Meshless LocalWeak form Method Based on a Combined Basis Function for Numerical Investigation of Brusselator Model and Spike Dynamics in the Gierer-Meinhardt System

In this paper, at first, a new combined shape function is proposed. Then, based on this shape function, the meshless local weak form method is applied to find the numerical solution of time-dependent non-linear Brusselator and Gierer- Meinhardt systems. The combined shape function inherits the properties of radial point interpolation (RPI), moving least squares (MLS) and moving Kriging (MK) shape functions and is controlled by control parameters, which take different values in the domain [0;1]. The combined shape function provides synchronic use of different shape functions and this leads to more flexibility in the used method. The main aim of this paper is to show that the combined basis function can be used as a shape function in meshless local weak form methods and leads to better results in solving the system of non-linear partial differential equations especially Brusselator and Gierer-Meinhardt systems. The numerical results confirm the good efficiency of the proposed method for solving non-linear Brusselator and Gierer-Meinhardt systems.