Curriculum Week 1. First-Order Ordinary Differential Equations

Total Page：16

File Type：pdf, Size：1020Kb

Curriculum Week 1. First-order ordinary differential equations, definition, physical examples. Initial value problems. Existence and uniqueness of the solution (Peano existence theorem and Picard-Lindelöf theorem). Lipschitz condition. Weeks 2-3. Directly integrable, autonomous and separable differential equations. Application and modeling, readioactive decay, physical eyxamples. The logistic model. Week 4. Homogeneous equations, reduction to separable form. Linear substitution. Exact differential equations. Week 5. First order linear differential equations. General solution of nonhomogeneous linear differential equations. Finding a particular solution, variation of parameters. Models of chemical reactions, examples. Week 6. Higher order linear differential equations. General solution, basis, particular solution. Linearity principle. Existence and uniqueness theorem for initial value problems. Linear independence of solutions, Wronski determinant. Weeks 7-8. Higher order homogeneous equations with constant coefficients. Characteristic polynomial, characteristic equation (real roots, complex roots, higher multiplicity). General solution of nonhomogeneous linear differential equations. Method of undetermined coefficients. Physical examples. Week 9. Systems of first order differential equations. Existence and uniqueness theorem. Linear systems. Existence and uniqueness in the linear case, linearity principle. Basis, general solution, Wronskian. Homogeneous systems with constant coefficients. Conversion of an n-th order differential equation to a system. Week 10. Laplace transforms, definition. Existence theorem for Laplace transforms. Properties of the Laplace transform. Application of the Laplace transform for the solution of higher order linear differential equations and linear systems. Power series method. Weeks 11-12. Phase portraits fo planar systems, the trace-determinant plane. Weeks 13-14. Qualitative methods for nolinear systems. Linearization of nonlinear systems. Criteria for stability. Liapunov’s method. First integrals. Liapunov functions. Applications, the Lotka-Volterra population model. Bendixson’s criterion, Bendixson-Dulac theorem. .

Copy Link

Recommended publications

Differential Equations We Will Use “Dsolve” to Get an Analytical Solution to the DE Y’(T) = 2Ty(T)
Differential Equations - Theory and Applications - Version: Fall 2017 AndrásDomokos , PhD California State University, Sacramento Contents Chapter 0. Introduction 3 Chapter 1. Calculus review. Differentiation and integration rules. 4 1.1. Derivatives 4 1.2. Antiderivatives and Indefinite Integrals 7 1.3. Definite Integrals 11 Chapter 2. Introduction to Differential Equations 15 2.1. Definitions 15 2.2. Initial value problems 20 2.3. Classifications of DEs 23 2.4. Examples of DEs modelling real-life phenomena 25 Chapter 3. First order differential equations solvable by analytical methods 27 3.1. Differential equations with separable variables 27 3.2. First order linear differential equations 31 3.3. Bernoulli's differential equations 36 3.4. Non-linear homogeneous differential equations 38 3.5. Differential equations of the form y0(t) = f(at + by(t) + c). 40 3.6. Second order differential equations reducible to first order differential equations 42 Chapter 4. General theory of differential equations of first order 45 4.1. Slope fields (or direction fields) 45 4.1.1. Autonomous first order differential equations. 49 4.2. Existence and uniqueness of solutions for initial value problems 53 4.3. The method of successive approximations 59 4.4. Numerical methods for Differential equations 62 4.4.1. The Euler's method 62 4.4.2. The improved Euler (or Heun) method 67 4.4.3. The fourth order Runge-Kutta method 68 4.4.4. NDSolve command in Mathematica 71 Chapter 5. Higher order linear differential equations 75 5.1. General theory 75 5.2. Linear and homogeneous DEs with constant coefficients 78 5.3.
[Show full text]
Reverse Mathematics & Nonstandard Analysis
CORE Metadata, citation and similar papers at core.ac.uk Provided by Ghent University Academic Bibliography REVERSE MATHEMATICS & NONSTANDARD ANALYSIS: TOWARDS A DISPENSABILITY ARGUMENT SAM SANDERS Abstract. Reverse Mathematics is a program in the foundations of math- ematics initiated by Harvey Friedman ([8, 9]) and developed extensively by Stephen Simpson ([19]). Its aim is to determine which minimal axioms prove theorems of ordinary mathematics. Nonstandard Analysis plays an impor- tant role in this program ([14, 25]). We consider Reverse Mathematics where equality is replaced by the predicate ≈, i.e. equality up to infinitesimals from Nonstandard Analysis. This context allows us to model mathematical practice in Physics particularly well. In this way, our mathematical results have impli- cations for Ontology and the Philosophy of Science. In particular, we prove the dispensability argument, which states that the very nature of Mathematics in Physics implies that real numbers are not essential (i.e. dispensable) for Physics (cf. the Quine-Putnam indispensability argument). There are good reasons to believe that nonstandard analysis, in some version or another, will be the analysis of the future. Kurt Gödel 1. Introducing Reverse Mathematics 1.1. Classical Reverse Mathematics. Reverse Mathematics is a program in Foundations of Mathematics founded around 1975 by Harvey Friedman ([8] and [9]) and developed intensely by Stephen Simpson and others; for an overview of the subject, see [19] and [20]. The goal of Reverse Mathematics is to determine what minimal axiom system is needed to prove a particular theorem from ordi- nary mathematics. By now, it is well-known that large portions of mathematics (especially so in analysis) can be carried out in systems far weaker than ZFC, the ùsual' background theory for mathematics.
[Show full text]
Ordinary Differential Equations Simon Brendle
Ordinary Differential Equations Simon Brendle Contents Preface vii Chapter 1. Introduction 1 x1.1. Linear ordinary differential equations and the method of integrating factors 1 x1.2. The method of separation of variables 3 x1.3. Problems 4 Chapter 2. Systems of linear differential equations 5 x2.1. The exponential of a matrix 5 x2.2. Calculating the matrix exponential of a diagonalizable matrix 7 x2.3. Generalized eigenspaces and the L + N decomposition 10 x2.4. Calculating the exponential of a general n × n matrix 15 x2.5. Solving systems of linear differential equations using matrix exponentials 17 x2.6. Asymptotic behavior of solutions 21 x2.7. Problems 24 Chapter 3. Nonlinear systems 27 x3.1. Peano's existence theorem 27 x3.2. Existence theory via the method of Picard iterates 30 x3.3. Uniqueness and the maximal time interval of existence 32 x3.4. Continuous dependence on the initial data 34 x3.5. Differentiability of flows and the linearized equation 37 x3.6. Liouville's theorem 39 v vi Contents x3.7. Problems 40 Chapter 4. Analysis of equilibrium points 43 x4.1. Stability of equilibrium points 43 x4.2. The stable manifold theorem 44 x4.3. Lyapunov's theorems 53 x4.4. Gradient and Hamiltonian systems 55 x4.5. Problems 56 Chapter 5. Limit sets of dynamical systems and the Poincaré- Bendixson theorem 57 x5.1. Positively invariant sets 57 x5.2. The !-limit set of a trajectory 60 x5.3. !-limit sets of planar dynamical systems 62 x5.4. Stability of periodic solutions and the Poincarémap 65 x5.5.
[Show full text]
Indian Institute of Science Education and Research
INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH THIRUVANANTHAPURAM An autonomous institution under the Ministry of Human Resource Development, Government of India CURRICULUM AND SYLLABUS FOR THE INTEGRATED Ph. D. PROGRAMME Contents Structure .................................................1 School of Biology Curriculum...........................1 School of Chemistry Curriculum.........................2 School of Mathematics Curriculum.......................3 School of Physics Curriculum............................4 Biology Syllabus...........................................5 Theory Courses........................................6 Laboratory Courses..................................... 17 Chemistry Syllabus ........................................ 21 Theory Courses........................................ 22 Laboratory Courses..................................... 40 Mathematics Syllabus ...................................... 47 Physics Syllabus ........................................... 61 Theory Courses........................................ 61 Laboratory Courses..................................... 73 v Structure School of Biology Curriculum SCHOOL OF BIOLOGY CURRICULUM Semester –I Semester -II Course Course Name L T P C Course Course Name L T P C BIO 311 Microbiology 3 0 0 3 BIO 321 Structural Biology 3 0 0 3 BIO 312 Advanced Genetics 3 0 0 3 BIO 322 Immunology 3 0 0 3 and Genomics BIO 313 Physiology 3 0 0 3 BIO 323 Cell Biology 3 0 0 3 BIO 314 Biochemistry 3 0 0 3 BIO 324 Molecular Biology 3 0 0 3 BIO 315 Advanced Biology 0 0
[Show full text]
MATH 8430 Fundamental Theory of Ordinary Differential Equations Lecture Notes Julien Arino Department of Mathematics University
MATH 8430 Fundamental Theory of Ordinary Differential Equations Lecture Notes Julien Arino Department of Mathematics University of Manitoba Fall 2006 Contents 1 General theory of ODEs 3 1.1 ODEs, IVPs, solutions . 3 1.1.1 Ordinary differential equation, initial value problem . 3 1.1.2 Solutions to an ODE . 4 1.1.3 Geometric interpretation . 8 1.2 Existence and uniqueness theorems . 9 1.2.1 Successive approximations . 9 1.2.2 Local existence and uniqueness – Proof by fixed point . 10 1.2.3 Local existence and uniqueness – Proof by successive approximations 13 1.2.4 Local existence (non Lipschitz case) . 16 1.2.5 Some examples of existence and uniqueness . 21 1.3 Continuation of solutions . 24 1.3.1 Maximal interval of existence . 27 1.3.2 Maximal and global solutions . 28 1.4 Continuous dependence on initial data, on parameters . 29 1.5 Generality of first order systems . 32 1.6 Generality of autonomous systems . 34 1.7 Suggested reading, Further problems . 34 2 Linear systems 35 2.1 Existence and uniqueness of solutions . 35 2.2 Linear systems . 36 2.2.1 The vector space of solutions . 37 2.2.2 Fundamental matrix solution . 38 2.2.3 Resolvent matrix . 41 2.2.4 Wronskian . 43 2.2.5 Autonomous linear systems . 43 2.3 Affine systems . 46 2.3.1 The space of solutions . 46 2.3.2 Construction of solutions . 46 2.3.3 Affine systems with constant coefficients . 47 2.4 Systems with periodic coefficients . 48 2.4.1 Linear systems: Floquet theory .
[Show full text]
Lecture Notes on Ordinary Differential Equations
Lecture notes on Ordinary Differential Equations S. Sivaji Ganesh Department of Mathematics Indian Institute of Technology Bombay May 20, 2016 ii Sivaji IIT Bombay Contents I Ordinary Differential Equations 1 1 Initial Value Problems 3 1.1 Existence of local solutions . 5 1.1.1 Preliminaries . 5 1.1.2 Peano’s existence theorem . 7 1.1.3 Cauchy-Lipschitz-Picard existence theorem . 10 1.2 Uniqueness . 14 1.3 Continuous dependence . 16 1.3.1 Sandwich theorem for IVPs / Comparison theorem . 17 1.3.2 Theorem on Continuous dependence . 18 1.4 Continuation . 19 1.4.1 Characterisation of continuable solutions . 21 1.4.2 Existence and Classification of saturated solutions . 22 1.5 Global Existence theorem . 24 Exercises . 26 Bibliography 31 iii iv Contents Sivaji IIT Bombay Part I Ordinary Differential Equations Chapter 1 Initial Value Problems In this chapter we introduce the notion of an initial value problem (IVP) for first order systems of ODE, and discuss questions of existence, uniqueness of solutions to IVP. We also discuss well-posedness of IVPs and maximal interval of existence for a given solution to the IVP. We complement the theory with examples from the class of first order scalar equations. ( ) The main hypotheses in the studies of IVPs is Hypothesis HIVPS , which will be in force throughout our discussion. ( ) Hypothesis HIVPS Let Ω ⊆ Rn be a domain and I ⊆ R be an open interval. Let f : I × Ω ! Rn ( ) 7! ( ) = ( ) be a continuous function defined by x, y f x, y where y y1,... yn . Let ( ) 2 I × Ω x0, y0 be an arbitrary point.
[Show full text]
Arxiv:1809.06430V1 [Math.AP] 17 Sep 2018 with Prescribed Initial Condition
Nonstandard existence proofs for reaction diffusion equations Connor Olson, Marshall Mueller, and Sigurd B. Angenent Abstract. We give an existence proof for distribution solutions to a scalar reaction diffusion equation, with the aim of illustrating both the differences and the common ingredients of the nonstandard and standard approaches. In particular, our proof shows how the operation of taking the standard part of a nonstandard real number can replace several different compactness the- orems, such as Ascoli's theorem and the Banach{Alaoglu theorem on weak∗- compactness of the unit ball in the dual of a Banach space. Contents 1. Introduction1 2. Distribution solutions4 3. The Cauchy problem for the heat equation6 4. The Cauchy problem for a Reaction Diffusion Equation 11 References 16 1. Introduction 1.1. Reaction diffusion equations. We consider the Cauchy problem for scalar reaction diffusion equations of the form @u @2u (1a) = D + f(u(x; t)); x 2 ; t ≥ 0 @t @x2 R arXiv:1809.06430v1 [math.AP] 17 Sep 2018 with prescribed initial condition (1b) u(x; 0) = u0(x): In the setting of reaction diffusion equations the function u(x; t) represents the density at location x 2 R and time t ≥ 0 of some substance which diffuses, and simultaneously grows or decays due to chemical reaction, biological mutation, or some other process. The term D@2u=@x2 in the PDE (1a) accounts for the change in u due to diffusion, while the nonlinear term f(u) accounts for the reaction rates. The prototypical example of such a reaction diffusion equation is the Fisher/KPP equation (see [9], [2]) in which the reaction term is given by f(u) = u − u2.
[Show full text]
Theory of Ordinary Differential Equations
Theory of Ordinary Differential Equations Existence, Uniqueness and Stability Jishan Hu and Wei-Ping Li Department of Mathematics The Hong Kong University of Science and Technology ii Copyright c 2004 by Jishan Hu and Wei-Ping Li Contents 1 Existence and Uniqueness 1 1.1 Some Basics 1 1.2 Uniqueness Theorem 6 1.3 Continuity 8 1.4 Existence Theorem 13 1.5 Local Existence Theorem and The Peano Theorem 19 1.5.1 Local Existence Theorem 19 1.5.2 The Peano Theorem 21 1.6 Linear Systems 23 1.7 Continuation of Solutions 27 1.8 Miscellaneous Problems 29 2 Plane Autonomous Systems 33 2.1 Plane Autonomous Systems 33 2.2 Linear Autonomous Systems 39 2.3 Complete Classification for Linear Autonomous Systems 44 2.4 Liapunov Direct Method 52 2.5 A Test For Instability 61 2.6 Nonlinear Oscillations 65 2.6.1 Undamped Oscillations 65 iii iv CONTENTS 2.6.2 Damped Oscillations 66 2.7 Miscellaneous Problems 67 1 Existence and Uniqueness 1.1 SOME BASICS A normal system of first order ordinary differential equations (ODEs) is 8 dx1 > = X1(x1; : : : ; xn; t); > dt <> . (1.1) . > > dx :> n = X (x ; : : : ; x ; t): dt n 1 n Many varieties of ODEs can be reduced to this form. For example, consider an n-th order ODE dnx dx dn−1x = F t; x; ;:::; : dtn dt dtn−1 dx dn−1x Let x = x; x = ; : : : ; x = . Then the ODE can be changed to 1 2 dt n dtn−1 8 dx1 > = x2; > dt > > . <> . dx > n−1 = x ; > dt n > > dx :> n = F (t; x ; : : : ; x ): dt 1 n 1 2 1 EXISTENCE AND UNIQUENESS Let us review some notations and facts on vectors and vector-valued functions.
[Show full text]
Ordinary Differential Equations
M. Sc (Math) Code No. 16MMC21C3 Semester-I Ordinary Differential Equations DIRECTORATE OF DISTANCE EDUCATION MAHARSHI DAYANAND UNIVERSITY, ROHTAK (A State University established under Haryana Act No. XXV of 1975) NAAC 'A +’ Grade Accredited University Author Dr. Jagbir Singh Assistant Professor, Department of Mathematics Maharshi Dayanand University, Rohtak Copyright © 2020, Maharshi Dayanand University, ROHTAK All Rights Reserved. No part of this publication may be reproduced or stored in a retrieval system or transmitted in any form or by any means; electronic, mechanical, photocopying, recording or otherwise, without the written permission of the copyright holder. Maharshi Dayanand University ROHTAK – 124 001 Paper Code: 16MMC21C3 Ordinary Differential Equations M. Marks = 100 Term End Examination = 80 Assignment = 20 Time = 3 Hours Course Outcomes Students would be able to: CO1 Apply differential equations to variety of problems in diversified fields of life. CO2 Learn use of differential equations for modeling and solving real life problems. CO3 Interpret the obtained solutions in terms of the physical quantities involved in the original problem under reference. CO4 Use various methods of approximation to get qualitative information about the general behaviour of the solutions of various problems. Section - I Preliminaries, ε-approximate solution, Cauchy-Euler construction of an ε-approximate solution of an initial value problem, Equicontinuous family of functions, Ascoli-Arzela Lemma, Cauchy-Peano existence theorem. Lipschitz condition,
[Show full text]
Differential Equations, Boundary Value Problems
Computational Astrophysics I: Introduction and basic concepts Helge Todt Astrophysics Institute of Physics and Astronomy University of Potsdam SoSe 2021, 17.6.2021 H. Todt (UP) Computational Astrophysics SoSe 2021, 17.6.2021 1 / 26 Differential equations H. Todt (UP) Computational Astrophysics SoSe 2021, 17.6.2021 2 / 26 Types of differential equationsI One can classify differential equations regarding their order, so the degree of the highest derivative. General form of a first-order differential equation: dy = f (y; t) (1) dt dy 8 5 for any arbitrary function f , e.g., dt = 2ty − t + sin(y). A second-order differential equation has the form: d 2y dy dy + λ = f (t; ; y) (2) dt2 dt dt and so on. H. Todt (UP) Computational Astrophysics SoSe 2021, 17.6.2021 3 / 26 Types of differential equationsII Reduction By introducing auxillary variables/functions, every higher order differential equation can be reduced to a set of first-order differential equations y (m)(x) = f (x; y(x); y (1)(x);:::; y (m−1)(x)) (3) ! z1(x) := y(x) (4) (1) z2(x) := y (x) (5) . (6) (m−1) zm(x) := y (x) (7) 2 0 3 2 3 z1 z2 0 6 . 7 6 . 7 ! z = 4 . 5 = 4 . 5 (8) 0 zm f (x; z1; z2;:::; zm) H. Todt (UP) Computational Astrophysics SoSe 2021, 17.6.2021 4 / 26 Types of differential equations III One can distinguish ordinary diffential equations (ODE), where only one independent variable is explicitly involved (typically time or location), e.g.: dP = −ρ(r) g(r) (9) dr partial differential equations (PDE), where the solution depends at least on two variables, e.g.: @2 @2 @2 ∆ρ = + + ρ(x; y; z) = f (x; y; z) (10) @x2 @y 2 @z2 The theory and (numerical) solution of PDEs is more complicated than for ODE.
[Show full text]
Ordinary Differential Equation Lecture Notes for the Post-Graduate, Sem 3
Paper XII : Ordinary Differential Equation Lecture notes for the Post-graduate, sem 3 Course Department of Mathematics Ramakrishna Mission Vidyamandira Belur Math, INDIA Course Instructor : Dr. Arnab Jyoti Das Gupta August, 2020 to January, 2021 2 Syllabus 1. Preliminaries { Initial Value problem and the equivalent integral equation, mth order equation in d-dimensions as a first order system, concepts of local existence, existence in the large and uniqueness of solutions withy examples. 2. Basic Theorems { Ascoli-Arzela Theorem. A Theorem on convergence of solutions of a family of initial-value problems. 3. Picard-Lindelof Theorem { Peano's existence Theorem and corollary. Maximal intervals of exis- tence. Extension Theorem and corollaries. Kamke's convergence Theorem. Kneser's Theorem (Statement only). 4. Differential inequalities and Uniqueness { Gronwall's inequality. Maximal and minimal solu- tions. Differential inequalities. A Theorem of Winter. Uniqueness Theorems. Nagumo's and Osgood's criteria. 5. Egres pointstand Lyapunov functions. Successive approximations. 6. Variation of constants, reduction to smaller sustems. Basic inequalities, constant coefficients. Floquet Theory. Adjoint systems, Higher order equations. 7. Linear second order equations { Preliminaries. Basic facts. Theorems of Sturm. Sturm Liou- vilee Boundary value Problems. References 1. P. Hartman, Ordinary Differential Equations, John Wiley (1964). 2. E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, NY (1955). 3. G.F. Simmons : Differential Equaitons. 4. W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value problems. 5. S. L. Ross, Differential Equation Contents 1 Existence and Uniqueness of solutions 5 1.1 Notations . 5 1.2 Initial Value problem . 6 1.3 Uniqueness of solutions .
[Show full text]
Ordinary Differential Equations
Ordinary Di®erential Equations Fabio Bagagiolo Dipartimento di Matematica, Universit`adi Trento [email protected] Contents 1 Introduction 3 1.1 Motivating examples . 4 1.2 Notations, de¯nitions and further considerations . 9 1.3 Solving by hands and the need of a general theory . 13 1.4 Frequently asked questions . 16 2 Existence and uniqueness theorems for the Cauchy problem 18 2.1 De¯nitions: local and global existence . 18 2.2 The integral representation . 19 2.3 The Peano existence theorem under continuity hypothesis . 20 2.3.1 A ¯rst proof: a delayed argument . 20 2.3.2 A (sketched) second proof: the Euler method . 21 2.3.3 An example of non existence . 23 2.3.4 An example of non uniqueness . 23 2.4 Local existence and uniqueness under Lipschitz continuity hypothesis . 24 2.5 Global existence and uniqueness under Lipschitz continuity hypothesis . 26 3 The linear case 30 3.1 Linear systems of ¯rst order equations . 30 3.2 Homogeneous linear systems of ¯rst order equations . 31 3.2.1 The fundamental set of solutions and the Wronskian . 31 3.3 Homogeneous systems of ¯rst order linear equations with constant coe±cients 34 3.3.1 The homogeneous linear equation of n-th order with constant coef- ¯cients . 36 3.4 Nonhomogeneous systems of ¯rst order linear equations . 42 3.4.1 The constants variation method . 43 3.4.2 The nonhomogeneous linear system with constant coe±cients . 44 3.4.3 The nonhomogeneous linear equation of n-order with constants co- e±cients.
[Show full text]