Interval Shannon Wavelet Collocation Method for Fractional Fokker-Planck Equation

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).

Construction of Interval Shannon Wavelet and Its Application in Solving Nonlinear Black-Scholes Equation

“…In addition, with the increasing of the parameters and , the errors of both Yan and Mei's method become larger and larger evidently. This can be explained by the condition number theory [23]. On the contrary, the interval wavelet method proposed in this paper is not sensitive to the parameters and , and the precision is the best among all above methods.…”

“…In theory, the increase of can improve the approximate precision. But it was pointed out in [23] that the increase of can also bring the increase of the condition number of the problem to be solved, and this can decrease the numerical precision greatly. In addition, the larger can also bring more calculation.…”

“…The performance of Mei's interval wavelet [4] relates to the external interpolation points ( ), and the choice scheme of was not given in [3]. It was pointed out in [23] that the bigger usually introduces bigger condition number which can decrease the precision greatly. This can be illustrated in Figures 1(d) and 1(e).…”

Coupling Technique of Haar Wavelet Transform and Variational Iteration Method for a Nonlinear Option Pricing Model

“…This relates to the condition number of the matrix from the finite-difference method [30,31]. It is no doubt that the choice of L can change the condition number of the system of algebraic equations discretized by the wavelet interpolation operator or the finite-difference method [32]. Therefore, the choice of L should take the condition number into account.…”

Shannon Wavelet Precision Integration Method for Pathologic Onion Image Segmentation Based on Homotopy Perturbation Technology

“…The approximation is compared with the wavelet reconstruction formula and the error of approximation is explicitly computed [13,14]. Furthermore, Shannon wavelet has been used to solve the fractional calculus problems in the recent years [15][16][17]. A perceived disadvantage of the Shannon scaling function is that it tends to zero quite slowly as | | → ∞.…”