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This paper is concerned with the partial difference equation A"m"+"1","n+A"m","n"+"1-A"m","n+p"m","nA"m"-"k","n"-"l=0,m,n=0,1,2,...,, where k and l are two positive integers, {p"m","n} is a real double sequence. Some new oscillation criteria for this equation are obtained.

This paper is concerned with the linear delay partial difference equation where σ and τ are positive integers, a ( i , j ), b ( i , j ), p ( i , j ) are real sequences defined on i ≥ σ, j ≥ τ. Sufficient conditions for this equation to be stable are derived. Stability of certain nonlinear partial difference equations is studied also.

This paper is concerned with the linear delay partial difference equation u(i,j+1)=a(i,j)u(i+1,j)+b(i,j)u(i,j)+p(i,j)u(i−σ,j−τ) where σ and τ are two nonnegative integers, a(i,j), b(i,j), p(i,j) are three real sequences defined on i ≥ 0, j ≥ 0. Some sufficient conditions for this equation to be exponentially asymptotically stable are derived.

We consider the following boundary value problems: (−1)n−py(n) = a(t)h(y), n ⩽ 2, t ∈(0,1), y(i)(0) = 0, 0 ⩾ i ⩾ p − 1, y(i)(1) = 0, 0 ⩾ i ⩾ n − p − 1, and (−1)n−pΔny = F(k,y,Δy,…,Δn−1y), n ⩽ 2, 0 ⩾ k ⩾ m, Δiy(0), 0 ⩾ i ⩾ p − 1, Δiy(m+p+1)=0, 0 ⩾ i ⩾ n − p − 1, where 1 ≤ p ≤ n − 1 is fixed. By employing fixed-point theorems for operators on a cone, existence criteria are developed for multiple (at least three) positive solutions of the boundary value problems. As an application, we also establish the existence of radial solutions of certain partial difference equations. Several examples are included to dwell upon the importance of the results obtained.

Quadratic identities for Ising model correlations on a general planar lattice are derived. For the ordinary Ising model, they imply the nonlinear partial difference equations of McCoy and Wu. At the critical temperature Hirota's discrete-time Toda equation is found.

This paper considers the problem of simulating a class of nonlinear parabolic distributed-parameter systems. The quasilinearization technique is used to reduce the simulation problem to that of solving a sequence of linear parabolic equation with space and time varying coefficients. Then the Walsh function approach is employed in order to reduce computational effort and to replace a sequence of partial difference equations by a sequence of linear algebraic equations only.

The purpose of this paper is not to solve a problem but to pose one that may be of some interest, depth, and consequence.Given that the positive integer n has the canonical representation n = Πhi=1 piαi the problem of finding the number F(n) = f(α1, α2, … αn) of ordered factorizations of n into positive nontrivial integral factors is equivalent to that of finding the number of ordered partitions of the vector (α1, α2, … αn) into nonzero vectors with nonnegative integral components.

In this remark, we shall show a counter example for the main result of the paper [S.T. Liu, Y.Q. Liu, Oscillation theorems for second-order nonlinear partial difference equations, J. Comput. Appl. Math. 132 (2001) 479–482].

Any mass equation M(N,Z) may be expressed as a special solution of an inhomogeneous partial difference equation. One only has to introduce an appropriate difference operator D and calculate κ(N,Z) = D M(N,Z). The inhomogeneous partial difference equation $${\text{D M}}\left( {{\text{N}},{\text{Z}}} \right){\text{ }} = {\text{ }}\kappa \left( {{\text{N}},{\text{Z}}} \right)$$ (1) contains the original mass equation as a special solution. The most general solution of eq. (1) will include solutions of the homogeneous equation which can, for example, be used to generate shell correction terms.

The motivation for this work was to attempt to understand the reason that certain classes of explicit numerical algorithms for computing solutions to partial differential equations develop intense local patches of instability after long times, even when all the linear stability criteria are satisfied. Now, numerical analysts are the academic world’s greatest plumbers, and so practitioners of the art of numerical computation have invented many ingenious schemes to circumvent these problems (see, for example [1]). However, beyond declaring that the instabilities are nonlinear in character and beyond a few careful analyses of the problem (for example [2], [3]), the numerical analyst has not investigated in any great detail the nature of the breakdown.

Publisher Summary This chapter discusses the use of the Peaceman–Rachford alternating direction implicit method for solving the finite difference analogs of boundary value problems involving linear elliptic partial differential equations. It describes the application of the Peaceman–Rachford method to the separable case. The method is then applied to the nonlinear differential equation.teh chapter presents the numerical results obtained from solving boundary value problems for various mesh sizes and for various values of n. It was found that the use of the so-called proper application of the Peaceman–Rachford method yielded fairly satisfactory results. However, it was also found that the more natural application of the method was generally more effective. Further investigations are now underway to explain why the proper application was generally less effective than the other procedure.

In this note, we introduce the framework of partial difference equations (PdEs) over graphs for analyzing the behavior of multi-agent systems equipped with decentralized control schemes. Both leaderless and leader-follower models are considered. PdEs mimic partial differential equations (PDEs) on graphs and can be studied by introducing concepts of functional analysis strongly inspired to the corresponding ones arising in PDEs theory. We generalize different models proposed in the literature by introducing errors in the agent dynamics and analyze agent coordination through the joint use of PdEs and automatic control tools. Moreover, for the simplest control schemes, we show that the resulting PdEs enjoy properties that are similar to those of well-known PDEs like the heat equation, thus allowing to exploit physical-based reasoning for conjecturing formation properties.

This paper deals with discrete two-dimensional behaviors which are described by linear systems of partial difference equations with constant coefficients. Within the behavioral framework a natural concept of interconnection has been introduced by J.C.Willems, called regular interconnec- tion. We investigate regular interconnections that yield finite dimensional behaviors, and prove that when a finite dimensional behavior can be achieved from a given behavior by regular interconnection then the controllable part of is rectifiable. We apply this result to characterize all stabilizable behaviors. Â(c) 2009 EUCA.

In this first part of a two-parts paper we introduce the framework of Partial difference Equations (PdEs) over graphs for analyzing the behavior of multi-agent systems equipped with decentralized control schemes. We generalize the Vicsek's model by introducing errors in the agent dynamics and analyze agent alignment in leaderless and leader-follower models through the joint use of PdEs and automatic control tools. Moreover, we show that the resulting PdEs enjoy properties that are similar to those of well-known Partial Differential Equations (PDEs) like the heat equation, thus allowing to exploit physical-based reasoning for conjecturing properties of the collective dynamics.

A relatively short proof of Ehrenpreis's semilocal quotient structure theorem is given. The chief tool is a representation formula for the z-transforms of solutions to systems of partial difference equations with constant coefficients.