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We propose a novel method to compute a finite difference stencil for Riesz derivative for artibitrary speed of convergence. This method is based on applying a pre-filter to the Grunwald-Letnikov type central difference stencil. The filter is obtained by solving for the inverse of a symmetric Vandemonde matrix and exploiting the relationship between the Taylors series coefficients and fast Fourier transform. The filter costs Oleft(N^{2}right) operations to evaluate for Oleft(h^{N}right) of convergence, where h is the sampling distance. The higher convergence speed should more than offset the overhead with the requirement of the number of nodal points for a desired error tolerance significantly reduced. The benefit of progressive generation of the stencil coefficients for adaptive grid size for dynamic problems with the Grunwald-Letnikov type difference scheme is also kept because of the application of filtering. The higher convergence rate is verified through numerical experiments.
We make the split of the integral fractional Laplacian as $(-Delta)^s u=(-Delta)(-Delta)^{s-1}u$, where $sin(0,frac{1}{2})cup(frac{1}{2},1)$. Based on this splitting, we respectively discretize the one- and two-dimensional integral fractional Laplaci
In this work, new finite difference schemes are presented for dealing with the upper-convected time derivative in the context of the generalized Lie derivative. The upper-convected time derivative, which is usually encountered in the constitutive equ
The present article is devoting a numerical approach for solving a fractional partial differential equation (FPDE) arising from electromagnetic waves in dielectric media (EMWDM). The truncated Bernoulli and Hermite wavelets series with unknown coeffi
The aim of this paper is to develop fast second-order accurate difference schemes for solving one- and two-dimensional time distributed-order and Riesz space fractional diffusion equations. We adopt the same measures for one- and two-dimensional prob
In this article, a numerical scheme is introduced for solving the fractional partial differential equation (FPDE) arising from electromagnetic waves in dielectric media (EMWDM) by using an efficient class of finite difference methods. The numerical s