Search

This paper focuses on chaotification problems for first-order partial difference equations. Two chaotification schemes of the difference equations via sine functions are established, and all the co...

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 l...

We study the existence and uniqueness for discrete Neumann and periodic problems. We consider both ordinary and partial difference equations involving sublinear perturbations. All the proofs are based on reformulating these discrete problems as a general singular algebraic system. Firstly, we use variational techniques (specifically, the Saddle Point Theorem) and prove the existence result based on a type of Landesman–Lazer condition. Then we show that for a certain class of bounded nonlinearities this condition is even necessary and therefore, we specify also the cases in which there does not exist any solution. Finally, the uniqueness is discussed.

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.

This paper is concerned with chaos in 2D first-order partial difference equations with finite or infinite system size. By reformulating them into certain ordinary difference equations, several criteria of chaos are established, where some are induced by chaos in the corresponding scalar ordinary difference equation, and the others are implied by nondegenerate and regular snap-back repellers and regular snap-back repellers, respectively. Two illustrative examples are discussed with computer simulations.

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.

In this paper it will be shown that the method of separation of variables for partial difference equations (as for instance described by Mickens [1] and Agarwal [2] ) can be applied to a much larger class of problems as is generally assumed. To show how the method should be applied two problems are treated and solved. In [1] and [2] these two problems were considered and claimed to be not solvable by applying the method of separation of variables.

Sturmian comparison theorems are derived for a class of discrete hyperbolic type equations.

A penalty function method is applied to the determination of optimal controls for systems described by partial difference equations. The use of this penalty function method known as the ϵ-technique allows us to incorporate the dynamical constraints of the distributed parameter system into a new objective function. The conditions for the solution which minimizes the new objective function and the interrelationships between this solution and the solution which minimizes the original objective function are given. These interrelationships also suggest computational schemes for the synthesis of optimal controls for linear and non-linear distributed systems.