Abstract
Metzler et al. introduced a fractional Fokker-Planck equation (FFPE) describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a Boltzmann thermal heat bath. In this paper, we present an interval Shannon wavelet numerical method for the FFPE. In this method, a new concept named “dynamic interval wavelet” is proposed to solve the problem that the numerical solution of the fractional PDE is usually sensitive to boundary conditions. Comparing with the traditional wavelet defined in the interval, the Newton interpolator is employed instead of the Lagrange interpolation operator, so, the extrapolation points in the interval wavelet can be chosen dynamically to restrict the boundary effect without increase of the calculation amount. In order to avoid unlimited increasing of the extrapolation points, both the error tolerance and the condition number are taken as indicators for the dynamic choice of the extrapolation points. Then, combining with the finite difference technology, a new numerical method for the time fractional partial differential equation is constructed. A simple Fokker-Planck equation is taken as an example to illustrate the effectiveness by comparing with the Grunwald-Letnikov central difference approximation (GL-CDA).
Highlights
Due to the fact that 1/f signal gains the increasing interests in the field of biomedical signal processing and engineering systems [1], the differential equations of fractional order appear more and more frequently in various research areas and engineering applications [2, 3]
We present an interval Shannon wavelet numerical method for the fractional Fokker-Planck equation (FFPE)
Fractional Fokker-Planck equation is a typical fractional PDE, which is often used to describe a subdiffusive behavior of a particle under the combined influence of external nonlinear force field, and a Boltzmann thermal heat bath
Summary
Due to the fact that 1/f signal gains the increasing interests in the field of biomedical signal processing and engineering systems [1], the differential equations of fractional order appear more and more frequently in various research areas and engineering applications [2, 3]. Li [9] proposed an analytical method taking the fractal time series as the solution to a differential equation of fractional order or a response of a fractional system or a fractional filter driven with a white noise in the domain of stochastic processes and gave the exact solution of impulse response to a class of fractional oscillators [10] According to this idea, Li and his coresearchers solved many problems in science and technology [11,12,13,14]. A dynamic interval Shannon wavelet collocation method for the fractional FPDs is proposed In this method, the relation between the parameter L and the wavelet approximation error was discussed based on the interpolation error theory, and an adaptive choice procedure on L was constructed. Based on the Grunwald-Letnikov definition of the fractional order derivative, we construct a Shannon wavelet numerical method for the fraction Fokker-Planck equation
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