We analyze the well-posedness of an anisotropic, nonlocal diffusion equation. Establishing an equivalence between weighted and unweighted anisotropic nonlocal diffusion operators in the vein of unified nonlocal vector calculus, we apply our analysis to a class of fractional-order operators and present rigorous estimates for the solution of the corresponding anisotropic anomalous diffusion equation. Furthermore, we extend our analysis to the anisotropic diffusion-advection equation and prove well-posedness for fractional orders $s ∈ [0.5, 1).$ We also present an application of the advection-diffusion equation to anomalous transport of solutes.

We develop an efficient and accurate spectral deferred correction (SDC) method for fractional differential equations (FDEs) by extending the algorithm in [14] for classical ordinary differential equations (ODEs). Specifically, we discretize the resulted Picard integral equation by the SDC method and accelerate the convergence of the SDC iteration by using the generalized minimal residual algorithm (GMRES). We first derive the correction matrix of the SDC method for FDEs and analyze the convergence region of the SDC method. We then present several numerical examples for stiff and non-stiff FDEs including fractional linear and nonlinear ODEs as well as fractional phase field models, demonstrating that the accelerated SDC method is much more efficient than the original SDC method, especially for stiff problems. Furthermore, we resolve the issue of low accuracy arising from the singularity of the solutions by using a geometric mesh, leading to highly accurate solutions compared to uniform mesh solutions at almost the same computational cost. Moreover, for long-time integration of FDEs, using the geometric mesh leads to great computational savings as the total number of degrees of freedom required is relatively small.

In this paper, we introduce a nonlocal model for linear steady Stokes system with physical no-slip boundary condition. We use the idea of volume constraint to enforce the no-slip boundary condition and prove that the nonlocal model is well-posed. We also show that and the solution of the nonlocal system converges to the solution of the original Stokes system as the nonlocality vanishes.

In this paper we consider the existence and stability of multi-spike solutions to the fractional Gierer-Meinhardt model with periodic boundary conditions. In particular we rigorously prove the existence of symmetric and asymmetric two-spike solutions using a Lyapunov-Schmidt reduction. The linear stability of these two-spike solutions is then rigorously analyzed and found to be determined by the eigenvalues of a certain $2 × 2$ matrix. Our rigorous results are complemented by formal calculations of $N$-spike solutions using the method of matched asymptotic expansions. In addition, we explicitly consider examples of one- and two-spike solutions for which we numerically calculate their relevant existence and stability thresholds. By considering a one-spike solution we determine that the introduction of fractional diffusion for the activator or inhibitor will respectively destabilize or stabilize a single spike solution with respect to oscillatory instabilities. Furthermore, when considering two-spike solutions we find that the range of parameter values for which asymmetric two-spike solutions exist and for which symmetric two-spike solutions are stable with respect to competition instabilities is expanded with the introduction of fractional inhibitor diffusivity. However our calculations indicate that asymmetric two-spike solutions are always linearly unstable.

In this paper, we present a dual-horizon nonlocal diffusion model, in which the influence area at each point consists of a standard sphere horizon and an irregular dual horizon and its geometry is determined by the distribution of the varying horizon parameter. We prove the mass conservation and maximum principle of the proposed nonlocal model, and establish its well-posedness and convergence to the classical diffusion model. Noticing that the dual horizon-related term in fact vanishes in the corresponding variational form of the model, we then propose a finite element discretization for its numerical solution, which avoids the difficulty of accurate calculations of integrals on irregular intersection regions between the mesh elements and the dual horizons. Various numerical experiments in two and three dimensions are also performed to illustrate the usage of the variable horizon and demonstrate the effectiveness of the numerical scheme.

In this paper, we develop an efficient Hermite spectral-Galerkin method for nonlocal diffusion equations in unbounded domains. We show that the use of the Hermite basis can de-convolute the troublesome convolutional operations involved in the nonlocal Laplacian. As a result, the “stiffness” matrix can be fast computed and assembled via the four-point stable recursive algorithm with $\mathcal{O}(N^2)$ arithmetic operations. Moreover, the singular factor in a typical kernel function can be fully absorbed by the basis. With the aid of Fourier analysis, we can prove the convergence of the scheme. We demonstrate that the recursive computation of the entries of the stiffness matrix can be extended to the two-dimensional nonlocal Laplacian using the isotropic Hermite functions as basis functions. We provide ample numerical results to illustrate the accuracy and efficiency of the proposed algorithms.

In this paper, we propose a hybrid spectral method for a type of nonlocal problems, nonlinear Volterra integral equations (VIEs) of the second kind. The main idea is to use the shifted generalized Log orthogonal functions (GLOFs) as the basis for the first interval and employ the classical shifted Legendre polynomials for other subintervals. This method is robust for VIEs with weakly singular kernel due to the GLOFs can efficiently approximate one-point singular functions as well as smooth functions. The well-posedness and the related error estimates will be provided. Abundant numerical experiments will verify the theoretical results and show the high-efficiency of the new hybrid spectral method.

To model wave propagation in inhomogeneous media with frequency dependent power-law attenuation, it is needed to use the fractional powers of symmetric coercive elliptic operators in space and the Caputo tempered fractional derivative in time. The model studied in this paper is semilinear stochastic space-time fractional wave equations driven by infinite dimensional multiplicative Gaussian noise and additive fractional Gaussian noise, because of the potential fluctuations of the external sources. The purpose of this work is to discuss the Galerkin finite element approximation for the semilinear stochastic fractional wave equation. First, the space-time multiplicative Gaussian noise and additive fractional Gaussian noise are discretized, which results in a regularized stochastic fractional wave equation while introducing a modeling error in the mean-square sense. We further present a complete regularity theory for the regularized equation. A standard finite element approximation is used for the spatial operator, and a mean-square priori estimates for the modeling error and the approximation error to the solution of the regularized problem are established. Finally, numerical experiments are performed to confirm the theoretical analysis.

In this paper, L1/local discontinuous Galerkin method seeking the numerical solution to the time-fractional Stokes equation is displayed, where the time-fractional derivative is in the sense of Caputo with derivative order $α ∈ (0, 1).$ Although the time-fractional derivative is used, its solution may be smooth since such examples can be easily constructed. In this case, we use the uniform L1 scheme to approach the temporal derivative and use the local discontinuous Galerkin (LDG) method to approximate the spatial derivative. If the solution has a certain weak regularity at the initial time, we use the non-uniform L1 scheme to discretize the time derivative and still apply LDG method to discretizing the spatial derivative. The numerical stability and error analysis for both situations are studied. Numerical experiments are also presented which support the theoretical analysis.

In this work, we revisit the adaptive L1 time-stepping scheme for solving the time-fractional Allen-Cahn equation in the Caputo’s form. The L1 implicit scheme is shown to preserve a variational energy dissipation law on arbitrary nonuniform time meshes by using the recent discrete analysis tools, i.e., the discrete orthogonal convolution kernels and discrete complementary convolution kernels. Then the discrete embedding techniques and the fractional Grönwall inequality are applied to establish an $L^2$ norm error estimate on nonuniform time meshes. An adaptive time-stepping strategy according to the dynamical feature of the system is presented to capture the multi-scale behaviors and to improve the computational performance.

Based on a nonlocal Laplacian operator, a novel edge detection method of the grayscale image is proposed in this paper. This operator utilizes the information of neighbor pixels for a given pixel to obtain effective and delicate edge detection. The nonlocal edge detection method is used as an initialization for solving the Allen-Cahn equation to achieve two-phase segmentation of the grayscale image. Efficient exponential time differencing (ETD) solvers are employed in the time integration, and finite difference method is adopted in space discretization. The maximum bound principle and energy stability of the proposed numerical schemes are proved. The capability of our segmentation method has been verified in numerical experiments for different types of grayscale images.

A survey is given of convergence results that have been proved when the L1 scheme is used to approximate the Caputo time derivative $D^α_t$ (where $0 < α < 1)$ in initial-boundary value problems governed by $D^α_tu − ∆u = f$ and similar equations, while taking into account the weak singularity that is present in typical solutions of such problems. Various aspects of these analyses are outlined, such as global and local convergence bounds and the techniques used to derive them, fast implementation of the L1 scheme, semilinear problems, multi-term time derivatives, $α$-robustness, a posteriori error analysis, and two modified L1 schemes that achieve better accuracy. Over fifty references are provided in the bibliography, more than half of which are from the period 2019-2022.

We consider numerical approximation of spectral fractional Laplace-Beltrami problems on closed surfaces. The proposed numerical algorithms rely on their Balakrishnan integral representation and consists a sinc quadrature coupled with standard finite element methods for parametric surfaces. Possibly up to a log term, optimal rate of convergence are observed and derived analytically when the discrepancies between the exact solution and its numerical approximations are measured in $L^2$ and $H^1.$ The performances of the algorithms are illustrated on different settings including the approximation of Gaussian fields on surfaces.