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In this work, we study the numerical simulation of the one‐dimensional reaction‐diffusion system known as the Gray‐Scott model. This model is responsible for the spatial pattern formation, which we often meet in nature as the result of some chemical reactions. We have used the trigonometric quartic B‐spline (T4B) functions for space discretization with the Crank‐Nicolson method for time integration to integrate the nonlinear reaction‐diffusion equation into a system of algebraic equations. The solutions of the Gray‐Scott model are presented with different wave simulations. Test problems are chosen from the literature to illustrate the stationary waves, pulse‐splitting waves, and self‐replicating waves.

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.

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