NUMERICAL SOLUTIONS OF REACTION-DIFFUSION EQUATION SYSTEMS WITH TRIGONOMETRIC QUINTIC B-SPLINE COLLOCATION ALGORITHM

Abstract

In this study, trigonometric quintic B-spline collocation method is constructed for computing numerical solutions of the reaction-diffusion system (RDS). Schnakenberg, Gray-Scott and Brusselator models are special cases of reaction-diffusion systems considered as examples in this paper. Crank-Nicolson formulae is used for the time discretization of the generalized RDS and the nonlinear terms in time-discretized form of RDS are linearized using the Taylor expansion. The fully integration of the generalized system is carried out using the collocation method based on the trigonometric quintic B-splines. The method is tested on different problems to illustrate the accuracy. The error norms are calculated for the linear problem whereas the relative error is given for nonlinear problems. Both simple and easy B-spline algorithms are illustrated to give the solutions of RDS and also the graphical representation of the efficient solutions are presented for the nonlinear RDSs. Combination of the quintic B-splines and the collocation method is shown to present numerical solutions of the RDS successfully. With the presented method, it is possible to get approximate solutions as well as their derivatives up to an order of four on the problem domain.