Table of Content

20 March 2020, Volume 2 Issue 1

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A High Order Formula to Approximate the Caputo Fractional Derivative

R. Mokhtari, F. Mostajeran

2020, 2(1):  1-29.  doi:10.1007/s42967-019-00023-y

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We present here a high-order numerical formula for approximating the Caputo fractional derivative of order α for 0 < α < 1. This new formula is on the basis of the third degree Lagrange interpolating polynomial and may be used as a powerful tool in solving some kinds of fractional ordinary/partial diferential equations. In comparison with the previous formulae, the main superiority of the new formula is its order of accuracy which is 4-α, while the order of accuracy of the previous ones is less than 3. It must be pointed out that the proposed formula and other existing formulae have almost the same computational cost. The efectiveness and the applicability of the proposed formula are investigated by testing three distinct numerical examples. Moreover, an application of the new formula in solving some fractional partial diferential equations is presented by constructing a fnite diference scheme. A PDE-based image denoising approach is proposed to demonstrate the performance of the proposed scheme.

A Discontinuous Galerkin Method with Penalty for One-Dimensional Nonlocal Difusion Problems

Qiang Du, Lili Ju, Jianfang Lu, Xiaochuan Tian

2020, 2(1):  31-55.  doi:10.1007/s42967-019-00024-x

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There have been many theoretical studies and numerical investigations of nonlocal diffusion (ND) problems in recent years. In this paper, we propose and analyze a new discontinuous Galerkin method for solving one-dimensional steady-state and time-dependent ND problems, based on a formulation that directly penalizes the jumps across the element interfaces in the nonlocal sense. We show that the proposed discontinuous Galerkin scheme is stable and convergent. Moreover, the local limit of such DG scheme recovers classical DG scheme for the corresponding local difusion problem, which is a distinct feature of the new formulation and assures the asymptotic compatibility of the discretization. Numerical tests are also presented to demonstrate the efectiveness and the robustness of the proposed method.

Numerical Algorithm for the Time-Caputo and Space-Riesz Fractional Difusion Equation

Yuxin Zhang, Hengfei Ding

2020, 2(1):  57-72.  doi:10.1007/s42967-019-00032-x

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In this paper, we develop a novel fnite-diference scheme for the time-Caputo and spaceRiesz fractional difusion equation with convergence order O(τ_2-α + h²). The stability and convergence of the scheme are analyzed by mathematical induction. Moreover, some numerical results are provided to verify the efectiveness of the developed diference scheme.

Local Discontinuous Galerkin Scheme for Space Fractional Allen-Cahn Equation

Can Li, Shuming Liu

2020, 2(1):  73-91.  doi:10.1007/s42967-019-00034-9

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This paper is concerned with the efcient numerical solution for a space fractional Allen-Cahn (AC) equation. Based on the features of the fractional derivative, we design and analyze a semi-discrete local discontinuous Galerkin (LDG) scheme for the initial-boundary problem of the space fractional AC equation. We prove the optimal convergence rates of the semi-discrete LDG approximation for smooth solutions. Finally, we test the accuracy and efciency of the designed numerical scheme on a uniform grid by three examples. Numerical simulations show that the space fractional AC equation displays abundant dynamical behaviors.

Finite Element Convergence for State-Based Peridynamic Fracture Models

Prashant K. Jha, Robert Lipton

2020, 2(1):  93-128.  doi:10.1007/s42967-019-00039-4

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We establish the a priori convergence rate for fnite element approximations of a class of nonlocal nonlinear fracture models. We consider state-based peridynamic models where the force at a material point is due to both the strain between two points and the change in volume inside the domain of the nonlocal interaction. The pairwise interactions between points are mediated by a bond potential of multi-well type while multi-point interactions are associated with the volume change mediated by a hydrostatic strain potential. The hydrostatic potential can either be a quadratic function, delivering a linear force-strain relation, or a multi-well type that can be associated with the material degradation and cavitation. We frst show the well-posedness of the peridynamic formulation and that peridynamic evolutions exist in the Sobolev space H². We show that the fnite element approximations converge to the H² solutions uniformly as measured in the mean square norm. For linear continuous fnite elements, the convergence rate is shown to be C_tΔt + C_sh²∕ε², where ε is the size of the horizon, h is the mesh size, and Δ_t is the size of the time step. The constants C_t and C_s are independent of Δt and h and may depend on ε through the norm of the exact solution. We demonstrate the stability of the semi-discrete approximation. The stability of the fully discrete approximation is shown for the linearized peridynamic force. We present numerical simulations with the dynamic crack propagation that support the theoretical convergence rate.

A Finite-Diference Approximation for the One- and Two-Dimensional Tempered Fractional Laplacian

Yaoqiang Yan, Weihua Deng, Daxin Nie

2020, 2(1):  129-145.  doi:10.1007/s42967-019-00035-8

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This paper provides a fnite-diference discretization for the one- and two-dimensional tempered fractional Laplacian and solves the tempered fractional Poisson equation with homogeneous Dirichlet boundary conditions. The main ideas are to, respectively, use linear and quadratic interpolations to approximate the singularity and non-singularity of the one-dimensional tempered fractional Laplacian and bilinear and biquadratic interpolations to the two-dimensional tempered fractional Laplacian. Then, we give the truncation errors and prove the convergence. Numerical experiments verify the convergence rates of the order O(h^2-2s).

An Indirect Finite Element Method for Variable-Coefcient Space-Fractional Difusion Equations and Its Optimal-Order Error Estimates

Xiangcheng Zheng, V. J. Ervin, Hong Wang

2020, 2(1):  147-162.  doi:10.1007/s42967-019-00037-6

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We study an indirect fnite element approximation for two-sided space-fractional difusion equations in one space dimension. By the representation formula of the solutions u(x) to the proposed variable coefcient models in terms of v(x), the solutions to the constant coeffcient analogues, we apply fnite element methods for the constant coefcient fractional difusion equations to solve for the approximations v_h(x) to v(x) and then obtain the approximations u_h(x) of u(x) by plugging v_h(x) into the representation of u(x). Optimal-order convergence estimates of u(x)-u_h(x) are proved in both L² and H^α∕2 norms. Several numerical experiments are presented to demonstrate the sharpness of the derived error estimates.