Point-Symmetric Extension-Based Interval Shannon-Cosine Spectral Method for Fractional PDEs

Qing Wang,Yanqiao Li,Ruyi Xing,Yangyang Wu,Shu-Li Mei

doi:10.1155/2020/4565036

Abstract

The approximation accuracy of the wavelet spectral method for the fractional PDEs is sensitive to the order of the fractional derivative and the boundary condition of the PDEs. In order to overcome the shortcoming, an interval Shannon-Cosine wavelet based on the point-symmetric extension is constructed, and the corresponding spectral method on the fractional PDEs is proposed. In the research, a power function of cosine function is introduced to modulate Shannon function, which takes full advantage of the waveform of the Shannon function to ensure that many excellent properties can be satisfied such as the partition of unity, smoothness, and compact support. And the interpolative property of Shannon wavelet is held at the same time. Then, based on the point-symmetric extension and the general variational theory, an interval Shannon-Cosine wavelet is constructed. It is proved that the first derivative of the approximated function with this interval wavelet function is continuous. At last, the wavelet spectral method for the fractional PDEs is given by means of the interval Shannon-Cosine wavelet. By means of it, the condition number of the discrete matrix can be suppressed effectively. Compared with Shannon and Shannon-Gabor wavelet quasi-spectral methods, the novel scheme has stronger applicability to the shockwave appeared in the solution besides the higher numerical accuracy and efficiency.

Highlights

In recent years, fractional calculus has been attracting more and more researchers in different fields of science and engineering and has been theoretically developed quickly over the last two decades [1,2,3,4]
It has been proved that the fractional-order differential equation models are more consistent with the biological phenomena [5] and hydrodynamics [6,7,8] than those of integer-orders. e Caputo and Riemann–Liouville fractional derivatives are the classical definition, and both of them have a kernel with singularity
Despite a few special fraction PDEs having analytical solution [15], most of them should be solved by the numerical method. e solution of the fractional PDEs is sensitive to the iterative step, and so it is disabled to be solved by the traditional numerical method directly

Summary

Introduction

Fractional calculus has been attracting more and more researchers in different fields of science and engineering and has been theoretically developed quickly over the last two decades [1,2,3,4]. Shannon wavelets have been constructed based on the sinc function. Where σ is the width parameter (or called window size) Both of the two Shannon-type wavelets are obtained by taking the Gaussian window to modulate the sinc function. Erefore, it is necessary to construct a novel window for sinc function, which can satisfy the partition of unity, so that it can be utilized to solve fractional PDEs efficiently [32]. A simplified Shannon-Cosine wavelet function is proposed and the corresponding interval wavelet is constructed based on the point-symmetric extension [34, 35]. The interval Shannon-Cosine wavelet is employed to construct a wavelet spectral method for the fractional PDEs

Simplified Shannon-Cosine Scaling Function

Normalization and Choice of the Support Domain Parameter N

Point-Symmetric Interval Wavelet

Application in Solving Fractional PDEs

Conclusions