Characteristic Lie algebra and classification of semidiscrete models

We study characteristic Lie algebras of semi-discrete chains and attempt to use this notion to classify Darboux-integrable chains. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 1, pp. 413–423, July, 2007.     

二维输运方程离散格式的对称性

讨论了二维柱几何非定态中子输运方程离散格式的对称性问题，在几何空间和相空间连续的情况下，证明了时间离散方程的一维球对称性；而在时间和相空间离散的情况下，阐述了格式不具有一维球对称性；对时间和相空间离散情况下的几何空间间断有限元方程，得到了左右对称性。     

One Case of Appearance of Positive Solutions of Delayed Discrete Equations

When mathematical models describing various processes are analysed, the fact of existence of a positive solution is often among the basic features. In this paper, a general delayed discrete equation

A General Version of the Retract Method for Discrete Equations

We study a problem concerning the compulsory behavior of the solutions of systems of discrete equations u（k ＋ 1） = F（k, u（k））, k ∈ N（a） = {a, a ＋ 1, a ＋ 2 }, a ∈ N,N= {0, 1,... } and F ： N（a） × R^n→R^n. A general principle for the existence of at least one solution with graph staying for every k ∈ N（a） in a previously prescribed domain is formulated. Such solutions are defined by means of the corresponding initial data and their existence is proved by means of retract type approach. For the development of this approach a notion of egress type points lying on the defined boundary of a given domain and with respect to the system considered is utilized. Unlike previous investigations, the boundary can contain points which are not points of egress type, too. Examples are inserted to illustrate the obtained result.     

Asymptotic numerical analysis of the diffusion‐discrete absorption equation

The diffusion‐discrete absorption (DDA) equation is considered. This equation contains the standard diffusion term and the discrete sorption expressed by a sum of large number of δ‐functions with the support at a non‐uniform mesh multiplied by the unknown function (concentration). The main result of the paper is the homogenization (continualization) of this equation when the small parameter is the characteristic step h of the mesh. The error estimates are proved for the difference of the exact solution of the DDA equation and the solution of the homogenized differential equation. Copyright © 2012 John Wiley & Sons, Ltd.     

Discrete Models of the Harmonic Oscillator and a Discrete Analogue of Gauss' Hypergeometric Equation

We consider a doubly infinite discrete-discrete version of the bispectral problem and uncover a possible discrete analog of Gauss' equation.     

Discrete Bessel functions and partial difference equations

We introduce a new class of discrete Bessel functions and discrete modified Bessel functions of integer order. After obtaining some of their basic properties, we show that these functions lead to fundamental solutions of the discrete wave equation and discrete diffusion equation.     

关于抛物型方程离散流量的收敛性

关于用差分方法求解具有间断系数的二阶抛型方程的问题, Ａ．Ｈ． ＴＮＸＯＨＯＢ与 Ａ．Ａ．Ｃａｍａｐｃ　从１９６１年山开始曾经作过详尽的研究,他们的结果都总结在专著［２］中,有关的文献也可以在该书中找到．他们指出在系数间断点处附近的网格点上格式的截断误差为Ｏ（１）,但差分格式的解在极大意义下收敛于原微分方程的连续解．他们在构造差分格式,并论证其收敛性时,充分利用了原微分方程中流连续的性质,但是却没有讨论差分格式中的离散流量的收敛性．８０年代 Ｔ．Ａ． Ｍａｎｔｅｎｆｆｅｌ, Ａ．Ｂ． Ｗｈｉｔｅ, Ｊｒ…     

An Estimate for the Attraction Domains of Difference Equations with Periodic Linear Terms

We consider the quasilinear systems of difference equations with periodic coefficients in linear terms. We obtain estimates for the attraction domain of the zero solution and establish inequalities for the norms of solutions. The results are stated in terms of Lyapunov-type matrix series.     

Discrete Zγ: Embedded Circle Patterns with the Square Grid Combinatorics and Discrete Painlevé Equations

We study a discrete analogue of the holomorphic map z . It is given by Schramm's circle pattern with the square grid combinatorics. We show that the corresponding circle patterns are embedded and described by special separatrix solutions of discrete Painlevé equations. We establish global properties of these solutions and of the discrete z .     

Boundary value problems of discrete generalized Emden-Fowler equation

By using the critical point theory, some sufficient conditions for the existence of the solutions to the boundary value problems of a discrete generalized Emden-Fowler equation are obtained. In a special case, a sharp condition is obtained for the existence of the boundary value problems of the above equation. For a linear case, by the discrete variational theory, a necessary and sufficient condition for the existence, uniqueness and multiplicity of the solutions is also established.     

一类偏积分微分方程二阶差分全离散格式

本给出了数值求解一类偏积分微分方程的二阶全离散差分格式．采用了Crank-Nicolson格式；积分项的离散利用了Lubieh的二阶卷积积分公式；给出了稳定性的证明，误差估计及收敛性的结果．     

A geometric method for solving a series of integral Poincaré-Steklov equations

Eigenvalues and eigenfunctions are explicitly found for a family of singular integral equations. It is shown how their discrete spectrum becomes continuous as the equation degenerates. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 343–353, March, 1998.     

On solutions of a discretized heat equation in discrete Clifford analysis

The main purpose of this paper was to study solutions of the heat equation in the setting of discrete Clifford analysis. More precisely we consider this equation with discrete space and continuous time. Thereby we focus on representations of solutions by means of dual Taylor series expansions. Furthermore we develop a discrete convolution theory, apply it to the inhomogeneous heat equation and construct solutions for the related Cauchy problem by means of heat polynomials.     

On the Solution of Some Non-Local Problems

This paper deals with two types of non-local problems for the Poisson equation in the disc. The first of them deals with the situation when the function value on the circle is given as a combination of unknown function values in the disc. The other type deals with the situation when a combination of the value of the function and its derivative by radius on the circle are given as a combination of unknown function values in the disc. The existence and uniqueness of the classical solution of these problems is proved. The solutions are constructed in an explicit form.     

利用离散正交性解一类二维抛物方程初值反问题

本文研究二维抛物方程初值反问题，将此初值反问题的求解化为一个非线性方程组的求解问题，引进二维离散正交性概念，利用离散正交性，选取观察点列。简化线性方程组的求解，并对计算误差作出估计。     

Solvability Criterion for the Constrained Discrete Lyapunov and Riccati Equations

Conditions for the solvability of the discrete Lyapunov and the discrete Riccati equations subject to linear equality constraints are derived. These problems arise naturally in the context of output min-max robust control. It is shown that the following problems are equivalent to one another: (a) the solvability of the constrained discrete Riccati equation; and (b) the existence of a feedback gain that guarantees the solvability of the constrained discrete Lyapunov equation of the resulting closed loop. A simple criterion for the existence of a solution to both problems is presented. These problems are shown to be related to the discrete positive real property.     

On Some Discrete Variational Problems

Making use of a discrete version of the P.-L. Lions concentration-compactness principle, we establish some results on the existence of nontrivial solutions for nonlinear stationary discrete equations of the Schrödinger type.     

New estimates of mixed finite element method for fourth‐order wave equation

The conforming bilinear mixed finite element method is established for a fourth‐order wave equation. By applying the interpolation and projection simultaneously, the superclose and superconvergence results of order O (h ²) for the original variable u and intermediate variable p  =? a ₁(X )Δu in H¹‐norm are achieved for the semi‐discrete and fully discrete schemes, respectively (where a ₁(X ) is the coefficient appeared in the equation). The distinct advantage of this method is that it can reduce the smoothness of solutions u ,u _t , and p compared with the existing literature for getting optimal estimates alone. At last, some numerical results are carried out to validate the theoretical analysis. Copyright © 2017 John Wiley & Sons, Ltd.