The numerical solution of Fokker–Planck equation with radial basis functions (RBFs) based on the meshless technique of Kansa׳s approach and Galerkin method

This paper describes two numerical methods based on radial basis functions (RBFs) for solving the time-dependent linear and nonlinear Fokker–Planck equations in two dimensions. These methods (RBFs) give a closed form for approximating the solution of partial differential equations. We approximate the linear and nonlinear Fokker–Planck equations with radial basis functions which are based on two techniques, one of them is Kansa׳s approach and another technique is the Galerkin method of Tau type [54]. In this work, we discretize the time variable with Crank–Nicolson method. For the space variable, we apply the radial basis functions which are Multiquadrics (MQ) and Inverse Quadric (IQ). Also, we employ another radial basis function which was introduced in [35]. These basis functions depend on constant (shape) parameter. As is well known, the shape parameter has a strong influence on the accuracy of the numerical solutions and thus we test and compare several different strategies to choose this parameter. Both techniques (Kansa׳s approach and Tau method) yield a linear system of algebraic equations say AX=b. The matrix A is usually very ill-conditioned. We apply QR decomposition technique for solving the linear system arising from our approximations. Finally, some test problems are presented to illustrate the efficiency of the new methods for the numerical solution of linear and nonlinear Fokker–Planck equations. Also, to show the good accuracy of the method of radial basis functions, we compute the errors using L∞L∞, root mean square (RMS) and L2 norms.