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A discrete multidimensional system is the set of solutions to a system of linear partial difference equations defined on the lattice Z n . This paper shows that it is determined by a unique coarsest sublattice, in the sense that the solutions of the system on this sublattice determine the solutions on Z n ; it is therefore the correct domain of definition of the discrete system. In turn, the defining sublattice is determined by a Galois group of symmetries that leave invariant the equations defining the system. These results find application in understanding properties of the system such as controllability and autonomy, and in its order reduction. [ABSTRACT FROM AUTHOR]

In this paper we define the algebraic entropy test for face-centered quad equations, which are equations defined on vertices of a quadrilateral plus an additional interior vertex. This notion of algebraic entropy is applied to a recently introduced class of these equations that satisfy a new form of multidimensional consistency called consistency-around-a-face-centered-cube (CAFCC), whereby the system of equations is consistent on a face-centered cubic unit cell. It is found that for certain arrangements of equations (or pairs of equations) in the square lattice, all known CAFCC equations pass the algebraic entropy test possessing either quadratic or linear growth. [ABSTRACT FROM AUTHOR]

Hirota's discrete Korteweg‐de Vries equation (dKdV) is an integrable partial difference equation on Z2, which approaches the Korteweg‐de Vries equation in a continuum limit. We find new transformations to other equations, including a second‐degree second‐order partial difference equation, which provide an unusual embedding into a three‐dimensional lattice. The consistency of the resulting system extends a property that has been widely used to study partial difference equations on multidimensional lattices. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new nonlinear Volterra-Fredholm type discrete inequalities with three independent variables. These inequalities can serve as handy tools in the study of class of nonlinear Volterra-Fredholm type sum-difference equations, partial difference equations, and its variants to analyze the qualitative and quantitative behavior of their solutions. Some applications are also given to convey the importance of our results. [ABSTRACT FROM AUTHOR]

In this paper, we provide an essentially complete answer to the question of minimal initial data required to solve an overdetermined system of linear partial difference equations with real constant coefficients using the notion of characteristic sets. A characteristic set is a special subset of the domain with the defining property that for every solution trajectory of the system of equations, the knowledge of the solution trajectory restricted to this set uniquely determines the trajectory over the whole domain. We emphasize the fact that subsets which are sublattices and unions of finitely many parallel translates of such sublattices are best suited to answer the question of the minimality of initial data. We first provide an algebraic characterization of a sublattice to be a characteristic sublattice. The main result of this paper provides conditions under which a system admits a union of a sublattice and finitely many parallel translates of it as a characteristic set; an important condition is the rank of the sublattice being equal to the Krull dimension of the system. For the condition when the rank of the sublattice is strictly less than the Krull dimension of the system, we show that neither the sublattice nor a finite union of sublattices can be a characteristic set. For the case when the rank of the sublattice is strictly greater than the Krull dimension of the system, a union of the sublattice and finitely many parallel translates of it is a characteristic set. But, unlike the case when the rank of the sublattice is equal to the Krull dimension of the system, in this case a proper sublattice of the given sublattice exists which along with its finitely many parallel translates now qualify as a characteristic set. We also show that for a given overdetermined system of partial difference equations, a characteristic set of the form given by a union of a sublattice and finitely many parallel translates of it always exists. [ABSTRACT FROM AUTHOR]

Under the exponential trichotomy condition we study the Hyers-Ulam stability for the linear partial difference equation:... where An is a k × k matrix whose elements are sequences of n, Bn,m is a k × k matrix whose elements are double sequences of m, n and f: Rk - Rk is a vector function. We also investigate the Hyers-Ulam stability in the case where the matrices An, Bn,m and the vector function f = fn,m are constant. [ABSTRACT FROM AUTHOR]

This paper deals with the relativistic, quantized electromagnetic and Dirac field equations in the arena of discrete phase space and continuous time. The mathematical formulation involves partial difference equations. In the consequent relativistic quantum electrodynamics, the corresponding Feynman diagrams and S # -matrix elements are derived. In the special case of electron–electron scattering (Møller scattering), the explicit second-order element f S (2) # i is deduced. Moreover, assuming the slow motions for two external electrons, the approximation of f S (2) # i yields a divergence-free Coulomb potential. [ABSTRACT FROM AUTHOR]

In this paper we consider the following system of partial linear homogeneous difference equations: xs(i+2,j)+asxs+1(i+1,j+1)+bsxs(i,j+2)=0,s=1,2,...,n−1,xn(i+2,j)+anx1(i+1,j+1)+bnxn(i,j+2)=0and the system of partial linear nonhomogeneous difference equations: ys(i+2,j)+asys+1(i+1,j+1)+bsys(i,j+2)=fs(i,j),s=1,2,...,n−1,yn(i+2,j)+any1(i+1,j+1)+bnyn(i,j+2)=fn(i,j)where n=2,3,..., xs(0,j)=ϕs(j), j=2,3,..., xs(1,j)=ψs(j),j=1,2,... (resp. ys(0,j)=ϕs(j), j=2,3,..., ys(1,j)=ψs(j),j=1,2,...) for the first system (resp. for the second system); as, bs, are real constants; fs:N2→R are known functions; ϕs(j),ψs(j) are given sequences; and s=1,2,...,n and the domain of the solutions of the above systems are the sets Nm={(i,j),i+j=m}, m=2,3,.... More precisely, we find conditions so that every solution of the first system converges to 0 as i→∞ uniformly with respect to j. Moreover, we study the asymptotic stability of the trivial solution of the first system. In addition, under some conditions on fs, we prove that every solution of the second system is bounded, and finally, we find conditions on fs so that every solution of the second system converges to 0 as i→∞ uniformly with respect to j. [ABSTRACT FROM AUTHOR]

In this paper we present a recurrent procedure to solve an inversion problem for monic bivariate Krawtchouk polynomials written in vector column form, giving its solution explicitly. As a by-product, a general connection problem between two vector column of monic bivariate Krawtchouk families is also explicitly solved. Moreover, in the non monic case and also for Krawtchouk families, several expansion formulas are given, but for polynomials written in scalar form. [ABSTRACT FROM AUTHOR]

In this paper, we consider a boundary value problem (BVP) for second-order nonlinear partial difference equations on finite lattice domains. Some conditions are established that ensure existence and uniqueness of solutions to the BVP under consideration. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The aim of this note is to find the explicit solution of four-point linear partial difference equations with variable coefficients. Three-point and four-point cases in quadrant and cone domains are considered and the boundary conditions, allowing the existence and the uniqueness of the solution, are detailed. The adopted procedure is a double iteration and the solution consists of a product of series, whose number of factors may be reduced successively by means of the indefinite sum operator of the difference calculus. [ABSTRACT FROM AUTHOR]

We construct a discrete model for the time-independent harmonic oscillator Schrödinger partial differential equation and demonstrate that it can be separated into N ordinary difference equations for the case of N -cartesian space coordinates. [ABSTRACT FROM AUTHOR]

We consider the boundary value problem δ2my(k-m)=f(y(k),δ2y(k-1),…,δ2iy(k-1),…,δ2(m-1)y(k-(m-1))) k∊{a+1,…,b+1}, δ2iy(a+1-m)=δ2iy(b+1+m-2i)=0, 0≤i≤m-1, where m ≥ 1 and (−1)m f Rm → [0, ∞) is continuous. By using Amann and Leggett-Williams'' fixed-point theorems, we develop growth conditions on f so that the boundary value problem has triple positive symmetric solutions. The results obtained are then applied in the investigation of radial solutions for certain partial difference equation subject to Lidstone type conditions. [Copyright &y& Elsevier]

This paper summarizes a series of results on the oscillation and stability of partial delay difference equations. [Copyright &y& Elsevier]

Motivated by the classical one-dimensional diffusion equation, utkuxx = f(x,t), x∊R, t≥0,u(x,0)=∊(x), x∊R, we find the unique solution of the partial difference equation δtu(x,t)-kδ2x,xu(x-1,t)=f(x,t), x∊Z, t≥0,u(x,0)=∊(x), x∊Z, where k ∊ (0, 1). Our approach is to construct the Green''s function for the appropriate partial difference operator, and almost immediately we establish an existence and uniqueness result for the discrete nonhomogenous problem. [Copyright &y& Elsevier]

We present new oscillation tests for the PΔE a u (m + 1 , n) + b u (m , n + 1) − c u (m , n) + p (m , n) u (m − τ , n − σ) = 0 , m , n = 0 , 1 , ⋯ , where a, b, c ∈ (0, ∞) and τ, σ ∈ {0, 1, ⋅⋅⋅}, and { p (m, n)} ⊂ [0, ∞). Our main result improves some of the well-known results in the literature. We also present a numerical example, where all the previous results in the literature fail to deliver an answer. [ABSTRACT FROM AUTHOR]

Due to the applications in many fields, there is great interest in studying partial difference equations involving functions with two or more discrete variables. In this paper, we deal with the existence of infinitely many solutions for a partial discrete Dirichlet boundary value problem with the p-Laplacian by using critical point theory. Moreover, under appropriate assumptions on the nonlinear term, we determine open intervals of the parameter such that at least two positive solutions and an unbounded sequence of positive solutions are obtained by using the maximum principle. We also show two examples to illustrate our results. [ABSTRACT FROM AUTHOR]