Frequent oscillatory solutions of a nonlinear partial difference equation

This paper is concerned with a class of nonlinear delay partial difference equations with variable coefficients, which may change sign. By making use of frequency measures, some new oscillatory criteria are established.     

一类非线性偏差分方程组的频率振动性

应用频率测度法,讨论了一类非线性偏差分方程组的频率振动性,得到了此类方程组频率振动的全新判别准则.最后给出例子对主要结果进行说明.     

Unsaturated solutions for partial difference equations with forcing terms

The concept of unsaturated infinite double sequence is introduced by making use of frequency measures. Unsaturated solutions are then studied for a partial difference equation. Conditions for all solutions to be unsaturated are obtained. Since unsaturated solutions are oscillatory, our results yield oscillation criteria.     

Existence of solutions for a nonlinear system with a parameter

In this paper, the existence of solutions for a system of nonlinear equations is considered. n² nonzero real solutions are obtained by using the critical point theory. Additionally, the Dirichlet boundary value problems of even order difference equations and partial difference equations are investigated.     

Estimate of solutions for differential and difference functional equations with applications to difference methods

The theorems on the estimate of solutions for nonlinear second-order partial differential functional equations mainly of parabolic type with Dirichlet’s condition and for the suitable explicit finite difference functional schemes are proved. The proofs are based on the comparison technique. The convergent difference method given is considered without an assumption of the global generalized Perron condition on the functional variable but with local one in some sense only. It is a consequence of our estimate theorems. The functional dependence is of the Volterra type.     

一个药物作用肿瘤生长自由边界问题整体解的存在唯一性

该文研究一个描述药物作用下肿瘤生长的数学模型,这个肿瘤模型是对Jackson模型的一个改进,其数学形式是由一个二阶非线性抛物型方程与两个一阶非线性偏微分方程组耦合而成的自由边界问题.通过运用抛物型方程的L~P理论与一阶偏微分方程的特征方法,并利用Banach不动点定理,证明了该问题存在唯一的整体经典解.     

泛函偏差分方程定性理论的发展

泛涵偏差分方程是90年代初发展起来的一个新的和重要的研究领域，本文就泛函偏差分方程定性理论的研究状况和发展做一简要的评述。     

A note on a three-point partial difference equation with variable coefficients

The explicit solution of four-point linear partial difference equations, provided with variable coefficients and with boundary conditions including the independent variables, was found in a previous note. In addition, the note explained the procedure to be used in case of boundary conditions also including the dependent variables. The aim of this note is to determine the explicit solution of a three-point equation of the above-mentioned second type, encountered in the study of differential difference equations with the method of the steps.     

变系数偏微分方程组一般解的构造

求解偏微分方程不仅有理论意义,而且有实用价值.本文利用共轭算子的性质给出构造偏微分方程组一般解的方法.     

Existence of some neutral partial differential equations with infinite delay

This study intends to investigate a class of quasi-linear partial neutral functional differential equations with infinite delay. We assume that the linear part generates an analytic compact semigroup and the nonlinear part satisfies certain conditions. A sufficient condition is given to ensure the existence of mild and classical solutions. Finally, an example is given to illustrate our abstract results.     

Relations between symmetries and conservation laws for difference systems

This paper presents a relation between divergence variational symmetries for difference variational problems on lattices and conservation laws for the associated Euler–Lagrange system provided by Noether's theorem. This inspires us to define conservation laws related to symmetries for arbitrary difference equations with or without Lagrangian formulations. These conservation laws are constrained by partial differential equations obtained from the symmetries generators. It is shown that the orders of these partial differential equations have been reduced relative to those used in a general approach. Illustrative examples are presented.     

A new stability criterion for linear discrete systems

This paper is concerned with the stability of difference equations. A criterion to decide whether a certain polynomial has all its zeros inside the unit circle is applied to multistep linear methods in order to obtain the absolute stability region, and it is shown how this region can be extended. Systems of linear difference equations are also considered and an extension to partial difference equations is discussed.     

Effect of Space Dimensions on Equilibrium Solutions of Cahn-Hilliard and Conservative Allen-Cahn Equations

In this study, we investigate the effect of space dimensions on the equilibrium solutions of the Cahn-Hilliard (CH) and conservative Allen-Cahn (CAC) equations in one, two, and three dimensions. The CH and CAC equations are fourth-order parabolic partial and second-order integro-partial differential equations, respectively. The former is used to model phase separation in binary mixtures, and the latter is used to model mean curvature flow with conserved mass. Both equations have been used for modeling various interface problems. To study the space-dimension effect on both the equations, we consider the equilibrium solution profiles for symmetric, radially symmetric, and spherically symmetric drop shapes. We highlight the different dynamics obtained from the CH and CAC equations. In particular, we find that there is a large difference between the solutions obtained from these equations in three-dimensional space.     

Symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients

We present a symbolic computation procedure for deriving various high order compact difference approximation schemes for certain three dimensional linear elliptic partial differential equations with variable coefficients. Based on the Maple software package, we approximate the leading terms in the truncation error of the Taylor series expansion of the governing equation and obtain a 19 point fourth order compact difference scheme for a general linear elliptic partial differential equation. A test problem is solved numerically to validate the derived fourth order compact difference scheme. This symbolic derivation method is simple and can be easily used to derive high order difference approximation schemes for other similar linear elliptic partial differential equations.     

完全耦合的正倒向随机微分方程及相应的偏微分方程系统

本文利用完全耦合的正倒向随机微分方程,对一类耦合了一个代数方程的二阶拟线性抛物型偏微分方程系统,给出概率表示。在适当的假设下,得到这类偏微分方程系统粘性解的存在唯一性结果。     

具有连续变量的非线性偏差分方程的振动性

This paper is concerned with the oscillation of nonlinear partial difference equa- tions with continuous variables and the corresponding dual equations.Several sufficient conditions are obtained for the oscillation of these equations.     

The method of approximate particular solutions for solving certain partial differential equations

A standard approach for solving linear partial differential equations is to split the solution into a homogeneous solution and a particular solution. Motivated by the method of fundamental solutions for solving homogeneous equations, we propose a similar approach using the method of approximate particular solutions for solving linear inhomogeneous differential equations without the need of finding the homogeneous solution. This leads to a much simpler numerical scheme with similar accuracy to the traditional approach. To demonstrate the simplicity of the new approach, three numerical examples are given with excellent results. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 506–522, 2012     

On a fractional impulsive partial stochastic integro-differential equation with state-dependent delay and optimal controls

This paper is mainly concerned with a new class of fractional impulsive partial stochastic integro-differential equations with state-dependent delay and optimal controls in Hilbert spaces. Firstly, a more appropriate concept for mild solutions is introduced. Secondly, existence and uniqueness of mild solutions are proved by means of stochastic analysis theory, fractional calculus and the fixed point technique combined with solution operator. The existence of optimal pairs of system governed by fractional impulsive partial stochastic integro-differential equations is also presented. Finally, an example is given for demonstration.     

Exact solutions to a finite-difference model of a nonlinear reaction-advection equation: Implications for numerical analysis

This article gives exact solutions to a finite-difference model of a nonlinear reaction-advection equation. We show that this partial difference equation and the corresponding stationary and spatially independent difference equations derived from this model give the best representation of the original partial differential equation. The relevance of this work to the elimination of chaotic behavior in numerical solutions of differential equations is discussed.     

An inverse problem of Bilayer textile thickness determination in dynamic heat and moisture transfer

An inverse problem of bilayer textile thickness determination in dynamic heat and moisture transfer is presented satisfying the heat–moisture comfort level of human body. Heat and mass transfer law in bilayer textiles is displayed by proving the existence and uniqueness of solution to the coupled partial differential equations with initial-boundary value conditions. The finite difference method is employed to derive the numerical solution to partial differential equations. The regularized solution of the inverse problem is reformulated into solving function minimum problem through the Tikhonov regularization method. The golden section method is applied to solve the direct search problem and achieve the optimal solution to the inverse problem. Numerical algorithm and its numerical results provide theoretical explanation for textile materials research and development.