An approach to solving linear constant coefficient difference and differential equations

Linear constant coefficient differential (or difference) equations

It is well known that purely sinusoidal signals satisfy a linear second-order constant coefficient differential equation. It is also well known that a broad class of orthogonal special functions such as the Legendre and Hermite polynomials satisfy the second-order Sturm-Liouville differential equation. Both sinusoidal and AM-FM models have been used for analysis and synthesis of speech signals. In this paper, we present a Sturm-Liouville differential and difference equation approach to both continuous and discrete time frequency modulation. Orthogonal modes of frequency modulation that are not distorted by the Sturm-Liouville operator are described.

Design of discrete-time matrix all-pass filters using subspace Nevanlinna pick interpolation

The method of undetermined coefficients is used to solve constant coefficient nonhomogeneous differential equations whose forcing function is itself the solution of a homogeneous constant coefficient differential equation. In this paper, we show that the classical methods for tackling constant coefficient equations, including the method of undetermined coefficients, generalize to much wider class linear differential equations which, for example, include Cauchy-Euler type equations. This general method includes an explicit construction of the fundamental solution sets of such equations. We also briefly consider where this method can be applied by producing the most general second and third order differential equations that are polynomial in a first order differential operator. In addition, we provide a number of constant coefficient, Cauchy-Euler, and novel examples.

Having developed classes that underpin linear algebra operations in an earlier chapter, we now demonstrate how to construct object-oriented libraries for scientific computing applications that utilise the functionality of these classes. We use the specific example of developing a library that uses the finite difference method to solve boundary value, second order, constant coefficient differential equations. We begin by considering ordinary differential equations. We state the theory that underpins the finite difference method for these equations, and then describe the classes that we will develop to allow this theory to be applied. We then briefly discuss how to develop a library for the solution of Poisson’s equation, an elliptic partial differential equation, in two spatial dimensions.

Addition theorems for solutions to linear homogeneous constant coefficient ordinary differential equations

Application of commutator calculus to the study of linear impulsive systems

This chapter considers systems of n first-order linear constant-coefficient ordinary differential equations, which arise from nth-order differential equations or from systems of coupled differential equations. As becomes readily apparent, the theoretical basis for solving such systems relies heavily upon matrix algebra theory. Appendix B contains a review of some of the more important concepts from linear algebra.

The aim of the present article is to investigate of solutions stability of linear autonomous differential equations with retarded argument. The investigation of stability can be reduced to the root location problem for the characteristic equation. For the linear differential equation with several delays it is obtained the necessary and sufficient conditions, for all the roots of the characteristic equation to have negative real part (and hence the zero solution to be asymptotically stable). For the scalar delay differential equation stability domains in the parameter plane are obtained. We investigate the boundedness conditions and construct a domain of stability for linear autonomous differential equation with several delays. We use D-partition method, argument principle and numerical methods to construct of stability domains. In this article, we investigate the solutions stability of linear autonomous differential equations with several delays. It is obtained the necessary and sufficient conditions, for all the roots of the characteristic equation to have negative real part. We investigate the boundedness conditions with the help of argument principle and construct a domain of stability for linear autonomous differential equation with two delays. We use D-partition method, argument principle and numerical methods to construct of stability domains for linear autonomous differential equation with two delays. In the D-partition method, we look for parameter values for which the characteristic equation has at least one zero on the imaginary axis. We consider some examples of equations with two delays. In these cases, the stability domains are limited by two straight lines and a finite number of parametrically defined curves.

This document describes how one can derive the solutions to a linear constant coefficient homogeneous differential equation with repeated roots in the characteristic equation with Abel's Theorem. y (n) (t) + p1(t)y (n−1) (t) + p2(t)y (n−2) (t) + · · · + pn−1(t)y ' (t) + pn(t)y(t) = 0 (1.1) is encountered by every differential equations student. We most often study second order equations, the case when n = 2, due to Newton's laws of motion and with ease one can generalize the theory to higher order. We consider the case when the coefficient functions pi(t) are constant. Under these conditions, one makes the ansatz y = e rt which transforms (1.1) into an algebraic equation, which is more easily solved. In particular, if pi = ai, with the above ansatz, we have the characteristic equation r n + a1r n−1 + a2r n−2 + · · · + an−1r + an = 0. (1.2) By factoring out the e rt , one finds that solving an algebraic equation yields solutions to the original differential equation. Most textbooks first handle distinct real roots and later come back to discuss when r = r1 is a repeated solution to (1.2). We show an approach that relies on Abel's Theorem and the idea of reduction of order can constructively establish that the repeated root r = r1 with multiplicity m gives rise to m linearly independent solutions of the form yi(t) = t i−1 e r1t for 1 ≤ i ≤ m.

We present some interesting facts connected with the following second-order difference equation: xn+2−qnxn=fn,n∈N0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$x_{n+2}-q_{n}x_{n}=f_{n},\\quad n\\in \\mathbb{N}_{0}, $$\\end{document} where (q_{n})_{ninmathbb{N}_{0}} and (f_{n})_{ninmathbb {N}_{0}} are given sequences of numbers. We give some sufficient conditions for the existence of a unique bounded solution to the difference equation and present an elegant proof based on a combination of theory of linear difference equations and the Banach fixed point theorem. We also deal with the equation by using theory of solvability of difference equations. A global convergence result of solutions to a linear first-order difference equation is given. Some comments on an abstract version of the linear first-order difference equation are also given.

LetP(D) be a linear partial differential operator onR n with constant coefficients and letV be a closed cone ofR n . Some cases are discussed where or, more generally, where, for ϱ>σ, Here Γ(σ) (Ω), σ≥1, Ω an open set ofR n , denotes the Gevrey space of type σ and γ ′ (R n ) the space of Beurling ultradistributions of type ϱ.

Neighborhood global learning based flower pollination algorithm and its application to unmanned aerial vehicle path planning

In this paper we consider periodic functional differential and integral equations. Massera proved that the existence of a bounded solution implies the existence of a periodic solution for linear (inhomogeneous) ordinary differential equations. Chow proved a similar result for linear functional differential equations with finite delay. We give a new proof for Chow's result that works for linear functional differential equations with infinite delay and also for integral equations. We also give examples that do not satisfy uniform boundedness or uniform ultimate boundedness, the conditions frequently used to prove that there is a periodic solution.

* Preface * Background Material * Computational Algebraic Analysis for Systems of Linear Constant Coefficients Differential Equations * The Cauchy-Fueter Systems and its Variations * Special First Order Systems in Clifford Analysis * Some First Order Linear Operators in Physics * Open Problems and Avenues for Further Research * References * Index