The Fokker-Planck-Kolmogorov (FPK) equation governs the probability density function (p.d.f.) of the dynamic response of a particular class of linear or nonlinear system to random excitation. An interval wavelet numerical method (IWNM) for nonlinear random systems is proposed using interval Shannon-Gabor wavelet interpolation operator. An FPK equation for nonlinear oscillators and a time fractional Fokker-Planck equation are taken as examples to illustrate its effectiveness and efficiency. Compared with the common wavelet collocation methods, IWNM can decrease the boundary effect greatly. Compared with the finite difference method for the time fractional Fokker-Planck equation, IWNM can improve the calculation precision evidently.

The Fokker-Planck equation describes the time evolution of the probability density function of the velocity of a particle, and can be generalized to other observables as well. It is named after Adriaan Fokker and Max Planck and is also known as the Kolmogorov forward equation (diffusion), named after Andrey Kolmogorov, who first introduced it in a 1931 paper. When applied to particle position distributions, it is better known as the Smoluchowski equation. The case with zero diffusion is known in statistical mechanics as Liouville equation. In order to describe subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a Boltzmann thermal heat bath, Metzler et al. introduced a fractional Fokker-Planck equation (FFPE) which was shown to obey generalized Einstein relations, and its stationary solution is the Boltzmann distribution [

Many methods for calculating nonlinear random response have been developed by numerous scholars over a long period of time. One type of these methods is the diffuseness process theory method, and the primarily one is Fokker-Planck equation method. In practice, the most difficult problem of using Fokker-Planck equation method is how to solve the equation. For general nonlinear system, it is very difficult to obtain the exact solution. Various numerical methods have been used to solve the Fokker-Planck equation, such as the weighted residual method [

Recently the wavelet analysis is getting high attention to many authors for the nonlinear dynamic system, not only for the signal analysis but also for developing new numerical methods for calculating the partial difference equations. Bertoluzza and Naldi presented a wavelet collocation method for solving the partial differential equations [^{2}-function, based on Shannon wavelet functions, is given [

Mei proposed a general construction method of the interval wavelet based on the general variational principle [

Consider a one dimension function

The two dimension basic function can be expressed by tensor product of the one dimension basic function. In two dimension scaling function space, to each

It is easy to know that the Shannon-Gabor interval wavelet is a linear combination of the Shannon-Gabor scaling function

It should be noted that the construction method of the interval wavelet function reveals the close relationship between the restricted variational principle and the interval interpolation wavelet.

Consider the following two-dimension stationary Fokker-Planck equation:

The unknown quantity

In general,

In terms of the theory of collocation method, substituting (

Using (

As the basic function is the Shannon scaling function which possess explicit expression, so it is easy to deduce the first and second derivative of the basic function

Instituting (

In the system of (

In this section the method was applied to the random vibration of a Duffing oscillator, the equation of motion for which is [

For the case

The numerical results

The comparison between numerical solution and exact solution.

The comparison between the wavelet and interval wavelet collocation method for FPK equation is shown in Figure

Error comparison between the wavelet and the interval wavelet numerical methods for FPK equation.

Shannon-Gabor wavelet numerical method (the maximum of the absolute error is

Interval Shannon-Gabor wavelet numerical method (the maximum of the absolute error is

Time fractional Fokker-Planck equations have been recently treated by a number of authors and are found to be a useful approach for the description of transport dynamics in complex systems that are governed by anomalous diffusion and nonexponential relaxation patterns. Fractional derivatives play a key role in modeling particle transport in anomalous diffusion. Wei [

Comparison between the finite difference method and the wavelet numerical method in solving the time fractional FPK equation (

Finite difference method

Wavelet numerical method

Compared with the finite difference method, the Shannon-Gabor wavelet numerical method possesses more excellent numerical properties in solving the FPK equations, which has been illustrated in solving the time-fractional FPK equations. Based on the interpolation properties of the Shannon-Gabor wavelet, the interval interpolation wavelet collocation method based on the wavelet interpolation technique has been developed to solve the two dimension Fokker-Planck equation in a finite domain in this paper. This new method can decrease the boundary effect evidently and then decrease the numerical error in the whole in the definition domain greatly. We believe that the method can be easily generalized to rectangular higher dimension case. The comparisons with other numerical algorithms show that the method is competitive and efficient. Furthermore, it should be noted that the method can also be used to solve general partial differential equation.