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In this paper, fast compact finite difference schemes are derived for fourth-order multi-term fractional sub-diffusion equations with initial singularity under the first Dirichlet boundary conditions. In contrast to the direct scheme, the fast algorithm adopted to approximate the Caputo derivative reduces the computation costs effectively. Sharp error estimate of the proposed scheme for linear models is rigorously presented by the energy method. Furthermore, to handle more intricate nonlinear models, a crucial Grönwall inequality is then deduced to analyse the stability and convergence of the linearized scheme. It is worth pointing out that the Grönwall inequality is also helpful in numerical analysis of multi-step schemes for other problems, such as, integro-differential equations with multiple fractional derivatives. Ultimately, numerical examples are provided to verify the efficiency of the established difference schemes. [ABSTRACT FROM AUTHOR]

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]

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 two dimensional fractional integro-differential equations, where the order of time fractional derivative α ∈ (1 , 2) and integral order γ ∈ (0 , 1). To overcome the difficulty caused by the two fractional terms, we transform the original equation using the method of integration by parts. A novel high order compact alternating direction implicit (ADI) difference scheme is then proposed to solve the equivalent model. By some skills and detailed analysis, the unconditional stability and convergence in H 1 norm are proved, with the accuracy order O (τ 2 + h 1 4 + h 2 4) , where τ , h 1 and h 2 are temporal and spatial step sizes, respectively. Finally, numerical results are presented to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

A high-order compact alternating direction implicit scheme is considered to solve the two-dimensional time-fractional integro-differential equation with weak singularity near the initial time in this paper. The L 1 formula and trapezoidal PI rule on nonuniform meshes, which greatly improve the temporal accuracy compared to the method on uniform grids, are adopted to approximate the Caputo derivative and the Riemann-Liouville integral, respectively. With the help of a modified discrete fractional Grönwall inequality and some crucial skills, the stability and convergence of the proposed scheme are analyzed. Numerical results confirm the sharpness of the error analysis. [ABSTRACT FROM AUTHOR]

In this paper, a new numerical approximation is discussed for the two dimensional multi-term time fractional integro-differential equation. The proposed technique is based on the high order orthogonal spline collocation (OSC) method for the spatial discretization, the classical L1 approximation for the Caputo fractional derivative, and an alternating direction implicit (ADI) method in time, combined with the second order fractional quadrature rule proposed by Lubich to approximate the integral term. Detailed analysis for the optimal error estimate of the proposed scheme is rigorously discussed. Numerical results are listed to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose a fast and efficient numerical method to solve the two-dimensional integro-differential equation with a weakly singular kernel. The numerical method are considered by finite difference approach for spatial discretization and alternating direction implicit (ADI) method in time, combined with the second-order fractional quadrature convolution rule introduced by Lubich and the classical L 1 approximation for Caputo fractional derivative. The detailed analysis shows that the proposed scheme is unconditionally stable and convergent with the convergence order O (τ min { 1 + α , 2 − α } + h 1 2 + h 2 2). Some numerical results are also given to confirm our theoretical prediction. [ABSTRACT FROM AUTHOR]