We present here a high-order numerical formula for approximating the Caputo fractional derivative of order $\alpha$ for $0<\alpha<$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 differential equations. In comparison with the previous formulae, the main superiority of the new formula is its order of accuracy which is $4-\alpha ,$ 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 effectiveness 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 differential equations is presented by constructing a finite difference scheme. A PDE-based image denoising approach is proposed to demonstrate the performance of the proposed scheme.
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 diffusion 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 effectiveness and the robustness of the proposed method.
In this paper, we develop a novel finite-difference scheme for the time-Caputo and space-Riesz fractional diffusion equation with convergence order $\mathcal {O}(\tau ^{2-\alpha } + h^2)$. The stability and convergence of the scheme are analyzed by mathematical induction. Moreover, some numerical results are provided to verify the effectiveness of the developed difference scheme.
This paper is concerned with the efficient 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 efficiency 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.
We establish the a priori convergence rate for finite 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 first show the well-posedness of the peridynamic formulation and that peridynamic evolutions exist in the Sobolev space $H^2$. We show that the finite element approximations converge to the $H^2$ solutions uniformly as measured in the mean square norm. For linear continuous finite elements, the convergence rate is shown to be $C_t \Delta t + C_s h^2/\epsilon ^2$, where $\epsilon $ is the size of the horizon, h is the mesh size, and $\Delta t$ is the size of the time step. The constants $C_t$ and $C_s$ are independent of $\Delta t$ and h and may depend on $\epsilon $ 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.
This paper provides a finite-difference 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 $O(h^{2-2s})$.
We study an indirect finite element approximation for two-sided space-fractional diffusion equations in one space dimension. By the representation formula of the solutions u(x) to the proposed variable coefficient models in terms of v(x), the solutions to the constant coefficient analogues, we apply finite element methods for the constant coefficient fractional diffusion 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 $L^2$ and $H^{\alpha /2}$ norms. Several numerical experiments are presented to demonstrate the sharpness of the derived error estimates.