Numerical algorithms for the time‐Caputo and space‐Riesz fractional Bloch‐Torrey equations

Abstract

In this paper, high-order numerical methods for time-Caputo and space-Riesz fractional Bloch-Torrey equations in oneand two-dimensional space are constructed, where the second-order backward fractional difference operator and the sixth-order fractional-compact difference operator are applied to approximate the time and space fractional derivatives, respectively. The stability and convergence of the methods are analyzed and it is shown that the convergence orders are higher than the earlier work. Finally, some numerical experiments are presented to demonstrate the effectiveness of the methods and confirm our theoretical results.

Integrated radial basis functions to simulate modified anomalous sub‐diffusion equation

“…Authors of [12] derived a compact finite difference scheme for the fractional sub-diffusion equation. Also see [13][14][15][16][17][18][19][20][21]. In Khodadadian and Heitzinger [22], a basis-adaptation method based on polynomial chaos expansion has been used for the stochastic nonlinear Poisson-Boltzmann equation.…”

“…FIGURE14 The diagram of exact and numerical solutions and the obtained error by integrated radial basis function (IRBF) method on Ω 5 with 𝜏 = 0.01, 𝛼 = 0.2, N = 17 × 17 at T = 0.5 for Test problem 5.2…”

Fast calculation based on a spatial two‐grid finite element algorithm for a nonlinear space–time fractional diffusion model

“…Because of the complexity of structure for this class of fractional models, people need to develop efficient numerical methods to achieve the numerical solutions. One can see some numerical studies such as difference algorithms for the advection–diffusion equation with space–time fractional derivatives [1], the space–time fractional diffusion or diffusion‐wave models [2, 3] and space–time fractional Bloch–Torrey models [4, 5], FE schemes for the telegraph equation with time–space fractional derivatives [6], FE method for 2D space and time fractional Bloch–Torrey equations [7], Galerkin FE methods for the 2D diffusion‐wave equations with time–space fractional derivatives [8, 9], fast solution algorithm based on the FE technique for a space–time fractional diffusion problem [10], Galerkin FE scheme for space–time fractional diffusion models [11, 12], fast iterative difference method for a time–space fractional convection‐diffusion model [13], difference method for time–space fractional differential equations [14], spectral method for the time–space fractional models [15, 16], reproducing kernel particle algorithm for 2D time–space fractional diffusion problems [17], several ADI algorithms for a time–space fractional diffusion model [18], fast difference algorithm for space–time FPDEs [19], fast FE algorithm for a fractional Allen–Cahn model including space–time derivatives [20], parallel algorithms for time–space fractional PDEs of parabolic type [21], discontinuous spectral element algorithms for space–time fractional advection problems [22], local discontinuous Galerkin method for space–time fractional convection‐diffusion models [23], the spectral method of a space–time fractional diffusion problem [24], and transcendental Bernstein series approach for space–time fractional telegraph model [25]. Here, we cannot list all the references on numerical methods for space–time fractional PDEs.…”