Yifa Tang

Two finite difference schemes for time fractional diffusion-wave equation

Time fractional diffusion-wave equations are generalizations of classical diffusion and wave equations which are used in modeling practical phenomena of diffusion and wave in fluid flow, oil strata and others. In this paper we construct two finite difference schemes to solve a class of initial-boundary value time fractional diffusion-wave equations based on its equivalent partial integro-differential equations. Under the weak smoothness conditions, we prove that our two schemes are convergent with first-order accuracy in temporal direction and second-order accuracy in spatial direction. Numerical experiments are carried out to demonstrate the theoretical analysis.

The Grünwald-Letnikov method for fractional differential equations

This paper is devoted to the numerical treatment of fractional differential equations. Based on the Grunwald-Letnikov definition of fractional derivatives, finite difference schemes for the approximation of the solution are discussed. The main properties of these explicit and implicit methods concerning the stability, the convergence and the error behavior are studied related to linear test equations. The asymptotic stability and the absolute stability of these methods are proved. Error representations and estimates for the truncation, propagation and global error are derived. Numerical experiments are given.

Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations

Abstract In this article, a class of two-dimensional Riesz space fractional diffusion equations is considered. Some fractional spaces are established and some equivalences between fractional derivative spaces and fractional Sobolev space are presented. By the Galerkin finite element method and backward difference method, a fully discrete scheme is obtained. According to Lax–Milgram theorem, the existence and uniqueness of the solution to the fully discrete scheme are investigated. The stability and convergence of the scheme are also derived. Finally, some numerical examples are given for verification of our theoretical analysis.

Symplectic and multi-symplectic methods for the nonlinear Schrödinger equation

The Hamiltonian and the multi-symplectic formulations of the nonlinear Schrodinger equation are considered. For the multi-symplectic formulation, a new six-point difference scheme which is equivalent to the multi-symplectic Preissman integrator is derived. Numerical experiments are also reported

Finite difference/finite element method for two-dimensional space and time fractional Bloch-Torrey equations

In this paper, a class of two-dimensional space and time fractional Bloch-Torrey equations (2D-STFBTEs) are considered. Some definitions and properties of fractional derivative spaces are presented. By finite difference method and Galerkin finite element method, a semi-discrete variational formulation for 2D-STFBTEs is obtained. The stability and convergence of the semi-discrete form are discussed. Then, a fully discrete scheme of 2D-STFBTEs is derived and the convergence is investigated. Finally, some numerical examples based on linear piecewise polynomials and quadratic piecewise polynomials are given to prove the correctness of our theoretical analysis.

The symplecticity of multi-step methods☆

Abstract For any linear multi-step scheme, Feng Kang defines the step-transition operator characterizing it and defines the symplecticity of the method for Hamiltonian systems through the operator. In this paper, the author gets a valuable expression of the step-transition operator (Lemma 1, Section 2) and proves a conjecture due to Feng-any linear multi-step scheme is non-sympletic (Theorem 1, Section 3). Similarly, an interesting result (in Theorem 2, Section 3) for a sort of generalized multi-step schemes is obtained. The results indicate that some novel approach is needed for the construction of sympletic multi-step methods.

Symplectic wavelet collocation method for Hamiltonian wave equations

This paper introduces a novel symplectic wavelet collocation method for solving nonlinear Hamiltonian wave equations. Based on the autocorrelation functions of Daubechies compactly supported scaling functions, collocation method is conducted for the spatial discretization, which leads to a finite-dimensional Hamiltonian system. Then, appropriate symplectic scheme is employed for the integration of the Hamiltonian system. Under the hypothesis of periodicity, the properties of the resulted space differentiation matrix are analyzed in detail. Conservation of energy and momentum is also investigated. Various numerical experiments show the effectiveness of the proposed method.

Two Mixed Finite Element Methods for Time-Fractional Diffusion Equations

Based on spatial conforming and nonconforming mixed finite element methods combined with classical L1 time stepping method, two fully-discrete approximate schemes with unconditional stability are first established for the time-fractional diffusion equation with Caputo derivative of order

Finite element method for two-dimensional space-fractional advection-dispersion equations

0<\alpha <1