A fifth order semidiscrete mKdV equation

In this paper, aiming to get more insight on the relation between the higher order semidiscrete mKdV equations and higher order mKdV equations, we construct a fifth order semidiscrete mKdV equation from the three known semidiscrete mKdV fluxes. We not only give its Lax pairs, Darboux transformation, explicit solutions and infinitely many conservation laws, but also show that their continuous limits yield the corresponding results for the fifth order mKdV equation. We thus conclude that the fifth order discrete mKdV equation is extremely an useful discrete scheme for the fifth order mtCdV equation.

A semidiscrete Gardner equation

We construct the Darboux transformations, exact solutions, and infinite number of conservation laws for a semidiscrete Gardner equation. A special class of solutions of the semidiscrete equation, called table-top solitons, are given. The dynamical properties of these solutions are also discussed.

Discrete formulation of mixed finite element methods for vapor deposition chemical reaction equations

The vapor deposition chemical reaction processes, which are of extremely extensive applications, can be classified as a mathematical model by the following governing nonlinear partial differential equations containing velocity vector, temperature field, pressure field, and gas mass field. The mixed finite element （MFE） method is employed to study the system of equations for the vapor deposition chemical reaction processes. The semidiscrete and fully discrete MFE formulations are derived. And the existence and convergence （error estimate） of the semidiscrete and fully discrete MFE solutions are demonstrated. By employing MFE method to treat the system of equations for the vapor deposition chemical reaction processes, the numerical solutions of the velocity vector, the temperature field, the pressure field, and the gas mass field can be found out simultaneously. Thus, these researches are not only of important theoretical means, but also of extremely extensive applied vistas.

Dynamics of Nerve Pulse Propagation in a Weakly Dissipative Myelinated Axon

We analytically derived the complex Ginzburg-Landau equation from the Liénard form of the discrete FitzHugh Nagumo model by employing the multiple scale expansions in the semidiscrete approximation. The complex Ginzburg-Landau equation now governs the dynamics of a pulse propagation along a myelinated nerve fiber where the wave dispersion relation is used to explain the famous phenomena of propagation failure and saltatory conduction. Stability analysis of the pulse soliton solution that mimics the action potential fulfills the Benjamin-Feir criteria for plane wave solutions. Finally, results from our numerical simulations show that as the dissipation along the myelinated axon increases, the nerve impulse broadens and finally degenerates to front solutions.

Convergence of Weak Galerkin Finite Element Method for Second Order Linear Wave Equation in Heterogeneous Media

Weak Galerkin finite element method is introduced for solving wave equation with interface on weak Galerkin finite element space(Pk(K),P_(k−1)(∂K),[P_(k−1)(K)]^(2)).Optimal order a priori error estimates for both space-discrete scheme and implicit fully discrete scheme are derived in L1(L2)norm.This method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal meshes.Finite element algorithm presented here can contribute to a variety of hyperbolic problems where physical domain consists of heterogeneous media.

ASYMPTOTIC ERROR EXPANSION AND DEFECT CORRECTION FOR SOBOLEV AND VISCOELASTICITY TYPE EQUATIONS

In this paper we study the higher accuracy methods - the extrapolation and defect correction for the semidiscrete Galerkin approximations to the solutions of Sobolev and viscoelasticity type equations. The global extrapolation and the correction approximations of third order, rather than the pointwise extrapolation results are presented.

ERROR ESTIMATES FOR TWO-SCALE COMPOSITE FINITE ELEMENT APPROXIMATIONS OF NONLINEAR PARABOLIC EQUATIONS

We study spatially semidiscrete and fully discrete two-scale composite nite element method for approximations of the nonlinear parabolic equations with homogeneous Dirichlet boundary conditions in a convex polygonal domain in the plane.This new class of nite elements,which is called composite nite elements,was rst introduced by Hackbusch and Sauter[Numer.Math.,75(1997),pp.447-472]for the approximation of partial di erential equations on domains with complicated geometry.The aim of this paper is to introduce an effcient numerical method which gives a lower dimensional approach for solving partial di erential equations by domain discretization method.The composite nite element method introduces two-scale grid for discretization of the domain,the coarse-scale and the ne-scale grid with the degrees of freedom lies on the coarse-scale grid only.While the ne-scale grid is used to resolve the Dirichlet boundary condition,the dimension of the nite element space depends only on the coarse-scale grid.As a consequence,the resulting linear system will have a fewer number of unknowns.A continuous,piecewise linear composite nite element space is employed for the space discretization whereas the time discretization is based on both the backward Euler and the Crank-Nicolson methods.We have derived the error estimates in the L^(∞)(L^(2))-norm for both semidiscrete and fully discrete schemes.Moreover,numerical simulations show that the proposed method is an efficient method to provide a good approximate solution.

Dynamics of a discrete predator-prey model with Holling-II functional response

In this paper,we use a semidiscretization method to derive a discrete predator–prey model with Holling type II,whose continuous version is stated in[F.Wu and Y.J.Jiao,Stability and Hopf bifurcation of a predator-prey model,Bound.Value Probl.129(2019)1–11].First,the existence and local stability of fixed points of the system are investigated by employing a key lemma.Then we obtain the sufficient conditions for the occurrence of the transcritical bifurcation and Neimark–Sacker bifurcation and the stability of the closed orbits bifurcated by using the Center Manifold theorem and bifurcation theory.Finally,we present numerical simulations to verify corresponding theoretical results and reveal some new dynamics.

Numerical Blow-up for a Nonlinear Heat Equation

This paper concerns the study of the numerical approximation for the following initialboundary value problem{ut-uzx=f（u）,t∈（0,1）,t∈（0,T） u（0,t）=0,t∈（0,1）,t∈（0,T） u（x,0）=u0（x）,x∈（0,1）where f（s） is a positive, increasing, C1 convex function for the nonnegative values of s, f（0） 〉0, f∞ds/f（s） 〈∞, u0∈C1（[0, 1]）, u0（0） = 0, u＇0（1）=0. We find some conditions under which the solution of a semidiscrete form of the above problem blows up in a finite time and estimate its semidiserete blow-up time. We also prove the convergence of the semidiscrete blow-up time to the theoretical one. A similar study has been also undertaken for a discrete form of the above problem. Finally, we give some numerical results to illustrate our analysis.