Asymptotic Behavior of Linear Advanced Difference Equations

In previous chapters, we have learned to solve a wide range of differential equations. Primarily, our focus has been on linear differential equations: first-order linear equations, higher-order linear equations with constant coefficients, and systems of linear equations with constant coefficients. Indeed, we have learned through a variety of techniques that under the proviso that a differential equation or system is linear, we can almost always find a solution.

II.A - Exact Methods for ODEs

The investigations presented in this paper consist of two parts; the first offers a solution, in a qualified sense, of the general linear equation in finite differences; and the second gives an analysis of the general linear differential equation with rational factors, so far as concerns its solution in series. The author observes that there does not at present exist any general method of solving linear equations in finite differences of an order higher than the first; and that with reference to such equations of the first order, we obtain insufficient forms which are intelligible only when the independent variable is an integer. It is in this qualified sense that the solutions proposed in this paper are to be taken; so that the first part of these investigations may be considered as an extension of this form of solution from the general equation of the first order to the general equation of the n th order.

Time-splitting pseudo-spectral domain decomposition method for the soliton solutions of the one- and multi-dimensional nonlinear Schrödinger equations

A Direct Discrete-time IDA-PBC Design Method for a Class of Underactuated Hamiltonian Systems

In the present paper we propose four systems of linear Partial Differential Equations that can be deduced from the nonlinear Schrödinger equation for the propagation of light in optical fibers in the frame of the recently-proposed Combined Regular-Logarithmic Perturbation method. The unknown function in the Schrödinger equation is the optical field envelope; it is a complex-valued function. Following the Combined Regular-Logarithmic Perturbation method, proposed by Secondini, Forestieri and Menyuk, we look for complex solutions of the Schrödinger equation in the form of a perturbed continuous wave that relates three unknown real-valued functions. Since the Schrödinger equation is complex, we split it into two real equations, both in the three real unknowns. We linearize these two equations and add a third linear equation that relates the same three unknown quantities. We propose four different choices for the third equation, therefore we obtain four di¤erent real systems of linear Partial Di¤erential Equations and we analyze the corresponding systems of Ordinary Differential Equations for the Fourier transforms of the unknowns. One of the four systems we obtain is equivalent to that studied by the quoted authors. We add to it other three choices that could be useful to model di¤erent situations. Again, we consider the real part of the Ordinary Differential Equations and we present solutions in recursive form. We also suggest solutions for the complex-valued Fourier transforms by using Bessel functions.

A definite theory exists for solving linear systems of ordinary differential equations with constant coefficients. “This theory is essentially a branch of linear algebra, and allows us to solve all autonomous linear equations”, ([4], p. 95). Furthermore, “In view of the fact that the solution of these equations does not, in principle, present any great difficulties, they are often considered to be of no great interest for theory, and in textbooks they are usually relegated to the position of simple exercises appended to the general theory of linear equations”, ([81], p. 41). Modern textbooks [8, 21, 26, 44, 45, 46, 50, 90] give a detailed and rigorous treatment of the standard methods available for linear problems, supplemented with numerous mechanical, electrical, and engineering applications. The voluminous literature accumulated on the general properties of the solutions of linear differential equations must therefore no doubt leave the impression that no further contribution could be made to the standard normal linear dynamic system with real and constant coefficients. This impression, however, will fade rather quickly as we go on. Paradoxically, some fundamental aspects of this simple dynamic system are still unnoticed.

In several applications in science and engineering, different types of matrix problems emerge from the discretization of partial differential equations. This thesis is devoted to the development of new algorithms to solve this kind of problems. In particular, when the matrices involved are sparse and non-symmetric. The new algorithms are based on the Induced Dimension Reduction method [IDR(s)]. IDR(s) is a Krylov subspace method originally proposed in 2008 to solve systems of linear equations. IDR(s) has received considerable attention due to its stable and fast convergence. It is, therefore, natural to ask if it is possible to extend IDR(s) to solve other matrix problems, and if so, to compare those extensions with other well-established methods. This work aims to answer these questions. The main matrix problems considered in this dissertation are: the standard eigenvalue problem, the quadratic eigenvalue problem, the solution of systems of linear equations, the solution of sequences of systems of linear equations, and linear matrix equations. We focus on examples that arise from the discretization of partial differential equations.

Stochastic partial differential equations constitute a relatively new subject. There have been a number of interesting papers in the last few years, but the field is still in its infancy if compared with the highly developed theory of stochastic ordinary differential equations. In this paper we report on a study of a non linear heat equation in a finite interval of space subject to a white noise forcing term. Due to the non linearity the equation, without the forcing term, exhibits several equilibrium configurations two of which are stable, actually asymptotically stable. The solution of the complete forced equation is a stochastic process in space and time with continuous sample paths whose behaviour we study in the limit of small noise. We obtain lower and upper bounds for the probability of large fluctuations and then apply our estimates to the calculation of the transition probability between the stable configurations (tunneling).

Elimination of fast variables

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.

Second-order exact ensemble averaged equation for linear stochastic differential equations with multiplicative randomness and random forcing is obtained by using the cumulant expansion ensemble averaging method and by taking the time dependent sure part of the multiplicative operator into account. It is shown that the satisfaction of the commutativity and the reversibility requirements proposed earlier for linear stochastic differential equations without forcing are necessary for the linear stochastic differential equations with forcing when the cumulant expansion ensemble averaging method is used. It is shown that the applicability of the operator equality, which is used for the separation of operators in the literature, is also subjected to the satisfaction of the commutativity and the reversibility requirements. The van Kampen’s lemma, which is proposed for the analysis of nonlinear stochastic differential equations, is modified in order to make the probability density function obtained through the lemma depend on the forcing terms too. The second-order exact ensemble averaged equation for linear stochastic differential equations with multiplicative randomness and random forcing is also obtained by using the modified van Kampen’s lemma in order to validate the correctness of the modified lemma. Second-order exact ensemble averaged equation for one dimensional convection diffusion equation with reaction and source is obtained by using the cumulant expansion ensemble averaging method. It is shown that the van Kampen’s lemma can yield the cumulant expansion ensemble averaging result for linear stochastic differential equations when the lemma is applied to the interaction representation of the governing differential equation. It is found that the ensemble averaged equations given for one the dimensional convection diffusion equation with reaction and source in the literature obtained by applying the lemma to the original differential equation are restricted with small sure part of multiplicative operator. Second-order exact differential equations for the evolution of the probability density function for the one dimensional convection diffusion equation with reaction and source and one dimensional nonlinear overland flow equation with source are obtained by using the modified van Kampen’s lemma. The equation for the evolution of the probability density function for one dimensional nonlinear overland flow equation with source given in the literature is found to be not second-order exact. It is found that the differential equations for the evolution of the probability density functions for various hydrological processes given in the literature are not second-order exact. The significance of the new terms found due to the second-order exact ensemble averaging performed on the one dimensional convection diffusion equation with reaction and source and during the application of the van Kampen’s lemma to the one dimensional nonlinear overland flow equation with source is investigated.

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.

Chapter 2 - Simultaneous linear algebraic equations

The partial Noether operators and first integrals of a general system of two linear second-order ordinary differential equations (ODEs) with variable coefficients are studied by means of a partial Lagrangian. The canonical form for the general system of two second-order ordinary differential equations is invoked and all cases of this system are discussed with respect to partial Noether operators. We also tabulate the results for the special case b(x) = c(x) of the system which was considered elsewhere using a Lagrangian and a partial Lagrangian. The first integrals are obtained explicitly by exploiting a Noether-like theorem with the help of partial Noether operators. This study gives a new way to construct first integrals for systems without a variational principle as not all linear equations have a Lagrangian. Physical applications to conservative and oscillator mechanical systems are given.