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In this paper, we study the two-dimensional Volterra integro-differential equations for viscoelastic rods and membranes in a bounded smooth domain. The memory kernel of this equation makes it not easy to construct an efficient numerical scheme and perform theoretical analysis. The numerical method is considered by the finite difference approach for spatial discretization and Crank–Nicolson (CN) alternating direction implicit (ADI) scheme in the time direction. The integral terms are approximated by the fractional convolution quadrature. We prove that the proposed method is unconditional stable and derive error estimates in L 2 norm. The literature reported on the ADI finite difference method of this model is extremely sparse. Numerical results support the theoretical analysis. [ABSTRACT FROM AUTHOR]

Balanced space‐fractional derivative is usually applied in modelling the state‐dependence, isotropy, and anisotropy in diffusion phenomena. In this paper, we introduce a class of space‐fractional reaction‐diffusion model with singular source term arising in combustion process. The fractional derivative employed in this model is defined in the sum of left‐sided and right‐sided Riemann‐Liouville fractional derivatives. With assistance of Kaplan's first eigenvalue method, we prove that the classic solution of this model may not be globally well‐defined, and the heat conduction governed by this model depends on the order of fractional derivative, the parameters in the equation, and the length of spatial interval. Finite difference method is implemented to solve this model, and an adaptive strategy is applied to improve the computational efficiency. The positivity, monotonicity, and stability of the numerical scheme are discussed. Numerical simulation and observation of the quenching and stationary solutions coincide the theoretical studies. [ABSTRACT FROM AUTHOR]

In this paper, a high order finite difference scheme for a two dimensional nonlinear fractional Klein–Gordon equation in polar coordinates is proposed. The difficulty caused by the variable coefficient and the nonlinear term in the equation is carefully handled. Stability and convergence are analyzed using the matrix form of the scheme. Numerical examples are provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we consider finite difference schemes for two dimensional time fractional diffusion-wave equations on an annular domain. The problem is formulated in polar coordinates and, therefore, has variable coefficients. A compact alternating direction implicit scheme with [ABSTRACT FROM AUTHOR]

In this paper, a high order compact (HOC) scheme for time fractional Fokker–Planck equations with variable convection is constructed. The scheme is studied using its matrix form by the energy method. We find that the difficulty arising from the variable coefficient can be overcome by simple modifications of the coefficient matrices. The scheme is shown to be stable and convergent with order τ 2 − α + h 4 which is higher than some recently studied schemes. Numerical examples are given to justify the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, compact finite difference schemes for the modified anomalous fractional sub-diffusion equation and fractional diffusion-wave equation are studied. Schemes proposed previously can at most achieve temporal accuracy of order which depends on the order of fractional derivatives in the equations and is usually less than two. Based on the idea of weighted and shifted Grünwald difference operator, we establish schemes with temporal and spatial accuracy order equal to two and four respectively. [ABSTRACT FROM AUTHOR]

In this paper, we study the finite difference/iterative method for the fractional telegraph equation with Hadamard derivatives. The model is first transformed into an equivalent fractional integro‐differential equation, then the technic of exponential type meshes is adopted. The resulting discrete coefficients of Hadamard fractional integral own several graceful properties, which are crucial in numerical analysis. To conquer the difficulty caused by weak regularity, we also propose iterative schemes. The error estimate is equal to O(logt)n+γ,1

In this paper, we study the numerical method for time-fractional KdV–Burgers' equation with initial singularity. The famous L 2 - 1 σ formula on graded meshes is adopted to approximate the Caputo derivative. Meanwhile, a nonlinear finite difference method on uniform grids is deduced for spatial discretization. The proposed method is second order in time and first order in space. With the help of the fractional Grönwall inequality, the unconditional stability and convergence of the current scheme are analyzed based on some skills. To raise the accuracy in spatial direction, a second order method is then carefully deduced. At last, theoretical results are verified by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we study a fast and linearized finite difference method to solve the nonlinear time-fractional wave equation with multi fractional orders. We first propose a discretization to the multi-term Caputo derivative based on the recently established fast L 2 - 1 σ formula and a weighted approach. Then we apply the discretization to construct a fully fast linearized discrete scheme for the nonlinear problem under consideration. The nonlinear term, which just fulfills the Lipschitz condition, will be evaluated on the previous time level. Therefore only linear systems are needed to be solved for obtaining numerical solutions. The proposed scheme is shown to have second-order unconditional convergence with respect to the discrete H 1 -norm. The proposed fast linearized method can be directly extended to solve the nonlinear distributed-order time-fractional wave problem. Numerical examples are provided to justify the efficiency. [ABSTRACT FROM AUTHOR]