Discrete doubly periodic and solitary wave solutions for the semi-discrete coupled mKdV equations

In this paper, the improved Jacobian elliptic function expansion approach is extended and applied to constructing discrete solutions of the semi-discrete coupled modified Korteweg de Vries （mKdV） equations with the aid of the symbolic computation system Maple. Some new discrete Jacobian doubly periodic solutions are obtained. When the modulus m →1, these doubly periodic solutions degenerate into the corresponding solitary wave solutions, including kink-type, bell-type and other types of excitations.

A Family of Integrable Rational Semi-Discrete Systems and Its Reduction

Within framework of zero curvature representation theory, a family of integrable rational semi-discrete systems is derived from a matrix spectral problem. The Hamiltonian forms of obtained semi-discrete systems are constructed by means of the discrete trace identity. The Liouville integrability for the obtained family is demonstrated. In the end, a reduced family of obtained semi-discrete systems and its Hamiltonian form are worked out.

SOLUTION OF 2D SHALLOW WATER EQUATIONS BY GENUINELY MULTIDIMENSIONAL SEMI-DISCRETE CENTRAL SCHEME

A numerical two-dimensional shallow water method was based on method for solving the equations was presented. This the third-order genuinely multidimensional semi-discrete central scheme for spatial discretization and the optimal third-order Strong Stability Preserving （SSP） Runge-Kutta method for time integration. The third-order compact Central Weighted Essentially Non-Oscillatory （CWENO） reconstruction was adopted to guarantee the non-oscillatory behavior of the presented scheme and improve the resolution. Two kinds of source terms were considered in this work. They were evaluated using different approaches. The resulting scheme does not require Riemann solvers or characteristic decomposition, hence it retains all the attractive features of central schemes such as simplicity and high resolution. To evaluate the performance of the presented scheme, several numerical examples were tested. The results demonstrate that our method is efficient, stable and robust.

Lie Symmetry Analysis of the Inhomogeneous Toda Lattice Equation via Semi-Discrete Exterior Calculus

In this work,the Lie point symmetries of the inhomogeneous Toda lattice equation are obtained by semi-discrete exterior calculus,which is a semi-discrete version of Harrison and Estabrook’s geometric approach.A four-dimensional Lie algebra and its one-,two-and three-dimensional subalgebras are given.Two similarity reductions of the inhomogeneous Toda lattice equation are obtained by using the symmetry vectors.

Solutions and Continuum Limits of Two Semi-Discrete Lattice Potential Korteweg-de Vries Equations

We derive bilinear forms and Casoratian solutions for two semi-discrete potential Korteweg-de Vries equations. Their continuum limits go to the counterparts of the continuous potential Korteweg-de Vries equation.

Semi-Discrete and Fully Discrete Weak Galerkin Finite Element Methods for a Quasistatic Maxwell Viscoelastic Model

This paper considers weak Galerkin finite element approximations on polygonal/polyhedral meshes for a quasistatic Maxwell viscoelastic model.The spatial discretization uses piecewise polynomials of degree k(k≥1)for the stress approximation,degree k+1 for the velocity approximation,and degree k for the numerical trace of velocity on the inter-element boundaries.The temporal discretization in the fully discrete method adopts a backward Euler difference scheme.We show the existence and uniqueness of the semi-discrete and fully discrete solutions,and derive optimal a priori error estimates.Numerical examples are provided to support the theoretical analysis.

Semi-Discrete and Fully Discrete Mixed Finite Element Methods for Maxwell Viscoelastic Model of Wave Propagation

Semi-discrete and fully discrete mixedfinite element methods are consid-ered for Maxwell-model-based problems of wave propagation in linear viscoelastic solid.This mixedfinite element framework allows the use of a large class of exist-ing mixed conformingfinite elements for elasticity in the spatial discretization.In the fully discrete scheme,a Crank-Nicolson scheme is adopted for the approximation of the temporal derivatives of stress and velocity variables.Error estimates of the semi-discrete and fully discrete schemes,as well as an unconditional stability result for the fully discrete scheme,are derived.Numerical experiments are provided to verify the theoretical results.

A Semi-Discretizing Method Based Efficient Model for Fluidelastic Instability Threshold Prediction of Tube Bundles

Fluidelastic instability is destructive in tube bundles subjected to cross flow.Flow channel model proposed by Leaver and Weaver is well used for modeling this problem.However,as the tube motion is supposed to be harmonic,it may not simulate the general dynamic behaviors of tubes.To improve this,a model with arbitrary tube motion is proposed by Hassan and Hayder.While,due to involving in the time delay term,the stability problem cannot be solved by the eigenvalue scheme,and time domain responses of the tube have to be obtained to assess the instability threshold.To overcome this weakness,a new approach based on semi-discretizing method(SDM)is proposed in this study to make the instability threshold be predicted by eigenvalues directly.The motion equation of tube is built with considering the arbitrary tube motion and the time delay between fluid flow and tube vibration.A time delay integral term is derived and the SDM is employed to construct a transfer matrix,which transforms the infinite dimensional eigenvalue problem into a finite one.Hence the stability problem become solvable accordingly.With the proposed method,the instability threshold of a typical square tube array model is predicted,and the influences of system parameters on stability are also discussed.With comparing with prior works,it shows significant efficiency improvement in prediction of the instability threshold of tube bundles.

An Implicit-Explicit Computational Method Based on Time Semi-Discretization for Pricing Financial Derivatives with Jumps

This paper considers pricing European options under the well-known of SVJ model of Bates and related computational methods. According to the no-arbitrage principle, we first derive a partial differential equation that the value of any European contingent claim should satisfy, where the asset price obeys the SVJ model. This equation is numerically solved by using the implicit- explicit backward difference method and time semi-discretization. In order to explain the validity of our method, the stability of time semi-discretization scheme is also proved. Finally, we use a simulation example to illustrate the efficiency of the method.

Numerical Solution of Advection Diffusion Equation Using Semi-Discretization Scheme

Numerical diffusion and oscillatory behavior characteristics are averted applying numerical solutions of advection-diffusion equation are themselves immensely sophisticated. In this paper, two numerical methods have been used to solve the advection diffusion equation. We use an explicit finite difference scheme for the advection diffusion equation and semi-discretization on the spatial variable for advection-diffusion equation yields a system of ordinary differential equations solved by Euler’s method. Numerical assessment has been executed with specified initial and boundary conditions, for which the exact solution is known. We compare the solutions of the advection diffusion equation as well as error analysis for both schemes.

两类加权空间间的积分算子与离散算子的有界性及算子范数估计

Using the weight coefficient method, we first discuss semi-discrete Hilbert-type inequalities, and then discuss boundedness of integral and discrete operators and operator norm estimates based on Hilbert-type inequalities in weighted Lebesgue space and weighted normed sequence space.

Nonlinear property of the visco-elastic-plastic material in the impact problem

In this paper a numerical investigation on the non-Newtonian flow problem is conducted, in order to shed further light on the mathematical and virtual test methods in the auto-crash safety analysis. The accurate mathematical prediction would supply ultimate research tool for the passive safety analysis in such a scale.

循环不等式概述

循环不等式概述张双奎，韩普宪，肖凤翔（平顶山教育学院）（许昌教育学院）（红光隆农技校）为循环不等式，其中和式（１）左边的和称为循环和．这个不等式是１９５４年美国数学家Ｈ．Ｓ．ｓｈａｐｉｒｏ［１］提出的．虽然不等式（１）貌似初等，但长期来，它却吸引了国...

Generating Solutions to Discrete sine-Gordon Equation from Modified B(a|¨)cklund Transformation

We modify the bilinear Biicklund transformation for the discrete sine-Gordon equation and derive variety of solutions by freely choosing parameters from the modified B^cklund transformation. Dynamics of solutions and continuum limits are also discussed.

Nonconforming Mixed Finite Element Method for Nonlinear Hyperbolic Equations

A nonconforming mixed finite element method for nonlinear hyperbolic equations is discussed. Existence and uniqueness of the solution to the discrete problem are proved. Priori estimates of optimal order are derived for both the displacement and the stress.

THE SEMI-DISCREIXTE METHOD FOR SOLVING HIGH-DIMENSION WAVE EQUATION

The article gives a semi-discrete method for solving high-dimension wave equationBy the method, high-dimension wave equation is converted by, means of diseretizationinto I-D wave equation system which is well-posed. The convergence of the semidijcrete method is given. The numerical calculating resulis show that the speed of convergence is high.

正确处理解题过程中的几个关系

正确处理解题过程中的几个关系林文晓（平顶山师专数学系）解题是学习数学的手段，也是学习数学的目的，提高学生的解题能力，是数学教学的任务之一．笔者通过长期的教学实践，体会到要使学生解题正确、圆满、简捷，并通过解题发展学生的思维能力，重要的是在教学过程中引...

化工专业的学生必须重视高等数学课的学习

化工专业的学生必须重视高等数学课的学习燕青芝（平顶山师专化学系）在从事化工专业的教学工作中，笔者发现所教的历届学生都存在着高等数学知识薄弱的问题．高等数学知识差，导致讲课受阻，使有些工程问题难以讲解透彻，学生理解费力，对教材的掌握达不到应有的深度。１...

An ETD Method for American Options under the Heston Model

A numerical method for American options pricing on assets under the Heston stochastic volatility model is developed.A preliminary transformation is applied to remove the mixed derivative term avoiding known numerical drawbacks and reducing computational costs.Free boundary is treated by the penalty method.Transformed nonlinear partial differential equation is solved numerically by using the method of lines.For full discretization the exponential time differencing method is used.Numerical analysis establishes the stability and positivity of the proposed method.The numerical convergence behaviour and effectiveness are investigated in extensive numerical experiments.

APPROXIMATION OF NONCONFORMING QUASI-WILSON ELEMENT FOR SINE-GORDON EQUATIONS

In this paper, nonconforming quasi-Wilson finite element approximation to a class of nonlinear sine-Gordan equations is discussed. Based on the known higher accuracy results of bilinear element and different techniques from the existing literature, it is proved that the inner product △↓（u - Ih^1u）, △↓vh） and the consistency error can be estimated as order O（h^2） in broken H^1 - norm/L^2 - norm when u ∈ H^3（Ω）/H^4（Ω）, where Ih^1u is the bilinear interpolation of u, Vh belongs to the quasi-Wilson finite element space. At the same time, the superclose result with order O（h^2） for semi-discrete scheme under generalized rectangular meshes is derived. Furthermore, a fully-discrete scheme is proposed and the corresponding error estimate of order O（h^2 ＋ τ^2） is obtained for the rectangular partition when u ∈ H^4（Ω）, which is as same as that of the bilinear element with ADI scheme and one order higher than that of the usual analysis on nonconforming finite elements.