Shannon–Cosine wavelet spectral method for solving fractional Fokker–Planck equations

Shuli Mei,Wanlin Gao

doi:10.1142/s0219691318500212

Abstract

The windowed-Shannon wavelet is not recommended generally as the window function will destroy the partition of unity of Shannon mother wavelet. A novel windowing scheme is proposed to overcome the shortcoming of the general windowed-Shannon function, and then, a novel and efficient Shannon–Cosine wavelet spectral method is provided for solving the fractional PDEs. Taking full advantage of the waveform of sinc function to hold the partition of unity, Shannon–Cosine wavelet is constructed, which is composed of Shannon wavelet and the trigonometric polynomials. It was proved that the proposed wavelet function meets the requirements of being a trial function and possesses many other excellent properties such as normalization, interpolation, two-scale relations, compact support domain, and so on. Therefore, it is a real wavelet function instead of a general Shannon–Gabor wavelet which is a kind of quasi-wavelet. Next, by means of the Shannon–Cosine wavelet collocation method, the corresponding algebraic equation system of the fractional Fokker–Planck equation can be obtained. Approximate solutions of the fractional Fokker–Plank equations are compared with the exact solutions. These calculations illustrate that the accuracy of the Shannon–Cosine wavelet collocation solutions is quite high even using a small number of grid points.

Highlights

In recent years, fractional calculus has been rediscovered by scientists and engineers and applied in an increasing number of fields, namely, in the areas of electromagnetism, control engineering, and signal processing.[7,12,14] Besides, using the Caputo derivative in the Ortigueira sense,the fractional derivative of the Riemann zeta function has been computed recently,[3] which plays an important role both in number theory and in several applications of quantum mechanics
The use of windowed or truncated sinc functions is not recommended because these fail to satisfy the partition of unity; this has the disturbing consequence that the reconstruction error will not vanish as the sampling step tends to zero
Encouraged by the achievements on the wavelet spectral method in recent years, we develop a Shannon–Cosine wavelet by introducing the trigonometric polynomials to improve the performance of Shannon wavelet

Summary

Introduction

Fractional calculus has been rediscovered by scientists and engineers and applied in an increasing number of fields, namely, in the areas of electromagnetism, control engineering, and signal processing.[7,12,14] Besides, using the Caputo derivative in the Ortigueira sense,the fractional derivative of the Riemann zeta function has been computed recently,[3] which plays an important role both in number theory and in several applications of quantum mechanics. Shannon–Cosine wavelet has the compact support domain and holds all the excellent properties of Shannon wavelet,[1,17] especially the partition of unity. It breaks the curse of the windowed sinc function. This provides a series of Shannon– Cosine wavelets with different support domain and smoothness

Construction of Shannon–Cosine Wavelet

Choice of the parameter

Properties of Shannon–Cosine Scaling Function

Approximation Error Analysis and Choice of Parameter N

Shannon–Cosine Spectral Method on Fractional Fokker–Planck Equations

Conclusions