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Complex physical processes, which could evolve in both spatial and temporal dimensions and be represented by partial difference equations, could also operate in a repetitive mode with iterative learning methods as suitable control laws. For these three-dimensional systems (of the spatial, temporal, and iterative dimensions), the stability in the iterative direction is critical for many applications, which can be analyzed and synthesized under the proposed concept of iterative dissipativity. The definition of iterative dissipativity, which is first introduced in this paper, encapsulates the dominant information in both the spatial and temporal dimensions, while also placing a particular emphasis on the iteration improvement. This property allows for the derivation of sufficient conditions for asymptotic stability in the iteration direction, which are represented by linear matrix inequality criteria that can be readily solved. Performance in both the spatial and temporal dimensions can also be satisfied under this iterative dissipativity concept, even in absence of real-time feedback. Moreover, the optimization solutions of the control parameters can be determined. Finally, a thermal process and a numeric example are presented to illustrate the effectiveness of the proposed iteratively dissipative learning control approach. [ABSTRACT FROM AUTHOR]

In this paper we discuss an example of classical integrable equation with rather unusual ‘B’-type Kadomtsev–Petviashvili (KP) soliton hierarchy.

Complex physical processes, which could evolve in both spatial and temporal dimensions and be represented by partial difference equations, could also operate in a repetitive mode with iterative learning methods as suitable control laws. For these three-dimensional systems (of the spatial, temporal, and iterative dimensions), the stability in the iterative direction is critical for many applications, which can be analyzed and synthesized under the proposed concept of iterative dissipativity. The definition of iterative dissipativity, which is first introduced in this paper, encapsulates the dominant information in both the spatial and temporal dimensions, while also placing a particular emphasis on the iteration improvement. This property allows for the derivation of sufficient conditions for asymptotic stability in the iteration direction, which are represented by linear matrix inequality criteria that can be readily solved. Performance in both the spatial and temporal dimensions can also be satisfied under this iterative dissipativity concept, even in absence of real-time feedback. Moreover, the optimization solutions of the control parameters can be determined. Finally, a thermal process and a numeric example are presented to illustrate the effectiveness of the proposed iteratively dissipative learning control approach.

In this paper, the consensus tracking problem for the linear and nonlinear partial difference multi-agent systems with switching communication topology and control delay is investigated. Based on relative local measurements of neighboring followers, while considering spatio-temporal discretization and initial state deviation, a discrete distributed consensus protocol with initial value learning is designed for each agent via D-type iterative learning approach. Through rigorous mathematical theoretical analysis, the necessary and sufficient conditions are obtained. Under the switching of the communication topology, these conditions ensure that the consensus tracking control of the MASs can be solved. After applying the designed protocol, in the sense of the L 2 norm and along the positive direction of the iteration axis, the consensus tracking error between any two agents can converge to zero. Finally, some simulation examples are used to demonstrate the validity of the protocol and theoretical results. • A novel partial difference MASs dynamic model is established to describe the discrete spatio-temporal evolution behavior. • A new distributed iterative learning approach with initial value learning for partial difference MASs is given. • The proposed control approach can solve the problem of switching topology and control delay simultaneously. [ABSTRACT FROM AUTHOR]

Under the exponential trichotomy condition we study the Hyers–Ulam stability for the linear partial difference equation: \[ x_{n+1,m}=A_nx_{n,m}+B_{n,m}x_{n,m+1}+f(x_{n,m}),\qquad n,m\in \mathbb{Z} \] where $A_n$ is a $k\times k$ matrix whose elements are sequences of $n$, $B_{n,m}$ is a $k\times k$ matrix whose elements are double sequences of $m,n$ and $f:\mathbb{R}^k\rightarrow \mathbb{R}^k$ is a vector function. We also investigate the Hyers–Ulam stability in the case where the matrices $A_n, B_{n,m}$ and the vector function $f=f_{n,m}$ are constant.

Apartial discrete Dirichlet boundary value problem involving mean curvature operator is concerned in this paper. Under proper assumptions on the nonlinear term, we obtain some feasible conditions on the existence of multiple solutions by the method of critical point theory. We further separately determine open intervals of the parameter to attain at least two positive solutions and an unbounded sequence of positive solutions with the help of the maximum principle.

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]

In this paper, iterative learning control (ILC) is employed in discrete spatial-temporal parabolic distributed parameter systems (DPSs), where the trial lengths vary randomly. A distributed ILC strategy is proposed, in which containing spatial variable, utilizes all past tracking information to improve current performance. Through rigorous theoretical analysis, the convergence of the system output error is proved under mathematical expectation along the iteration axis. Finally, the proposed method is applied to numerical simulation to illustrate its effectiveness.

Abstract In this paper, we present an existence criterion for multiple positive solutions of nonlinear neutral delay partial difference equations. Such equations can be regarded as a discrete analog of neutral delay partial differential equations. Our main result relies on fixed point index theory. An example is constructed to show the applicability of the obtained result.

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 article reveals the fact that considerable study time invested into acquisition of proficiency in infinitesimal calculus leaves behind only the ability to tackle problems of order four or less. Today's problems reaching into orders of thousands are being handled by clever computer‐based algorithms seldom addressed in high school or college. Yet, the calculus teaching will persist as long as physics and some other natural science teachers start their lectures with a differential equation. An alternative method, preparing high school students for serious problem‐solving abilities, is presented with examples. References provided, many published by this author, do apply the method to a broad range of natural sciences. [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 mainly propose improvements of the logarithmic difference lemma for meromorphic functions in several complex variables and then investigate meromorphic solutions of partial difference equations from the viewpoint of Nevanlinna theory. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a combined form of the discrete Sumudu transform method with the discrete homotopy perturbation method to solve linear and nonlinear partial difference equations. This method is called the discrete homotopy perturbation Sumudu transform method(DHPSTM). The results reveal that the introduced method is very efficient, simple and can be applied to other linear and nonlinear difference equations. The nonlinear terms can be easily handled by use of He's polynomials. [ABSTRACT FROM AUTHOR]

In this paper we give the closed form expressions of some higher order nonlinear rational partial difference equations in the form Xn,m=Xn−r,m−rΨ+∏i=1rXn−i,m−i where n,m∈N and the initial values Xn, t, Xt,m−r are real numbers with t∈{0,−1,−2,…,−r+1} such that ∏j=0r−1Xj−r+1,i+j−r+1≠−Ψ and ∏j=0r−1Xi+j−r+2,j−r+1≠−Ψ, i∈N0. We will use a new method to prove the results by using what we call ‘piecewise double mathematical induction’ which we introduce here for the first time as a generalization of many types of mathematical induction. As a direct consequences, we investigate and conclude the explicit solutions of some higher order ordinary difference equations.

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.

Partial Differential Equations (PDEs)-based morphology offers a wide range of continuous operators to address various image processing problems. Most of these operators are formulated as Hamilton-Jacobi equations or curve evolution level set and morphological flows. In our previous works, we have proposed a simple method to solve PDEs on point clouds using the framework of PdEs (Partial difference Equations) on graphs. In this paper, we propose to apply a large class of morphological-based operators on graphs for processing raw 3D point clouds and extend their applications for the processing of colored point clouds of geo-informatics 3D data. Through illustrations, we show that this simple framework can be used in the resolution of many applications for geo-informatics purposes. [ABSTRACT FROM AUTHOR]

In this paper we give the closed form expressions of some higher order nonlinear rational partial difference equations in the form X n , m = X n − r , m − r Ψ + ∏ i = 1 r X n − i , m − i where n , m ∈ N and the initial values X n, t , X t , m − r are real numbers with t ∈ { 0 , − 1 , − 2 , … , − r + 1 } such that ∏ j = 0 r − 1 X j − r + 1 , i + j − r + 1 ≠ − Ψ and ∏ j = 0 r − 1 X i + j − r + 2 , j − r + 1 ≠ − Ψ , i ∈ N 0 . We will use a new method to prove the results by using what we call ‘piecewise double mathematical induction’ which we introduce here for the first time as a generalization of many types of mathematical induction. As a direct consequences, we investigate and conclude the explicit solutions of some higher order ordinary difference equations. [ABSTRACT FROM AUTHOR]

In this paper we propose an adaptation of the $$\infty $$ -Poisson equation on weighted graphs, and propose a finer expression of the $$\infty $$ -Laplace operator with gradient terms on weighted graphs, by making the link with the biased version of the tug-of-war game. Using this formulation, we propose a hybrid $$\infty $$ -Poisson Hamilton-Jacobi equation, and we show the link between this version of the $$\infty $$ -Poisson equation and the adaptation of the eikonal equation on weighted graphs. Our motivation is to use this extension to compute distances on any discrete data that can be represented as a weighted graph. Through experiments and illustrations, we show that this formulation can be used in the resolution of many applications in image, 3D point clouds, and high-dimensional data processing using a single framework. [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 a recent papers,some authors applied Nevanlinna theory to prove some results on complex difference equations reminiscent of the classical Malmquist theorem in complex differential equations. We will mainly investigate Malmquist theorem of a type of systems of complex partial difference equations on C, improvements and extensions of such results are presented in this paper. [ABSTRACT FROM AUTHOR]

In this paper we introduce a working generalization of the theory of Gröbner bases for algebras of partial difference polynomials with constant coefficients. One obtains symbolic (formal) computation for systems of linear or non-linear partial difference equations arising, for instance, as discrete models or by the discretization of systems of differential equations. From an algebraic viewpoint, the algebras of partial difference polynomials are free objects in the category of commutative algebras endowed with the action by endomorphisms of a monoid isomorphic to ℕr. Then, the investigation of Gröbner bases in this context contributes also to the current research trend consisting in studying polynomial rings under the action of suitable symmetries that are compatible with effective methods. Since the algebras of difference polynomials are not Noetherian, we propose in this paper a theory for grading them that provides a Noetherian subalgebra filtration. This implies that the variants of Buchberger's algorithm we developed for difference ideals terminate in the finitely generated graded case when truncated up to some degree. Moreover, even in the non-graded case, we provide criterions for certifying completeness of eventually finite Gröbner bases when they are computed within sufficiently large bounded degrees. We generalize also the concepts of homogenization and saturation, and related algorithms, to the context of difference ideals. The feasibility of the proposed methods is shown by an implementation in Maple that is the first to provide computations for systems of non-linear partial difference equations. We make use of a test set based on the discretization of concrete systems of non-linear partial differential equations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the linear delay partial difference equation Am,n=∑i=1upiAm−ki,n−li+∑j=1vqjAm+τj,n+σj, where pi and qj are r×r matrices, Am,n=(am,n1,am,n2, …,am,nr)T, ki, li , τj and σj are nonnegative integers, u and v are positive integers. Sufficient and necessary conditions for all solutions of this equation to be oscillatory componentwise are obtained.

Partial Differential Equations (PDEs)-based morphology offers a wide range of continuous operators to address various image processing problems. Most of these operators are formulated as Hamilton–Jacobi equations or curve evolution level set and morphological flows. In our previous works, we have proposed a simple method to solve PDEs on point clouds using the framework of PdEs (Partial difference Equations) on graphs. In this paper, we propose to apply a large class of morphological-based operators on graphs for processing raw 3D point clouds and extend their applications for the processing of colored point clouds of geo-informatics 3D data. Through illustrations, we show that this simple framework can be used in the resolution of many applications for geo-informatics purposes.

A true Tree Calculus is being developed to make a joint study of the two statistics 'eoc' (end of minimal chain) and 'pom' (parent of maximum leaf) on the set of secant trees. Their joint distribution restricted to the set {eoc-pom ≤ 1} is shown to satisfy two partial difference equation systems, to be symmetric and to be expressed in the form of an explicit three-variable generating function. [ABSTRACT FROM AUTHOR]

Following Poupard's study of strictly ordered binary trees with respect to two parameters, namely, ‘end of minimal chain’ and ‘parent of maximum leaf’ a true tree calculus is being developed to solve a partial difference equation system and then make a joint study of those two statistics. Their joint distribution is shown to be symmetric and to be expressed in the form of an explicit three-variable generating function. [ABSTRACT FROM PUBLISHER]