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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 Laplacian with the inhomogeneous Dirichlet boundary condition and give the corresponding truncation errors with the help of the interpolation estimate. Moreover, the suitable corrections are proposed to guarantee the convergence in solving the inhomogeneous fractional Dirichlet problem and an $mathcal{O}(h^{1+alpha-2s})$ convergence rate is obtained when the solution $uin C^{1,alpha}(bar{Omega}^{delta}_{n})$, where $n$ is the dimension of the space, $alphain(max(0,2s-1),1]$, $delta$ is a fixed positive constant, and $h$ denotes mesh size. Finally, the performed numerical experiments confirm the theoretical results.
We study finite element approximations of the nonhomogeneous Dirichlet problem for the fractional Laplacian. Our approach is based on weak imposition of the Dirichlet condition and incorporating a nonlocal analogous of the normal derivative as a Lagr
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
Fourth-order differential equations play an important role in many applications in science and engineering. In this paper, we present a three-field mixed finite-element formulation for fourth-order problems, with a focus on the effective treatment of
This research was mainly conducted to explore the possibility of formulating an efficient algorithm to find roots of nonlinear equations without using the derivative of the function. The Weerakoon-Fernando method had been taken as the base in this pr
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