to solve it numerically. Gazdag and Canosa [2] proposed an accurate space derivative method which uses finite Fourier series to evaluate the space derivatives for solving Fisher’s equation. Ablowitz and Zeppetella [3] have established the explicit solution for Fisher’s equation. Twizell et al. [4] applied explicit finite difference methods to solve Fisher’s equation. Tang and Weber [5] demonstrated the use of Petrov–Galerkin finite element technique to discuss the numerical solution. Mickens [6] constructed a new class of finite-difference scheme for Fisher’s equation. Carey and Shen [7] imple- mented the least square finite element method. Qiu and Sloan [8] studied the numerical solution of Fisher’s equa- tion by using moving mesh method. Rizwan [9] compared the finite difference scheme and the nodal integral method for Fisher’s equation. Sinc collocation method was used to solve the equation numerically by Al-Khaled [10]. Wazwaz and Gorguis [11] obtained the exact solution of Fisher’s equation by using Adomian decomposition method. Olmos and Shizgal [12] applied a spectral method based on Chebyshev–Lobatto points to develop a pseudospectral solution of Fisher’s equation. In recent years, various *Corresponding author: Gurpreet Singh Bhatia, Department of Mathematics, Lovely Professional University, Phagwara, Punjab 144411, India, E-mail: gurpreetsidakbhatia@gmail.com https://orcid.org/0000-0001-8180-952X Geeta Arora, Department of Mathematics, Lovely Professional University, Phagwara, Punjab 144411, India, E-mail: geetadma@gmail.com IJNSNS 2019; aop Brought to you by | Chalmers University of Technology Authenticated Download Date | 10/7/19 10:31 AM