In this paper, a new alternating direction implicit Galerkin--Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component discretized by Crank--Nicolson method. detailed implementation of presented. stability and convergence analysis strictly proven, which shows that derived stable convergent order $2$ in time. An optimal error estimate also obtained introducing orthogonal projector. present extended to...

Recently, many models are formulated in terms of fractional derivatives, such as control processing, viscoelasticity, signal and anomalous diffusion. In the present paper, we further study important properties Riemann‐Liouville (RL) derivative, one mostly used derivatives. Some Caputo derivative which have not been discussed elsewhere simultaneously mentioned. The partial derivatives also introduced. These discussions beneficial understanding calculus modeling equations science engineering.

We study synchronization for two unidirectionally coupled networks. This is a substantial generalization of several recent papers investigating inside network. derive analytically criterion the networks which have same (inside) topological connectivity. Then numerical examples are given fit theoretical analysis. In addition, calculations with different connections presented and interesting desynchronization alternately appear increasing value coupling strength.

In this paper, two finite difference/element approaches for the time-fractional subdiffusion equation with Dirichlet boundary conditions are developed, in which time direction is approximated by fractional linear multistep method and space element method. The methods unconditionally stable convergent of order $O(\tau^q+h^{r+1})$ $L^2$ norm, where $q=2-\beta$ or 2 when analytical solution to sufficiently smooth, $\beta\,(0<\beta<1)$ derivative, $\tau$ $h$ step sizes space, respectively, $r$...

In this review paper, the finite difference methods (FDMs) for fractional differential equations are displayed. The considered mainly include kinetic of diffusion or dispersion with time, space and time-space derivatives. some way, these numerical have similar form as case classical equations, which can be seen generalizations FDMs typical equations. And tools, such von Neumann analysis method, energy method Fourier extended to accordingly. At same techniques improving accuracy reducing...

This article aims to fill in the gap of second-order accurate schemes for time-fractional subdiffusion equation with unconditional stability. Two fully discrete are first proposed space discretized by finite element method and time fractional linear multistep methods. These two methods unconditionally stable maximum global convergence order $O(\tau+h^{r+1})$ $L^2$ norm, where $\tau$ $h$ step sizes space, respectively, $r$ is degree piecewise polynomial space. The average rates also...

Fractional finite difference methods are useful to solve the fractional differential equations. The aim of this article is prove stability and convergence Euler method, Adams method high order based on convolution formula by using generalized discrete Gronwall inequality. Numerical experiments also presented, which verify theoretical analysis.

In this paper, the spectral approximations are used to compute fractional integral and Caputo derivative. The effective recursive formulae based on Legendre, Chebyshev Jacobi polynomials developed approximate integral. And succinct scheme for approximating derivative is also derived. collocation method proposed solve initial value problems boundary problems. Numerical examples provided illustrate effectiveness of derived methods.

Variable-order fractional diffusion equation model is a recently developed and promising approach to characterize time-dependent or concentration-dependent anomalous diffusion, process in inhomogeneous porous media. To further study the properties of variable-order time subdiffusion models, efficient numerical schemes are urgently needed. This paper investigates for equations finite domain. Three difference including explicit scheme, implicit scheme Crank–Nicholson studied. Stability...

In this paper, synchronization between two discrete-time networks, called “outer synchronization” for brevity, is theoretically and numerically studied. First, a sufficient criterion outer coupled networks which have the same connection topologies derived analytically. Numerical examples are also given they in line with theoretical analysis. Additionally, numerical investigations of different analyzed as well. The involved results show that these matrices can reach synchronization.

In this review paper, we are mainly concerned with the finite difference methods, Galerkin element and spectral methods for fractional partial differential equations (FPDEs), which divided into time-fractional, space-fractional, space-time-fractional (PDEs). Besides, fast algorithms FPDEs included in order to stimulate more efficient high-dimensional FPDEs.