Analysis of local discontinuous Galerkin method for time–space fractional convection–diffusion equations

A Numerical Scheme Based on the Chebyshev Functions to Find Approximate Solutions of the Coupled Nonlinear Sine-Gordon Equations with Fractional Variable Orders

“…For instance, [32] defined a new method based on the linearization formula to find the fuzzy approximate solution of fractional differential equations under uncertainty, a new spectral tau method for solving equations of Kelvin-Voigt equations is proposed in [33], and [34] studied a numerical method based on the Bernstein functions to solve the linear cable equation of variable order. An implicit RBF meshless method [35], spectral method [36], homotopy analysis scheme [37], meshless method [38], collocation and finite difference-collocation methods [39], Ritz-Galerkin method [40], new collocation scheme for solving fractional partial differential equations [41], Sinc-Chebyshev collocation scheme [42], numerical method based on the shifted Chebyshev functions [43], numerical method based on the shifted Legendre functions [44], piecewise integroquadratic spline interpolation and finite difference method [45], wavelet scheme [46], Legendre wavelet method [47], Bernoulli wavelet method [48], and Lagrange multiplier scheme [49] and other methods [50][51][52][53][54]. Several methods are presented for solving variable-order fractional equations.…”

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.…”