Composite generalized Laguerre spectral method for nonlinear Fokker–Planck equations on the whole line

Abstract

In this paper, we propose a composite Laguerre spectral method for the nonlinear Fokker-Planck equations modelling the relaxation of fermion and boson gases. A composite Laguerre spectral scheme is constructed. Its convergence is proved. Numerical results show the efficiency of this approach and coincide well with theoretical analysis. Some results on the Laguerre approximation and techniques used in this paper are also applicable to other nonlinear problems on the whole line.

Multi-Domain Decomposition Pseudospectral Method for Nonlinear Fokker–Planck Equations

“…In the end of this section, we introduce some inverse inequalities which will be used in the sequel (cf. [12,37]).…”

“…We can use the composite Laguerre spectral method with mode N to solve problem (1), see Wang [37]. Let r > 1 , for any 0 ≤ t ≤ T , we have that (cf.…”

“…For cases of certain nonlinear problems defined on unbounded domains; see [6,10,11,17,18,23,31,34,36,40,43,44]. Wang [37] considered spectral methods for the nonlinear Fokker-Planck equation defined on the whole line using the scaled generalized Laguerre functions coupled with the domain decomposition. The scaling parameter offers great flexibility to match the asymptotic behaviors of exact solutions for differential equations of degenerate type at |v| → ∞ .…”

“…In this paper, we consider the Fokker-Planck equations modeling the relaxation of fermion and boson gases (cf. [37]):…”

Analytical solution of stochastic differential equation by multilayer perceptron neural network approximation of Fokker–Planck equation

“…They can be used to solve a wide variety of phenomena such as sound, heat, fluid dynamics [5][6][7]. Much research has been devoted to find the numerical solution of FPE, different standard methods for solving PDEs such as finite difference, finite volume, finite element, and spectral methods have been employed to solve FPE [8][9][10][11][12][13][14][15][16][17]. In a different approach, the solution of the FPE was approximated by RBF networks [18].…”