A meshless technique based on the local radial basis functions collocation method for solving parabolic–parabolic Patlak–Keller–Segel chemotaxis model

Abstract

In this paper local radial basis functions (LRBFs) collocation method is proposed for solving the (Patlak-) Keller–Segel model. We use the Crank–Nicolson difference scheme for the time derivative to obtain a finite difference scheme with respect to the time variable for the Keller–Segel model. Then we use the local radial basis functions (LRBFs) collocation method to approximate the spatial derivative. We obtain the numerical results for the mentioned model. As we know, recently some approaches presented for preventing the blow up of the cell density. In the current paper we use the multiquadric (MQ) radial basis function. The aim of this paper is to show that the meshless methods based on the local RBFs collocation approach are also suitable for solving models that have the blow up of the cell density. Also, six test problems are given that show the acceptable accuracy and efficiency of the proposed schemes.