Theory Of Difference Equations Numerical Methods And Applications

Abstract. Recently the creation matrix, intimately related to the Pascal matrix and its generalizations, has been used to develop matrix representations of special polynomials, in particular Appell polynomials. In this paper we describe a matrix approach to polynomials in several hypercomplex variables based on special block matrices whose structures simulate the creation matrix and the Pascal matrix. We apply the approach to hypercomplex Laguerre polynomials, although it can be used for other Appell sequences, too.

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“…The matrix H is a block version of the creation matrix H and verifies similar properties to that matrix [9], namely…”

Section: Particular Block Matricesmentioning

confidence: 99%

Laguerre Polynomials in Several Hypercomplex Variables and Their Matrix Representation

Malonek

Tomaz

2011

Computational Science and Its Applications - ICCSA 2011

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“…In this subsection we review a few facts and theorems from theory of difference equations. For more details see [1,11,26] and references therein.…”

Section: Theory Of Difference Equationsmentioning

confidence: 99%

A Theory of Anharmonic Lattice Statics for Analysis of Defective Crystals

2006

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This paper develops a theory of anharmonic lattice statics for the analysis of defective complex lattices. This theory differs from the classical treatments of defects in lattice statics in that it does not rely on harmonic and homogenous force constants. Instead, it starts with an interatomic potential, possibly with infinite range as appropriate for situations with electrostatics, and calculates the equilibrium states of defects. In particular, the present theory accounts for the differences in the force constants near defects and in the bulk. The present formulation reduces the analysis of defective crystals to the solution of a system of nonlinear difference equations with appropriate boundary conditions. A harmonic problem is obtained by linearizing the nonlinear equations, and a method for obtaining analytical solutions is described in situations where one can exploit symmetry. It is then extended to the anharmonic problem using modified Newton-Raphson iteration. The method is demonstrated for model problems motivated by domain walls in ferroelectric materials.

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“…Due to its wide application in many fields such as science, economics, neural networks, ecology and cybernetics, the theory of nonlinear difference equations has been widely studied since the 1970s; see, for example, [1,6,21,22]. At the same time, boundary value problems (BVPs) of difference equations have received much attention from many authors; see [2-5, 7-13, 15, 18-20, 25-31] and the references therein.…”

Section: Introductionmentioning

confidence: 99%