Differential Equations and Dynamical Systems Classnotes for Math 645 University of Massachusetts V3
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Bessel's Differential Equation Mathematical Physics
R. I. Badran Bessel's Differential Equation Mathematical Physics Bessel's Differential Equation: 2 Recalling the DE y n y 0 which has a sinusoidal solution (i.e. sin nx and cos nx) and knowing that these solutions can be treated as power series, we can find a solution to the Bessel's DE which is written as: 2 2 2 x y xy (x p )y 0 . The solution is represented by a series. This series very much look like a damped sine or cosine. It is called a Bessel function. To solve the Bessel' DE, we apply the Frobenius method by mn assuming a series solution of the form y an x . n0 Substitute this solution into the DE equation and after some mathematical steps you may find that the indicial equation is 2 2 m p 0 where m= p. Also you may get a1=0 and hence all odd a's are zero as well. The recursion relation can be found as a a n n2 (n m 2)2 p 2 with (n=0, 1, 2, 3, ……etc.). Case (1): m= p (seeking the first solution of Bessel DE) We get the solution 2 4 p x x y a x 1 2(2p 2) 2 4(2p 2)(2p 4) This can be rewritten as: x p2n J (x) y a (1)n p 2n n0 2 n!( p n)! 1 Put a 2 p p! The Bessel function has the factorial form x p2n J (x) (1)n p 2n p , n0 n!( p n)!2 R. -
The Standard Form of a Differential Equation
Fall 06 The Standard Form of a Differential Equation Format required to solve a differential equation or a system of differential equations using one of the command-line differential equation solvers such as rkfixed, Rkadapt, Radau, Stiffb, Stiffr or Bulstoer. For a numerical routine to solve a differential equation (DE), we must somehow pass the differential equation as an argument to the solver routine. A standard form for all DEs will allow us to do this. Basic idea: get rid of any second, third, fourth, etc. derivatives that appear, leaving only first derivatives. Example 1 2 d d 5 yx()+3 ⋅ yx()−5yx ⋅ () 4x⋅ 2 dx dx This DE contains a second derivative. How do we write a second derivative as a first derivative? A second derivative is a first derivative of a first derivative. d2 d d yx() yx() 2 dx dxxd Step 1: STANDARDIZATION d Let's define two functions y0(x) and y1(x) as y0()x yx() and y1()x y0()x dx Then this differential equation can be written as d 5 y1()x +3y ⋅ 1()x −5y ⋅ 0()x 4x dx This DE contains 2 functions instead of one, but there is a strong relationship between these two functions d y1()x y0()x dx So, the original DE is now a system of two DEs, d d 5 y1()x y0()x and y1()x +3y ⋅ 1()x −5y ⋅ 0()x 4x⋅ dx dx The convention is to write these equations with the derivatives alone on the left-hand side. d y0()x y1()x dx This is the first step in the standardization process. -
Exponential Stability for the Resonant D’Alembert Model of Celestial Mechanics
DISCRETE AND CONTINUOUS Website: http://aimSciences.org DYNAMICAL SYSTEMS Volume 12, Number 4, April 2005 pp. 569{594 EXPONENTIAL STABILITY FOR THE RESONANT D'ALEMBERT MODEL OF CELESTIAL MECHANICS Luca Biasco and Luigi Chierchia Dipartimento di Matematica Universit`a\Roma Tre" Largo S. L. Murialdo 1, 00146 Roma (Italy) (Communicated by Carlangelo Liverani) Abstract. We consider the classical D'Alembert Hamiltonian model for a rotationally symmetric planet revolving on Keplerian ellipse around a ¯xed star in an almost exact \day/year" resonance and prove that, notwithstanding proper degeneracies, the system is stable for exponentially long times, provided the oblateness and the eccentricity are suitably small. 1. Introduction. Perturbative techniques are a basic tool for a deep understand- ing of the long time behavior of conservative Dynamical Systems. Such techniques, which include the so{called KAM and Nekhoroshev theories (compare [1] for gen- eralities), have, by now, reached a high degree of sophistication and have been applied to a great corpus of di®erent situations (including in¯nite dimensional sys- tems and PDE's). KAM and Nekhoroshev techniques work under suitable non{ degeneracy assumptions (e.g, invertibility of the frequency map for KAM or steep- ness for Nekhoroshev: see [1]); however such assumptions are strongly violated exactly in those typical examples of Celestial Mechanics, which { as well known { were the main motivation for the Dynamical System investigations of Poincar¶e, Birkho®, Siegel, Kolmogorov, Arnold, Moser,... In this paper we shall consider the day/year (or spin/orbit) resonant planetary D'Alembert model (see [10]) and will address the problem of the long time stability for such a model. -
Adding Integral Action for Open-Loop Exponentially Stable Semigroups and Application to Boundary Control of PDE Systems A
1 Adding integral action for open-loop exponentially stable semigroups and application to boundary control of PDE systems A. Terrand-Jeanne, V. Andrieu, V. Dos Santos Martins, C.-Z. Xu Abstract—The paper deals with output feedback stabilization of boundary regulation is considered for a general class of of exponentially stable systems by an integral controller. We hyperbolic PDE systems in Section III. The proof of the propose appropriate Lyapunov functionals to prove exponential theorem obtained in the context of hyperbolic systems is given stability of the closed-loop system. An example of parabolic PDE (partial differential equation) systems and an example of in Section IV. hyperbolic systems are worked out to show how exponentially This paper is an extended version of the paper presented in stabilizing integral controllers are designed. The proof is based [16]. Compare to this preliminary version all proofs are given on a novel Lyapunov functional construction which employs the and moreover more general classes of hyperbolic systems are forwarding techniques. considered. Notation: subscripts t, s, tt, ::: denote the first or second derivative w.r.t. the variable t or s. For an integer n, I I. INTRODUCTION dn is the identity matrix in Rn×n. Given an operator A over a ∗ The use of integral action to achieve output regulation and Hilbert space, A denotes the adjoint operator. Dn is the set cancel constant disturbances for infinite dimensional systems of diagonal matrix in Rn×n. has been initiated by S. Pojohlainen in [12]. It has been extended in a series of papers by the same author (see [13] II. -
CORSO ESTIVO DI MATEMATICA Differential Equations Of
CORSO ESTIVO DI MATEMATICA Differential Equations of Mathematical Physics G. Sweers http://go.to/sweers or http://fa.its.tudelft.nl/ sweers ∼ Perugia, July 28 - August 29, 2003 It is better to have failed and tried, To kick the groom and kiss the bride, Than not to try and stand aside, Sparing the coal as well as the guide. John O’Mill ii Contents 1Frommodelstodifferential equations 1 1.1Laundryonaline........................... 1 1.1.1 Alinearmodel........................ 1 1.1.2 Anonlinearmodel...................... 3 1.1.3 Comparingbothmodels................... 4 1.2Flowthroughareaandmore2d................... 5 1.3Problemsinvolvingtime....................... 11 1.3.1 Waveequation........................ 11 1.3.2 Heatequation......................... 12 1.4 Differentialequationsfromcalculusofvariations......... 15 1.5 Mathematical solutions are not always physically relevant . 19 2 Spaces, Traces and Imbeddings 23 2.1Functionspaces............................ 23 2.1.1 Hölderspaces......................... 23 2.1.2 Sobolevspaces........................ 24 2.2Restrictingandextending...................... 29 2.3Traces................................. 34 1,p 2.4 Zero trace and W0 (Ω) ....................... 36 2.5Gagliardo,Nirenberg,SobolevandMorrey............. 38 3 Some new and old solution methods I 43 3.1Directmethodsinthecalculusofvariations............ 43 3.2 Solutions in flavours......................... 48 3.3PreliminariesforCauchy-Kowalevski................ 52 3.3.1 Ordinary differentialequations............... 52 3.3.2 Partial differentialequations............... -
Differential Equations and Linear Algebra
Chapter 1 First Order Equations 1.1 Four Examples : Linear versus Nonlinear A first order differential equation connects a function y.t/ to its derivative dy=dt. That rate of change in y is decided by y itself (and possibly also by the time t). Here are four examples. Example 1 is the most important differential equation of all. dy dy dy dy 1/ y 2/ y 3/ 2ty 4/ y2 dt D dt D dt D dt D Those examples illustrate three linear differential equations (1, 2, and 3) and a nonlinear differential equation. The unknown function y.t/ is squared in Example 4. The derivative y or y or 2ty is proportional to the function y in Examples 1, 2, 3. The graph of dy=dt versus y becomes a parabola in Example 4, because of y2. It is true that t multiplies y in Example 3. That equation is still linear in y and dy=dt. It has a variable coefficient 2t, changing with time. Examples 1 and 2 have constant coefficient (the coefficients of y are 1 and 1). Solutions to the Four Examples We can write down a solution to each example. This will be one solution but it is not the complete solution, because each equation has a family of solutions. Eventually there will be a constant C in the complete solution. This number C is decided by the starting value of y at t 0, exactly as in ordinary integration. The integral of f.t/ solves the simplest differentialD equation of all, with y.0/ C : D dy t 5/ f.t/ The complete solution is y.t/ f.s/ds C : dt D D C Z0 1 2 Chapter 1. -
Exponential Stability of Distributed Parameter Systems Governed by Symmetric Hyperbolic Partial Differential Equations Using Lyapunov’S Second Method ∗
ESAIM: COCV 15 (2009) 403–425 ESAIM: Control, Optimisation and Calculus of Variations DOI: 10.1051/cocv:2008033 www.esaim-cocv.org EXPONENTIAL STABILITY OF DISTRIBUTED PARAMETER SYSTEMS GOVERNED BY SYMMETRIC HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS USING LYAPUNOV’S SECOND METHOD ∗ Abdoua Tchousso1, 2, Thibaut Besson1 and Cheng-Zhong Xu1 Abstract. In this paper we study asymptotic behaviour of distributed parameter systems governed by partial differential equations (abbreviated to PDE). We first review some recently developed results on the stability analysis of PDE systems by Lyapunov’s second method. On constructing Lyapunov functionals we prove next an asymptotic exponential stability result for a class of symmetric hyper- bolic PDE systems. Then we apply the result to establish exponential stability of various chemical engineering processes and, in particular, exponential stability of heat exchangers. Through concrete examples we show how Lyapunov’s second method may be extended to stability analysis of nonlin- ear hyperbolic PDE. Meanwhile we explain how the method is adapted to the framework of Banach spaces Lp,1<p≤∞. Mathematics Subject Classification. 37L15, 37L45, 93C20. Received November 2, 2005. Revised September 4, 2006 and December 7, 2007. Published online May 30, 2008. 1. Introduction Lyapunov’s second method, called also Lyapunov’s direct method, is one of efficient and classical methods for studying asymptotic behavior and stability of the dynamical systems governed by ordinary differential equations. It dates back to Lyapunov’s thesis published at the beginning of the twentieth century [22]. The applications of the method and its theoretical development have been marked by the works of LaSalle and Lefschetz in the past [19,35] and numerous successful achievements later in the control of mechanical, robotic and aerospace systems (cf. -
The Calculus of Trigonometric Functions the Calculus of Trigonometric Functions – a Guide for Teachers (Years 11–12)
1 Supporting Australian Mathematics Project 2 3 4 5 6 12 11 7 10 8 9 A guide for teachers – Years 11 and 12 Calculus: Module 15 The calculus of trigonometric functions The calculus of trigonometric functions – A guide for teachers (Years 11–12) Principal author: Peter Brown, University of NSW Dr Michael Evans, AMSI Associate Professor David Hunt, University of NSW Dr Daniel Mathews, Monash University Editor: Dr Jane Pitkethly, La Trobe University Illustrations and web design: Catherine Tan, Michael Shaw Full bibliographic details are available from Education Services Australia. Published by Education Services Australia PO Box 177 Carlton South Vic 3053 Australia Tel: (03) 9207 9600 Fax: (03) 9910 9800 Email: [email protected] Website: www.esa.edu.au © 2013 Education Services Australia Ltd, except where indicated otherwise. You may copy, distribute and adapt this material free of charge for non-commercial educational purposes, provided you retain all copyright notices and acknowledgements. This publication is funded by the Australian Government Department of Education, Employment and Workplace Relations. Supporting Australian Mathematics Project Australian Mathematical Sciences Institute Building 161 The University of Melbourne VIC 3010 Email: [email protected] Website: www.amsi.org.au Assumed knowledge ..................................... 4 Motivation ........................................... 4 Content ............................................. 5 Review of radian measure ................................. 5 An important limit .................................... -
Stability Analysis for a Class of Partial Differential Equations Via
1 Stability Analysis for a Class of Partial terms of differential matrix inequalities. The proposed Differential Equations via Semidefinite method is then applied to study the stability of systems q Programming of inhomogeneous PDEs, using weighted norms (on Hilbert spaces) as LF candidates. H Giorgio Valmorbida, Mohamadreza Ahmadi, For the case of integrands that are polynomial in the Antonis Papachristodoulou independent variables as well, the differential matrix in- equality formulated by the proposed method becomes a polynomial matrix inequality. We then exploit the semidef- Abstract—This paper studies scalar integral inequal- ities in one-dimensional bounded domains with poly- inite programming (SDP) formulation [21] of optimiza- nomial integrands. We propose conditions to verify tion problems with linear objective functions and poly- the integral inequalities in terms of differential matrix nomial (sum-of-squares (SOS)) constraints [3] to obtain inequalities. These conditions allow for the verification a numerical solution to the integral inequalities. Several of the inequalities in subspaces defined by boundary analysis and feedback design problems have been studied values of the dependent variables. The results are applied to solve integral inequalities arising from the using polynomial optimization: the stability of time-delay Lyapunov stability analysis of partial differential equa- systems [23], synthesis control laws [24], [29], applied to tions. Examples illustrate the results. optimal control law design [15] and system analysis [11]. Index Terms—Distributed Parameter Systems, In the context of PDEs, preliminary results have been PDEs, Stability Analysis, Sum of Squares. presented in [19], where a stability test was formulated as a SOS programme (SOSP) and in [27] where quadratic integrands were studied. -
MTHE / MATH 237 Differential Equations for Engineering Science
i Queen’s University Mathematics and Engineering and Mathematics and Statistics MTHE / MATH 237 Differential Equations for Engineering Science Supplemental Course Notes Serdar Y¨uksel November 26, 2015 ii This document is a collection of supplemental lecture notes used for MTHE / MATH 237: Differential Equations for Engineering Science. Serdar Y¨uksel Contents 1 Introduction to Differential Equations ................................................... .......... 1 1.1 Introduction.................................... ............................................ 1 1.2 Classification of DifferentialEquations ............ ............................................. 1 1.2.1 OrdinaryDifferentialEquations ................. ........................................ 2 1.2.2 PartialDifferentialEquations .................. ......................................... 2 1.2.3 HomogeneousDifferentialEquations .............. ...................................... 2 1.2.4 N-thorderDifferentialEquations................ ........................................ 2 1.2.5 LinearDifferentialEquations ................... ........................................ 2 1.3 SolutionsofDifferentialequations ................ ............................................. 3 1.4 DirectionFields................................. ............................................ 3 1.5 Fundamental Questions on First-Order Differential Equations ...................................... 4 2 First-Order Ordinary Differential Equations .................................................. -
Linear Differential Equations Math 130 Linear Algebra
where A and B are arbitrary constants. Both of these equations are homogeneous linear differential equations with constant coefficients. An nth-order linear differential equation is of the form Linear Differential Equations n 00 0 Math 130 Linear Algebra anf (t) + ··· + a2f (t) + a1f (t) + a0f(t) = b: D Joyce, Fall 2013 that is, some linear combination of the derivatives It's important to know some of the applications up through the nth derivative is equal to b. In gen- of linear algebra, and one of those applications is eral, the coefficients an; : : : ; a1; a0, and b can be any to homogeneous linear differential equations with functions of t, but when they're all scalars (either constant coefficients. real or complex numbers), then it's a linear differ- ential equation with constant coefficients. When b What is a differential equation? It's an equa- is 0, then it's homogeneous. tion where the unknown is a function and the equa- For the rest of this discussion, we'll only consider tion is a statement about how the derivatives of the at homogeneous linear differential equations with function are related. constant coefficients. The two example equations 0 Perhaps the most important differential equation are such; the first being f −kf = 0, and the second 00 is the exponential differential equation. That's the being f + f = 0. one that says a quantity is proportional to its rate of change. If we let t be the independent variable, f(t) What do they have to do with linear alge- the quantity that depends on t, and k the constant bra? Their solutions form vector spaces, and the of proportionality, then the differential equation is dimension of the vector space is the order n of the equation. -
Introduction to Lyapunov Stability Analysis
Tools for Analysis of Dynamic Systems: Lyapunov’s Methods Stanisław H. Żak School of Electrical and Computer Engineering ECE 680 Fall 2013 1 A. M. Lyapunov’s (1857--1918) Thesis 2 Lyapunov’s Thesis 3 Lyapunov’s Thesis Translated 4 Some Details About Translation 5 Outline Notation using simple examples of dynamical system models Objective of analysis of a nonlinear system Equilibrium points Lyapunov functions Stability Barbalat’s lemma 6 A Spring-Mass Mechanical System x---displacement of the mass from the rest position 7 Modeling the Mass-Spring System Assume a linear mass, where k is the linear spring constant Apply Newton’s law to obtain Define state variables: x1=x and x2=dx/dt The model in state-space format: 8 Analysis of the Spring-Mass System Model The spring-mass system model is linear time-invariant (LTI) Representing the LTI system in standard state-space format 9 Modeling of the Simple Pendulum The simple pendulum 10 The Simple Pendulum Model Apply Newton’s second law J mglsin where J is the moment of inertia, 2 J ml Combining gives g sin l 11 State-Space Model of the Simple Pendulum Represent the second-order differential equation as an equivalent system of two first-order differential equations First define state variables, x1=θ and x2=dθ/dt Use the above to obtain state–space model (nonlinear, time invariant) 12 Objectives of Analysis of Nonlinear Systems Similar to the objectives pursued when investigating complex linear systems Not interested in detailed solutions, rather one seeks to characterize the system behavior---equilibrium points and their stability properties A device needed for nonlinear system analysis summarizing the system behavior, suppressing detail 13 Summarizing Function (D.G.