In this work, we develop an efficient iterative scheme for a class of nonlocal evolution models involving a Caputo fractional derivative of order $\alpha (0,1)$ in time. The fully discrete scheme is obtained using the standard Galerkin method with conforming piecewise linear finite elements in space and corrected high-order BDF convolution quadrature in time. At each time step, instead of solving the linear algebraic system exactly, we employ a multigrid iteration with a Gauss–Seidel smoother to approximate the solution efficiently. Illustrative numerical results for nonsmooth problem data are presented to demonstrate the approach.
In this paper, we consider numerical solutions of fractional ordinary differential equations with the Caputo–Fabrizio derivative, and construct and analyze a high-order time-stepping scheme for this equation. The proposed method makes use of quadratic interpolation function in sub-intervals, which allows to produce fourth-order convergence. A rigorous stability and convergence analysis of the proposed scheme is given. A series of numerical examples are presented to validate the theoretical claims. Traditionally a scheme having fourth-order convergence could only be obtained by using block-by-block technique. The advantage of our scheme is that the solution can be obtained step by step, which is cheaper than a block-by-block-based approach.
The aim of this article is to review our recent work on nonlocal dynamics of non-Gaussian systems arising in a gene regulatory network. We have used the mean exit time, escape probability and maximal likely trajectory to quantify dynamical behaviors of a stochastic differential system with non-Gaussian $\alpha$-stable Lévy motions, to examine how the non-Gaussianity index and noise intensity affect the gene transcription processes.
The aim of this paper is to obtain the numerical solutions of generalized space-fractional Burgers’ equations with initial-boundary conditions by the Jacobi spectral collocation method using the shifted Jacobi–Gauss–Lobatto collocation points. By means of the simplified Jacobi operational matrix, we produce the differentiation matrix and transfer the space-fractional Burgers’ equation into a system of ordinary differential equations that can be solved by the fourth-order Runge–Kutta method. The numerical simulations indicate that the Jacobi spectral collocation method is highly accurate and fast convergent for the generalized space-fractional Burgers’ equation.