Symmetries of a （2＋1）-Dimensional Toda-like Lattice

Lie group technique for solving differential-difference equations is applied to a new (2+1)-dimensional Toda-like lattice. An infinite dimensional Lie algebra and the corresponding commutation relations are obtained.

Explicit Solutions for Generalized (2+1)-Dimensional Nonlinear Zakharov-Kuznetsov Equation

The exact solutions of the generalized (2+1)-dimensional nonlinear Zakharov-Kuznetsov (Z-K) equationare explored by the method of the improved generalized auxiliary differential equation.Many explicit analytic solutionsof the Z-K equation are obtained.The methods used to solve the Z-K equation can be employed in further work toestablish new solutions for other nonlinear partial differential equations.

Use of Spreadsheets as an Introductory Tool for Solving Fluid-Structure Interaction Problem

A simple and intuitive manner for solving fluid-structure interaction problem has been developed using Microsoft Excel spreadsheets. By eliminating the need of previous knowledge of any programming language, the method appears as an interesting introduction to numerical solutions of partial differential equations, due to the direct and didactic way that it is developed. Proposed procedure enables the analysis of tridimensional geometries using the finite difference method and can be extended to other differential equations or boundary conditions. The author＇s objective in this paper is to develop a simple and reliable preliminary method for solving acoustic fluid-structure interaction problems with application to dam-reservoir interaction phenomena and also contribute in the educational growth for undergraduate students that are beginning research in such matter.

The Two Patterns of High Exact Difference in Convection Equation

This paper gives two patterns of high exact d ifference in solving convection equation. The error of cut section is to O(Δ t 2+Δx 4).

Numerical Simulation and Experimental Verification of Temperature Distribution of Piezoelectric Stack with Heating and Thermal Insulation Device

This paper discusses the temperature field distribution of piezoelectric stack with heating and thermal insulation device in cryogenic temperature environment. Firstly,the model of the piezoelectric damper is simplified and established by using partial-differential heat conduction equation. Secondly,the two-dimensional Du Fort-Frankel finite difference scheme is used to discretize the thermal conduction equation,and the numerical solution of the transient temperature field of piezoelectric stack driven by heating film at different positions is obtained by programming iteration. Then,the cryogenic temperature cabinet is used to simulate the low temperature environment to verify the numerical analysis results of the temperature field. Finally,the finite difference results are compared with the finite results and the experimental data in steady state and transient state,respectively. Comparison shows that the results of the finite difference method are basically consistent with the finite element and the experimental results,but the calculation time is shorter. The temperature field distribution results obtained by the finite difference method can verify the thermal insulation performance of the heating system and provide data basis for the temperature control of piezoelectric stack.

Analysis of unsteady supercavitating flow around a wedge

Supercavitating flow around a slender symmetric wedge moving at variable velocity in static fluid has been studied. Singular integral equation for the flow has been founded through distributing the sources and sinks on the symmetrical axis. The supereavity length at each moment is determined by solving the singular integral equation with finite difference method. The supercavity shape at each moment is obtained by solving the partial differential equation with variable coefficient. For the case that the wedge takes the impulse and uniformly variable motion, numerical results of time history of the supercavity length and shape are presented. The calculated results indicate that the shape and the length of the supercavity vary in a similar way to the case that the wedge takes variable motion, and there is a time lag in unsteady supercavitating flow induced by the variation of wedge velocity.

Finite Difference Scheme for Solving Parabolic Partial Differential Equations with Random Variable Input under Mean Square Sense

This study deal with seven points finite difference method to find the approximation solutions in the area of mean square calculus solutions for linear random parabolic partial differential equations. Several numerical examples are presented to show the ability and efficiency of this method.

Optimal l~∞ error estimates of finite difference methods for the coupled Gross-Pitaevskii equations in high dimensions

Due to the difficulty in obtaining the a priori estimate,it is very hard to establish the optimal point-wise error bound of a finite difference scheme for solving a nonlinear partial differential equation in high dimensions(2D or 3D).We here propose and analyze finite difference methods for solving the coupled GrossPitaevskii equations in two dimensions,which models the two-component Bose-Einstein condensates with an internal atomic Josephson junction.The methods which we considered include two conservative type schemes and two non-conservative type schemes.Discrete conservation laws and solvability of the schemes are analyzed.For the four proposed finite difference methods,we establish the optimal convergence rates for the error at the order of O(h^2+τ~2)in the l~∞-norm(i.e.,the point-wise error estimates)with the time stepτand the mesh size h.Besides the standard techniques of the energy method,the key techniques in the analysis is to use the cut-off function technique,transformation between the time and space direction and the method of order reduction.All the methods and results here are also valid and can be easily extended to the three-dimensional case.Finally,numerical results are reported to confirm our theoretical error estimates for the numerical methods.

Approximate Solutions of Nonlinear Fractional Kolmogorov-Petrovskii-Piskunov Equations Using an Enhanced Algorithm of the Generalized Two-Dimensional Differential Transform Method

By constructing the iterative formula with a so-called convergence-control parameter, the generalized two-dimensional differential transform method is improved. With the enhanced technique, the nonlinear fractional Kolmogorov-Petrovskii-Piskunov equations are dealt analytically and approximate solutions are derived. The results show that the employed approach is a promising tool for solving many nonlinear fractional partial differential equations. The algorithm described in this work is expected to be employed to solve more problems in fractional calculus.

Stability and convergence of the variable directions difference scheme for one nonlinear two-dimensional model

The system of two-dimensional nonlinear partial differential equations is considered. This system describes the vein formation in meristematic tissues of young leaves. Variable directions difference scheme is constructed and investigated. Absolute stability regarding space and time steps of scheme is shown. The convergence statement for the constructed scheme is proved. Rate of convergence is given. Various numerical experiments are carried out and results of some of them are considered in this paper. Comparison of numerical experiments with the results of the theoretical investigation is given too.

Some recent advances on vertex centered finite volume element methods for elliptic equations

In this paper, we report our recent advances on vertex centered finite volume element methods （FVEMs） for second order partial differential equations （PDEs）. We begin with a brief review on linear and quadratic finite volume schemes. Then we present our recent advances on finite volume schemes of arbitrary order. For each scheme, we first explain its construction and then perform its error analysis under both HI and L2 norms along with study of superconvergence properties.

Integrable discretization of soliton equations via bilinear method and Backlund transformation

We present a systematic procedure to derive discrete analogues of integrable PDEs via Hirota’s bilinear method.This approach is mainly based on the compatibility between an integrable system and its B¨acklund transformation.We apply this procedure to several equations,including the extended Korteweg-deVries(Kd V)equation,the extended Kadomtsev-Petviashvili(KP)equation,the extended Boussinesq equation,the extended Sawada-Kotera(SK)equation and the extended Ito equation,and obtain their associated semidiscrete analogues.In the continuum limit,these differential-difference systems converge to their corresponding smooth equations.For these new integrable systems,their B¨acklund transformations and Lax pairs are derived.