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This paper proposes and analyzes a tempered fractional integrodifferential equation in three-dimensional (3D) space. The Crank-Nicolson (CN) method and trapezoidal convolution quadrature rule are used to approximate the time derivative and tempered fractional integral term respectively, and finite difference/compact difference approaches combined with corresponding alternating direction implicit (ADI) algorithms are employed for spatial discretizations, which obtains the fully discrete CN ADI finite difference/compact difference schemes. Then, convergence analysis of two kinds of ADI schemes is derived via the energy argument and the generating function of Toeplitz matrices. Provided numerical examples confirm our theoretical estimates. [ABSTRACT FROM AUTHOR]

We present an alternating direction implicit scheme for the numerical solution of the distributed-order fractional integro-differential equation with two weakly singular kernels. Orthogonal spline collocation with piecewise Hermite bicubics is used for spatial discretization. The weighted and shifted Grünwald difference formula is employed to discretize the distributed-order time-fractional derivative combined with the second-order quadrature convolution rule introduced by Lubich for the Riemann–Liouville fractional integral. We prove the stability of the proposed scheme and derive error estimates. We also describe an efficient implementation of the scheme and show the numerical results demonstrating the accuracy and convergence rates in L 2 norm. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the numerical solution of the three‐dimensional (3D) nonlinear tempered fractional integrodifferential equation which is subject to the initial and boundary conditions. The backward Euler (BE) method in association with the first‐order convolution quadrature rule is employed to discretize this equation for time, and the Galerkin finite element method is applied for space, which is combined with an alternating direction implicit (ADI) algorithm, in order to reduce the computational cost for solving the three‐dimensional nonlocal problem. Then a fully discrete BE ADI Galerkin finite element scheme can be obtained by linearizing the non‐linear term. Thereafter we prove a positive‐type lemma, from which the stability and convergence of the proposed numerical scheme are derived based on the energy method. Numerical experiments are performed to verify the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

We consider the initial boundary value problem for a fractional evolution integro-differential equation with a weakly singular kernel arising from viscoelastic rods and membranes in a bounded smooth domain. In our numerical scheme, an arbitrary-order orthogonal spline collocation (OSC) method is used for spatial approximation and combined with the Crank-Nicolson (CN) alternating direction implicit (ADI) method for time approximation, and the integral term is approximated via the product integration (PI) rule. The detailed stability and convergence of the CN ADI OSC method in L 2 norm are proved. Finally, some numerical simulations are provided to illustrate our theoretical error estimates. In addition, the comparison between the non-uniform mesh and the uniform mesh shows the effectiveness of the non-uniform mesh when the solution is not sufficiently smooth. [ABSTRACT FROM AUTHOR]

A novel numerical technique is considered for the solution of a multi-term time-fractional diffusion equation. The orthogonal spline collocation method is used for in space, and a finite difference method in time. The stability and convergence are provided. The numerical experiments for one and two dimensional problems support our theoretical analysis. [ABSTRACT FROM AUTHOR]