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

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.

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

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691318500212 ◽

2018 ◽

Vol 16 (03) ◽

pp. 1850021 ◽

Cited By ~ 4

Author(s):

Shuli Mei ◽

Wanlin Gao

Keyword(s):

Spectral Method ◽

Planck Equation ◽

Partition Of Unity ◽

Equation System ◽

Wavelet Function ◽

Gabor Wavelet ◽

Trigonometric Polynomials ◽

Sinc Function ◽

Shannon Wavelet ◽

Fokker Planck

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.

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A unified Petrov–Galerkin spectral method for fractional PDEs

Computer Methods in Applied Mechanics and Engineering ◽

10.1016/j.cma.2014.10.051 ◽

2015 ◽

Vol 283 ◽

pp. 1545-1569 ◽

Cited By ~ 67

Author(s):

Mohsen Zayernouri ◽

Mark Ainsworth ◽

George Em Karniadakis

Keyword(s):

Spectral Method ◽

Fractional Pdes ◽

Galerkin Spectral Method

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Pseudo Spectral Method Based on Symmetric Extension

2018 7th European Workshop on Visual Information Processing (EUVIP) ◽

10.1109/euvip.2018.8611666 ◽

2018 ◽

Cited By ~ 1

Author(s):

Izumi Ito

Keyword(s):

Spectral Method ◽

Symmetric Extension ◽

Pseudo Spectral Method ◽

Pseudo Spectral

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Arbitrarily Oriented Phase Randomization of Design Ground Motions by Continuous Wavelets

Infrastructures ◽

10.3390/infrastructures6100144 ◽

2021 ◽

Vol 6 (10) ◽

pp. 144

Author(s):

Haoyu Xie ◽

Riki Honda

Keyword(s):

Wavelet Transform ◽

Dynamic Analysis ◽

Frequency Domain ◽

Ground Motions ◽

Gabor Wavelet ◽

Wavelet Basis ◽

Time Frequency ◽

Continuous Wavelets ◽

Shannon Wavelet ◽

Design Selection

For dynamic analysis in seismic design, selection of input ground motions is of huge importance. In the presented scheme, complex Continuous Wavelet Transform (CWT) is utilized to simulate stochastic ground motions from historical records of earthquakes with phase disturbance arbitrarily localized in time-frequency domain. The complex arguments of wavelet coefficients are determined as phase spectrum and an innovative formulation is constructed to improve computational efficiency of inverse wavelet transform with a pair of random complex arguments introduced and make more candidate wavelets available in the article. The proposed methodology is evaluated by numerical simulations on a two-degree-of-freedom system including spectral analysis and dynamic analysis with Shannon wavelet basis and Gabor wavelet basis. The result shows that the presented scheme enables time-frequency range of disturbance in time-frequency domain arbitrarily oriented and complex Shannon wavelet basis is verified as the optimal candidate mother wavelet for the procedure in case of frequency information maintenance with phase perturbation.

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The Coupling of Pseudo-Spectral Method and Domain Decomposition Method for the Elastostatic Problem

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.580-583.3071 ◽

2014 ◽

Vol 580-583 ◽

pp. 3071-3074

Author(s):

Rui Ding ◽

Fu Jun Chen ◽

Quan Shen ◽

Ling Liu

Keyword(s):

Domain Decomposition ◽

Spectral Method ◽

Decomposition Method ◽

Original Problem ◽

Domain Decomposition Method ◽

Partition Of Unity ◽

High Accuracy ◽

Pseudo Spectral Method ◽

Pseudo Spectral ◽

Elastostatic Problem

It presents the coupling of Pseudo-spectral method (PS) and domain decomposition method (DDM) for the elastostatic problem. First, the original problem is decomposed into several sub-problems by DDM. Next combining the advantage of easy programming and high accuracy of Pseudo-spectral method, we can solve these sub-problems in parallel by PS. Finally the global numerical solution is obtained by the partition of unity approximation. Some numerical experiments illustrate the effectiveness and accuracy of our method.

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Linear Visco-Resistive Computations of Magnetohydrodynamic Waves: I. The Code and Test Cases

International Astronomical Union Colloquium ◽

10.1017/s0252921100025926 ◽

1994 ◽

Vol 144 ◽

pp. 503-505

Author(s):

R. Erdélyi ◽

M. Goossens ◽

S. Poedts

Keyword(s):

Resonant Absorption ◽

Numerical Code ◽

Mhd Waves ◽

Test Cases ◽

Numerical Accuracy ◽

Element Discretization ◽

Flux Tubes ◽

Magnetohydrodynamic Waves ◽

Viscosity Coefficients ◽

Magnetic Flux Tubes

AbstractThe stationary state of resonant absorption of linear, MHD waves in cylindrical magnetic flux tubes is studied in viscous, compressible MHD with a numerical code using finite element discretization. The full viscosity tensor with the five viscosity coefficients as given by Braginskii is included in the analysis. Our computations reproduce the absorption rates obtained by Lou in scalar viscous MHD and Goossens and Poedts in resistive MHD, which guarantee the numerical accuracy of the tensorial viscous MHD code.

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