Numerical Solutions of Reaction-Diffusion Equation Systems with Trigonometric Quintic B-spline Collocation Algorithm

Abstract

In this study, the numerical solutions of reaction-diffusion systems are investigated via the trigonometric quintic B-spline nite element collocation method. These equations appear in various disciplines in order to describe certain physical facts, such as pattern formation, autocatalytic chemical reactions and population dynamics. The Schnakenberg, Gray-Scott and Brusselator models are special cases of reaction-diffusion systems considered as numerical examples in this paper. For numerical purposes, Crank-Nicolson formulae are used for the time discretization and the resulting system is linearized by Taylor expansion. In the finite element method, a uniform partition of the solution domain is constructed for the space discretization. Over the mentioned mesh, dirac-delta function and trigonometric quintic B-spline functions are chosen as the weighted function and the bases functions, respectively. Thus, the reaction-diffusion system turns into an algebraic system which can be represented by a matrix equation so that the coeffcients are block matrices containing a certain number of non-zero elements in each row. The method is tested on different problems. To illustrate the accuracy, error norms are calculated in the linear problem whereas the relative error is given in other nonlinear problems. Subject to the character of the nonlinear problems, the occurring spatial patterns are formed by the trajectories of the dependent variables. The degree of the base polynomial allows the method to be used in high-order differential equation solutions. The algorithm produces accurate results even when the time increment is larger. Therefore, the proposed Trigonometric Quintic B-spline Collocation method is an effective method which produces acceptable results for the solutions of reaction-diffusion systems.

Numerical solution to the Gray-Scott Reaction-Diffusion equation using Hyperbolic B-spline

“…In biology, the RD system explains the pattern of stripes of leopards, snakes, zebras, seashells, and jaguars [2] and creates a model for replicating the hepatitis B virus contamination with spatial dependence [3]. Nonlinear terms are attained by most of the RD system which makes it difficult to elucidate analytically [4]. Furthermore, the self-replicating configurations have also been witnessed for this system [5,6,7].…”

“…Then many researchers analyze GSM numerically as well as analytically [11][12][13][14]. Also by implementing the different methods numerical solution is obtained for GSM by numerous researchers such as Rasheed and Manaa [15] used finite difference and successive approximation method, Galerkin finite element method by Jiwari and Yadav [16], the combination of higher-order schemes were implemented by Owolabi et al [17], Onarcan et al [18] solved GSM numerically by using quintic trigonometric B-spline, Korkmaz et al [19] used differential quadrature along with implicit Rosenbrock of third-fourth order and exponential B-spline. Hale [20] attained explicit nontrivial still patterns in the unidimensional Gray-Scott Reaction-diffusion model for cubic autocatalysis.…”