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Point-Symmetric Extension-Based Interval Shannon-Cosine Spectral Method for Fractional PDEs
Discrete Dynamics in Nature and Society ( IF 1.4 ) Pub Date : 2020-06-02 , DOI: 10.1155/2020/4565036
Ruyi Xing 1 , Yanqiao Li 2 , Qing Wang 2 , Yangyang Wu 2 , Shu-Li Mei 2
Affiliation  

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.

中文翻译:

基于点对称扩展的区间香农-余弦谱方法求解分数阶偏微分方程

小波谱方法对分数PDE的近似精度对分数导数的阶数和PDE的边界条件敏感。为了克服这一缺点,构造了一种基于点对称扩展的区间香农-余弦小波,并提出了相应的分数PDE谱方法。在研究中,引入余弦函数的幂函数来调制香农函数,充分利用了香农的波形可以确保满足许多优异性能的功能,例如统一的分隔,光滑度和紧凑的支撑。并同时保持了香农小波的插值性质。然后,基于点对称扩展和一般变分理论,构造了区间香农-余弦小波。证明了具有该区间小波函数的近似函数的一阶导数是连续的。最后,利用区间香农-余弦小波给出了分数阶偏微分方程的小波谱方法。借助于此,可以有效地抑制离散矩阵的条件数。与Shannon和Shannon-Gabor小波拟谱方法相比,
更新日期:2020-06-02
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