A practical method based on distributed approximating functionals ͑DAFs͒ is proposed for numerically solving a general class of nonlinear time-dependent Fokker-Planck equations. The method relies on a numerical scheme that couples the usual path-integral concept to the DAF idea. The high accuracy and reliability of the method are illustrated by applying it to an exactly solvable nonlinear Fokker-Planck equation, and the method is compared with the accurate K-point Stirling interpolation formula finite-difference method. The approach is also used successfully to solve a nonlinear self-consistent dynamic mean-field problem for which both the cumulant expansion and scaling theory have been found by Drozdov and Morillo ͓Phys. Rev. E 54, 931 ͑1996͔͒ to be inadequate to describe the occurrence of a long-lived transient bimodality. The standard interpretation of the transient bimodality in terms of the ''flat'' region in the kinetic potential fails for the present case. An alternative analysis based on the effective potential of the Schrödinger-like Fokker-Planck equation is suggested. Our analysis of the transient bimodality is strongly supported by two examples that are numerically much more challenging than other examples that have been previously reported for this problem.

Lagrange distributed approximating functionals (LDAFs) are proposed as the basis for a new, collocation-type method for accurately approximating functions and their derivatives both on and off discrete grids. Example applications are presented to illustrate the use of LDAFs for solving the Schrödinger equation and Fokker-Planck equation. LDAFs are constructed by combining the DAF concept with the Lagrange interpolation scheme. [S0031-9007(97)03702-2]

In this paper, we present a class of distributed approximating functionals ͑DAF's͒ for solving various problems in the sciences and engineering. Previous DAF's were specifically constructed to avoid interpolation in order to achieve the ''well-tempered'' limit, in which the same order error is made both on and off the grid points. These DAF's are constructed by combining the DAF concept with various interpolation schemes. The approach then becomes the same as the ''moving least squares'' method, but the specific ''interpolating DAF's'' obtained are new, to our knowledge. These interpolating DAF's are illustrated using Lagrange interpolation ͑the ''LDAF''͒ and a Gaussian weight function. Four numerical tests are used to illustrate the LDAF's: differentiation on and off a grid, fitting a function off a grid, time-dependent quantum dynamical evolution, and solving nonlinear Burgers' equation. ͓S1063-651X͑98͒10204-0͔

The distributed approximating functional method is applied to the solution of the Fokker–Planck equations. The present approach is limited to the standard eigenfunction expansion method. Three typical examples, a Lorentz Fokker–Planck equation, a bistable diffusion model and a Henon–Heiles two-dimensional anharmonic resonating system, are considered in the present numerical testing. All results are in excellent agreement with those of established methods in the field. It is found that the distributed approximating functional method yields the accuracy of a spectral method but with a local method’s simplicity and flexibility for the eigenvalue problems arising from the Fokker–Planck equations.

The Fokker-Planck equation is solved by the method of distributed approximating functionals via forward time propagation. Numerical schemes involving higher-order terms in ⌬t are discussed for the time discretization. Three typical examples ͑a Wiener process, an Ornstein-Uhlenbeck process, and a bistable diffusion model͒ are used to test the accuracy and reliability of the present approach, which provides solutions that are accurate up to ten significant figures while using a small number of grid points and a reasonably large time increment. Two sets of solutions for the bistable system, one computed using the eigenfunction expansion of a preceding paper and the other using the present time-dependent treatment, agree to no fewer than five significant figures. It is found that the distributed approximating functional method, while simple in its implementation, yields the most accurate numerical solutions yet available for the Fokker-Planck equation.