Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) Collocation Method for Solving Linear and Nonlinear Fokker-Planck Equations

• Published in: Communications in theoretical physics Vol. 69; no. 5; pp. 519 - 531
• Main Authors: Parand, K., Latifi, S., Moayeri, M. M., Delkhosh, M.
• Format: Journal Article
• Language: English
• Published: Chinese Physical Society and IOP Publishing Ltd 01.05.2018
• Subjects: Crank-Nicolson technique; Fokker-Planck equations; Generalized Lagrange functions; Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) collocation
• ISSN: 0253-6102
• DOI: 10.1088/0253-6102/69/5/519

In this study, we have constructed a new numerical approach for solving the time-dependent linear and nonlinear Fokker-Planck equations. In fact, we have discretized the time variable with Crank-Nicolson method and for the space variable, a numerical method based on Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) collocation method is applied. It leads to in solving the equation in a series of time steps and at each time step, the problem is reduced to a problem consisting of a system of algebraic equations that greatly simplifies the problem. One can observe that the proposed method is simple and accurate. Indeed, one of its merits is that it is derivative-free and by proposing a formula for derivative matrices, the difficulty aroused in calculation is overcome, along with that it does not need to calculate the General Lagrange basis and matrices; they have Kronecker property. Linear and nonlinear Fokker-Planck equations are given as examples and the results amply demonstrate that the presented method is very valid, effective, reliable and does not require any restrictive assumptions for nonlinear terms.