SYSTEMS OF ODES. mg sin ( (x)) dx 2 =
|
|
- Leslie Miller
- 5 years ago
- Views:
Transcription
1 SYSTEMS OF ODES Consider the pendulum shown below. Assume the rod is of neglible mass, that the pendulum is of mass m, and that the rod is of length `. Assume the pendulum moves in the plane shown, and assume there is no friction in the motion about its pivot point. Let (x) denote the position of the pendulum about the vertical line thru the pivot, with measured in radians and x measured in units of time. Then Newton's second law implies m`d dx = mg sin ( (x))
2 Introduce Y 1 (x) = (x) and Y (x) = 0 (x). The function Y (x) is called the angular velocity. We can now write Y 0 1 (x) = Y (x); Y 1 (0) = (0) Y 0 (x) = g` sin (Y 1(x)) ; Y (0) = 0 (0) This is a simultaneous system of two dierential equations in two unknowns. We often write this in vector form. Introduce " # Y1 (x) Y(x) = Y (x) Then Y 0 Y (x) (x) = 4 g ` sin (Y 1(x)) " # Y1 (0) Y(0) = Y 0 = = Y (0) 5 " (0) 0 (0) #
3 Introduce z f (x; z) = 4 g ` sin (z 5 ; z = 1) Then our dierential equation problem Y 0 Y (x) (x) = 4 g ` sin (Y 1(x)) " # Y1 (0) Y(0) = Y 0 = = Y (0) can be written in the familiar form 5 " z1 z " (0) 0 (0) # # Y 0 (x) = f (x; Y(x)) ; Y(0) = Y 0 (1) We can convert any higher order dierential equation into a system of rst order dierential equations, and we can write them in the vector form (1).
4 Lotka-Volterra predator-prey model. Y1 0 = AY 1[1 BY ]; Y 1 (0) = Y 1;0 Y 0 = CY () [DY 1 1]; Y (0) = Y ;0 with A; B; C; D > 0. x denotes time, Y 1 (x) is the number of prey (e.g., rabbits) at time x, and Y (x) the number of predators (e.g., foxes). If there is only a single type of predator and a single type of prey, then this model is often a good approximation of reality. Again write and dene f (x; z) = Y(x) = " Y1 (x) Y (x) " Az1 [1 Bz ] Cz [Dz 1 1] # # ; z = " z1 although there is no explicit dependence on x. Then system () can be written as z Y 0 (x) = f (x; Y(x)) ; Y(0) = Y 0 #
5 GENERAL SYSTEMS OF ODES An initial value problem for a system of m dierential equations has the form Y 0 1 (x) = f 1(x; Y 1 (x); : : : ; Y m (x)); Y 1 (x 0 ) = Y 1;0.. Ym(x) 0 = f m (x; Y 1 (x); : : : ; Y m (x)); Y m (x 0 ) = Y m;0 () Introduce Y(x) = f(x; z) = Y 1 (x). Y m (x) Then () can be written as 7 5 ; Y 0 = f 1 (x; z 1 ; : : : ; z m ). f m (x; z 1 ; : : : ; z m ) 6 4 Y 1;0. Y m;0 Y 0 (x) = f (x; Y(x)) ; Y(0) = Y
6 LINEAR SYSTEMS Of special interest are systems of the form Y 0 (x) = AY(x) + G(x); Y(0) = Y 0 (4) with A a square matrix of order m and G(x) a column vector of length m with functions G i (x) as components. Using the notation introduced for writing systems, f(x; z) = Az + G(x); z R m This equation is the analogue for studying systems of ODEs that the model equation y 0 = y + g(x) is for studying a single dierential equation.
7 Consider EULER'S METHOD FOR SYSTEMS Y 0 (x) = f (x; Y(x)) ; Y(0) = Y 0 to be a systems of two equations Y1 0(x) = f 1(x; Y 1 (x); Y (x)); Y 1 (0) = Y 1;0 Y 0(x) = f (5) (x; Y 1 (x); Y (x)); Y (0) = Y ;0 Denote its solution be [Y 1 (x); Y (x)]. Following the earlier derivations for Euler's method, we can use Taylor's theorem to obtain Y 1 (x n+1 ) = Y 1 (x n ) + hf 1 (x n ; Y 1 (x n ); Y (x n )) + h Y 00 1 ( n) (6) Y (x n+1 ) = Y (x n ) + hf (x n ; Y 1 (x n ); Y (x n )) + h Y 00 ( n) Dropping the remainder terms, we obtain Euler's method for problem (5), y 1;n+1 = y 1;n + hf 1 (x n ; y 1;n ; y ;n ); y 1;0 = Y 1;0 y ;n+1 = y ;n + hf (x n ; y 1;n ; y ;n ); y ;0 = Y ;0 for n = 0; 1; ; : : :
8 ERROR ANALYSIS If Y 1 (x), Y (x) are twice continuously dierentiable, and if the functions f 1 (x; z 1 ; z ) and f (x; z 1 ; z ) are suciently dierentiable, then it can be shown that max Y 1 (x n ) y 1;n ch x 0 xb (7) Y (x n ) y ;n ch max x 0 xb for a suitable choice of c 0. The theory depends on generalizations of the proof used with Euler's method for a single equation. One needs to assume that there is a constant K > 0 such that kf (x; z) f (x; w)k 1 K kz wk 1 (8) for x 0 x b, z; w R. Recall the denition of the norm kk 1 from Chapter 6.
9 The role in the single variable theory is replaced by the Jacobian matrix F(x; z) = 6 4 It is possible to show 1 (x; z 1 ; z 1 (x; z 1 ; z (x; z 1 ; (x; z 1 ; z K = is suitable for showing (8). max x 0 xb zr kf(x; z)k (9) All of this work generalizes to problems of any order m. Then we require kf (x; z) f (x; w)k 1 K kz wk 1 (10) with x 0 x b, z; w R m. often obtained using K = max x 0 xb zr m kf(x; z)k 1 The choice of K is where F(x; z) is the m m generalization of (9).
10 The Euler method in all cases can be written in the dimensionless form with y 0 = Y 0. y n+1 = y n + hf(x n ; y n ); n 0 It can be shown that if (10) is satised, and if Y(x) is twice-continuously dierentiable on [x 0 ; b], then max ky(x n) y n k 1 ch (11) x 0 xb for some c 0 and for all small values of h. In addition, we can show there is a vector function D(x) for which Y(x) y h (x) = D(x)h + O(h ); x 0 x n b for x = x 0 ; x 1 ; : : : ; b. Here y h (x) shows the dependence of the solution on h, and y h (x) = y n for x = x 0 + nh. This justies the use of Richardson extrapolation, leading to Y(x) y h (x) = y h (x) y h (x) + O(h )
11 NUMERICAL EXAMPLE. Consider solving the initial value problem Y Y 00 + Y 0 + Y = 4 sin(x); Y (0) = Y 0 (0) = 1; Y 00 (0) = 1 Reformulate it as (1) Y1 0 = Y Y 1 (0) = 1 Y 0 = Y Y (0) = 1 Y 0 1 Y Y 4 sin(x); Y (0) = 1 (1) The solution of (1) is Y (x) = cos(x) + sin(x), and the solution of (1) can be generated from it using Y 1 (x) = Y (x).
12 The results for Y 1 (x) = sin(x)+cos(x) are given in the following table, for stepsizes h = 0:1 and h = 0:05. The Richardson error estimate is quite accurate. x y(x) y(x) y h (x) y(x) y h (x) Ratio 0:4915 8:78E 4:5E :1 4 1: :9E 1 6:86E :0 6 0: :19E :49E :1 8 0:8486 1:56E 1 7:56E :1 10 1:809 8:9E 4:14E :0 x y(x) y(x) y h (x) y h (x) y h (x) 0:4915 4:5E 4:5E 4 1: :86E 7:05E 6 0:68075 :49E :70E 8 0:8486 7:56E 7:99E 10 1:809 4:14E 4:5E
13 OTHER METHODS Other numerical methods apply to systems in the same straightforward manner. by using the vector form Y 0 (x) = f (x; Y(x)) ; Y(0) = Y 0 (14) for a system, there is no apparent change in the numerical method. For example, the following Runge- Kutta method for solving a single dierential equation, y n+1 = y n + h [f(x n; y n )+f(x n+1 ; y n +hf(x n ; y n ))]; n 0, generalizes as follows for solving (14): y n+1 = y n + h [f(x n; y n )+f(x n+1 ; y n +hf(x n ; y n ))]; n 0. This can then be decomposed into components if needed. For a system of order, we have y j;n+1 = y j;n + h h fj (x n ; y 1;n ; y ;n ) for n 0 and j = 1;. +f j xn+1 ; y 1;n + hf 1 (x n ; y 1;n ; y ;n ); y ;n + hf (x n; y 1;n ; y ;n ) i
EXAMPLE OF ONE-STEP METHOD
EXAMPLE OF ONE-STEP METHOD Consider solving y = y cos x, y(0) = 1 Imagine writing a Taylor series for the solution Y (x), say initially about x = 0. Then Y (h) = Y (0) + hy (0) + h2 2 Y (0) + h3 6 Y (0)
More informationFourth Order RK-Method
Fourth Order RK-Method The most commonly used method is Runge-Kutta fourth order method. The fourth order RK-method is y i+1 = y i + 1 6 (k 1 + 2k 2 + 2k 3 + k 4 ), Ordinary Differential Equations (ODE)
More informationLogistic Map, Euler & Runge-Kutta Method and Lotka-Volterra Equations
Logistic Map, Euler & Runge-Kutta Method and Lotka-Volterra Equations S. Y. Ha and J. Park Department of Mathematical Sciences Seoul National University Sep 23, 2013 Contents 1 Logistic Map 2 Euler and
More informationChapter 8. Numerical Solution of Ordinary Differential Equations. Module No. 1. Runge-Kutta Methods
Numerical Analysis by Dr. Anita Pal Assistant Professor Department of Mathematics National Institute of Technology Durgapur Durgapur-71309 email: anita.buie@gmail.com 1 . Chapter 8 Numerical Solution of
More informationEAD 115. Numerical Solution of Engineering and Scientific Problems. David M. Rocke Department of Applied Science
EAD 115 Numerical Solution of Engineering and Scientific Problems David M. Rocke Department of Applied Science Transient Response of a Chemical Reactor Concentration of a substance in a chemical reactor
More informationODE Runge-Kutta methods
ODE Runge-Kutta methods The theory (very short excerpts from lectures) First-order initial value problem We want to approximate the solution Y(x) of a system of first-order ordinary differential equations
More informationAN OVERVIEW. Numerical Methods for ODE Initial Value Problems. 1. One-step methods (Taylor series, Runge-Kutta)
AN OVERVIEW Numerical Methods for ODE Initial Value Problems 1. One-step methods (Taylor series, Runge-Kutta) 2. Multistep methods (Predictor-Corrector, Adams methods) Both of these types of methods are
More informationPart 1: Overview of Ordinary Dierential Equations 1 Chapter 1 Basic Concepts and Problems 1.1 Problems Leading to Ordinary Dierential Equations Many scientic and engineering problems are modeled by systems
More informationInitial Value Problems
Numerical Analysis, lecture 13: Initial Value Problems (textbook sections 10.1-4, 10.7) differential equations standard form existence & uniqueness y 0 y 2 solution methods x 0 x 1 h h x 2 y1 Euler, Heun,
More informationCoupled differential equations
Coupled differential equations Example: dy 1 dy 11 1 1 1 1 1 a y a y b x a y a y b x Consider the case with b1 b d y1 a11 a1 y1 y a1 ay dy y y e y 4/1/18 One way to address this sort of problem, is to
More information1 The pendulum equation
Math 270 Honors ODE I Fall, 2008 Class notes # 5 A longer than usual homework assignment is at the end. The pendulum equation We now come to a particularly important example, the equation for an oscillating
More informationChap. 20: Initial-Value Problems
Chap. 20: Initial-Value Problems Ordinary Differential Equations Goal: to solve differential equations of the form: dy dt f t, y The methods in this chapter are all one-step methods and have the general
More informationBackward error analysis
Backward error analysis Brynjulf Owren July 28, 2015 Introduction. The main source for these notes is the monograph by Hairer, Lubich and Wanner [2]. Consider ODEs in R d of the form ẏ = f(y), y(0) = y
More informationChapter 11 ORDINARY DIFFERENTIAL EQUATIONS
Chapter 11 ORDINARY DIFFERENTIAL EQUATIONS The general form of a first order differential equations is = f(x, y) with initial condition y(a) = y a We seek the solution y = y(x) for x > a This is shown
More informationLECTURE 15 + C+F. = A 11 x 1x1 +2A 12 x 1x2 + A 22 x 2x2 + B 1 x 1 + B 2 x 2. xi y 2 = ~y 2 (x 1 ;x 2 ) x 2 = ~x 2 (y 1 ;y 2 1
LECTURE 5 Characteristics and the Classication of Second Order Linear PDEs Let us now consider the case of a general second order linear PDE in two variables; (5.) where (5.) 0 P i;j A ij xix j + P i,
More informationReview Higher Order methods Multistep methods Summary HIGHER ORDER METHODS. P.V. Johnson. School of Mathematics. Semester
HIGHER ORDER METHODS School of Mathematics Semester 1 2008 OUTLINE 1 REVIEW 2 HIGHER ORDER METHODS 3 MULTISTEP METHODS 4 SUMMARY OUTLINE 1 REVIEW 2 HIGHER ORDER METHODS 3 MULTISTEP METHODS 4 SUMMARY OUTLINE
More informationMath 265 (Butler) Practice Midterm III B (Solutions)
Math 265 (Butler) Practice Midterm III B (Solutions). Set up (but do not evaluate) an integral for the surface area of the surface f(x, y) x 2 y y over the region x, y 4. We have that the surface are is
More informationThis practice exam is intended to help you prepare for the final exam for MTH 142 Calculus II.
MTH 142 Practice Exam Chapters 9-11 Calculus II With Analytic Geometry Fall 2011 - University of Rhode Island This practice exam is intended to help you prepare for the final exam for MTH 142 Calculus
More informationCS520: numerical ODEs (Ch.2)
.. CS520: numerical ODEs (Ch.2) Uri Ascher Department of Computer Science University of British Columbia ascher@cs.ubc.ca people.cs.ubc.ca/ ascher/520.html Uri Ascher (UBC) CPSC 520: ODEs (Ch. 2) Fall
More informationMini project ODE, TANA22
Mini project ODE, TANA22 Filip Berglund (filbe882) Linh Nguyen (linng299) Amanda Åkesson (amaak531) October 2018 1 1 Introduction Carl David Tohmé Runge (1856 1927) was a German mathematician and a prominent
More informationSystems of Differential Equations: General Introduction and Basics
36 Systems of Differential Equations: General Introduction and Basics Thus far, we have been dealing with individual differential equations But there are many applications that lead to sets of differential
More information2.1 NUMERICAL SOLUTION OF SIMULTANEOUS FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS. differential equations with the initial values y(x 0. ; l.
Numerical Methods II UNIT.1 NUMERICAL SOLUTION OF SIMULTANEOUS FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS.1.1 Runge-Kutta Method of Fourth Order 1. Let = f x,y,z, = gx,y,z be the simultaneous first order
More informationSecond Order ODEs. CSCC51H- Numerical Approx, Int and ODEs p.130/177
Second Order ODEs Often physical or biological systems are best described by second or higher-order ODEs. That is, second or higher order derivatives appear in the mathematical model of the system. For
More informationComputational Methods CMSC/AMSC/MAPL 460. Ordinary differential equations
/0/0 Computational Methods CMSC/AMSC/MAPL 460 Ordinar differential equations Ramani Duraiswami Dept. of Computer Science Several slides adapted from Profs. Dianne O Lear and Eric Sandt TAMU Higher Order
More informationOur goal is to solve a general constant coecient linear second order. this way but that will not always happen). Once we have y 1, it will always
October 5 Relevant reading: Section 2.1, 2.2, 2.3 and 2.4 Our goal is to solve a general constant coecient linear second order ODE a d2 y dt + bdy + cy = g (t) 2 dt where a, b, c are constants and a 0.
More informationComputation Fluid Dynamics
Computation Fluid Dynamics CFD I Jitesh Gajjar Maths Dept Manchester University Computation Fluid Dynamics p.1/189 Garbage In, Garbage Out We will begin with a discussion of errors. Useful to understand
More informationExercises, module A (ODEs, numerical integration etc)
FYTN HT18 Dept. of Astronomy and Theoretical Physics Lund University, Sweden Exercises, module A (ODEs, numerical integration etc) 1. Use Euler s method to solve y (x) = y(x), y() = 1. (a) Determine y
More informationsecond order Runge-Kutta time scheme is a good compromise for solving ODEs unstable for oscillators
ODE Examples We have seen so far that the second order Runge-Kutta time scheme is a good compromise for solving ODEs with a good precision, without making the calculations too heavy! It is unstable for
More informationMath 266: Ordinary Differential Equations
Math 266: Ordinary Differential Equations Long Jin Purdue University, Spring 2018 Basic information Lectures: MWF 8:30-9:20(111)/9:30-10:20(121), UNIV 103 Instructor: Long Jin (long249@purdue.edu) Office
More informationConsider an ideal pendulum as shown below. l θ is the angular acceleration θ is the angular velocity
1 Second Order Ordinary Differential Equations 1.1 The harmonic oscillator Consider an ideal pendulum as shown below. θ l Fr mg l θ is the angular acceleration θ is the angular velocity A point mass m
More informationOrdinary Differential Equations
Ordinary Differential Equations Michael H. F. Wilkinson Institute for Mathematics and Computing Science University of Groningen The Netherlands December 2005 Overview What are Ordinary Differential Equations
More informationModule Two: Differential Calculus(continued) synopsis of results and problems (student copy)
Module Two: Differential Calculus(continued) synopsis of results and problems (student copy) Srikanth K S 1 Syllabus Taylor s and Maclaurin s theorems for function of one variable(statement only)- problems.
More informationNumerical Methods for ODEs. Lectures for PSU Summer Programs Xiantao Li
Numerical Methods for ODEs Lectures for PSU Summer Programs Xiantao Li Outline Introduction Some Challenges Numerical methods for ODEs Stiff ODEs Accuracy Constrained dynamics Stability Coarse-graining
More information2007 Summer College on Plasma Physics
1856-18 2007 Summer College on Plasma Physics 30 July - 24 August, 2007 Numerical methods and simulations. Lecture 2: Simulation of ordinary differential equations. B. Eliasson Institut fur Theoretische
More informationExamination paper for TMA4215 Numerical Mathematics
Department of Mathematical Sciences Examination paper for TMA425 Numerical Mathematics Academic contact during examination: Trond Kvamsdal Phone: 93058702 Examination date: 6th of December 207 Examination
More informationIntroduction to standard and non-standard Numerical Methods
Introduction to standard and non-standard Numerical Methods Dr. Mountaga LAM AMS : African Mathematic School 2018 May 23, 2018 One-step methods Runge-Kutta Methods Nonstandard Finite Difference Scheme
More informationSection x7 +
Difference Equations to Differential Equations Section 5. Polynomial Approximations In Chapter 3 we discussed the problem of finding the affine function which best approximates a given function about some
More informationModeling and Experimentation: Compound Pendulum
Modeling and Experimentation: Compound Pendulum Prof. R.G. Longoria Department of Mechanical Engineering The University of Texas at Austin Fall 2014 Overview This lab focuses on developing a mathematical
More informationMath 128A Spring 2003 Week 12 Solutions
Math 128A Spring 2003 Week 12 Solutions Burden & Faires 5.9: 1b, 2b, 3, 5, 6, 7 Burden & Faires 5.10: 4, 5, 8 Burden & Faires 5.11: 1c, 2, 5, 6, 8 Burden & Faires 5.9. Higher-Order Equations and Systems
More information8 Singular Integral Operators and L p -Regularity Theory
8 Singular Integral Operators and L p -Regularity Theory 8. Motivation See hand-written notes! 8.2 Mikhlin Multiplier Theorem Recall that the Fourier transformation F and the inverse Fourier transformation
More informationGrade: The remainder of this page has been left blank for your workings. VERSION E. Midterm E: Page 1 of 12
First Name: Student-No: Last Name: Section: Grade: The remainder of this page has been left blank for your workings. Midterm E: Page of Indefinite Integrals. 9 marks Each part is worth 3 marks. Please
More informationTHE TRAPEZOIDAL QUADRATURE RULE
y 1 Computing area under y=1/(1+x^2) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Trapezoidal rule x THE TRAPEZOIDAL QUADRATURE RULE From Chapter 5, we have the quadrature
More information6.3. Nonlinear Systems of Equations
G. NAGY ODE November,.. Nonlinear Systems of Equations Section Objective(s): Part One: Two-Dimensional Nonlinear Systems. ritical Points and Linearization. The Hartman-Grobman Theorem. Part Two: ompeting
More informationOrdinary Differential Equations: Initial Value problems (IVP)
Chapter Ordinary Differential Equations: Initial Value problems (IVP) Many engineering applications can be modeled as differential equations (DE) In this book, our emphasis is about how to use computer
More informationOrdinary Differential Equations (ODEs)
Ordinary Differential Equations (ODEs) 1 Computer Simulations Why is computation becoming so important in physics? One reason is that most of our analytical tools such as differential calculus are best
More informationResearch Computing with Python, Lecture 7, Numerical Integration and Solving Ordinary Differential Equations
Research Computing with Python, Lecture 7, Numerical Integration and Solving Ordinary Differential Equations Ramses van Zon SciNet HPC Consortium November 25, 2014 Ramses van Zon (SciNet HPC Consortium)Research
More informationContents. 2 Partial Derivatives. 2.1 Limits and Continuity. Calculus III (part 2): Partial Derivatives (by Evan Dummit, 2017, v. 2.
Calculus III (part 2): Partial Derivatives (by Evan Dummit, 2017, v 260) Contents 2 Partial Derivatives 1 21 Limits and Continuity 1 22 Partial Derivatives 5 23 Directional Derivatives and the Gradient
More informationNumerical Methods for Initial Value Problems; Harmonic Oscillators
Lab 1 Numerical Methods for Initial Value Problems; Harmonic Oscillators Lab Objective: Implement several basic numerical methods for initial value problems (IVPs), and use them to study harmonic oscillators.
More informationLecture 32: Taylor Series and McLaurin series We saw last day that some functions are equal to a power series on part of their domain.
Lecture 32: Taylor Series and McLaurin series We saw last day that some functions are equal to a power series on part of their domain. For example f(x) = 1 1 x = 1 + x + x2 + x 3 + = ln(1 + x) = x x2 2
More informationIntegration, Separation of Variables
Week #1 : Integration, Separation of Variables Goals: Introduce differential equations. Review integration techniques. Solve first-order DEs using separation of variables. 1 Sources of Differential Equations
More informationLecture 2: Review of Prerequisites. Table of contents
Math 348 Fall 217 Lecture 2: Review of Prerequisites Disclaimer. As we have a textbook, this lecture note is for guidance and supplement only. It should not be relied on when preparing for exams. In this
More informationAnalyse 3 NA, FINAL EXAM. * Monday, January 8, 2018, *
Analyse 3 NA, FINAL EXAM * Monday, January 8, 08, 4.00 7.00 * Motivate each answer with a computation or explanation. The maximum amount of points for this exam is 00. No calculators!. (Holomorphic functions)
More informationAlgorithmic Lie Symmetry Analysis and Group Classication for Ordinary Dierential Equations
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 1 / 25 Algorithmic Lie Symmetry Analysis and Group Classication for Ordinary Dierential Equations Dmitry A. Lyakhov 1 1 Computational Sciences
More informationDepartment of Applied Mathematics and Theoretical Physics. AMA 204 Numerical analysis. Exam Winter 2004
Department of Applied Mathematics and Theoretical Physics AMA 204 Numerical analysis Exam Winter 2004 The best six answers will be credited All questions carry equal marks Answer all parts of each question
More informationDegree Master of Science in Mathematical Modelling and Scientific Computing Mathematical Methods I Thursday, 12th January 2012, 9:30 a.m.- 11:30 a.m.
Degree Master of Science in Mathematical Modelling and Scientific Computing Mathematical Methods I Thursday, 12th January 2012, 9:30 a.m.- 11:30 a.m. Candidates should submit answers to a maximum of four
More information3.3. SYSTEMS OF ODES 1. y 0 " 2y" y 0 + 2y = x1. x2 x3. x = y(t) = c 1 e t + c 2 e t + c 3 e 2t. _x = A x + f; x(0) = x 0.
.. SYSTEMS OF ODES. Systems of ODEs MATH 94 FALL 98 PRELIM # 94FA8PQ.tex.. a) Convert the third order dierential equation into a rst oder system _x = A x, with y " y" y + y = x = @ x x x b) The equation
More informationStudy Guide/Practice Exam 3
Study Guide/Practice Exam 3 This study guide/practice exam covers only the material since exam. The final exam, however, is cumulative so you should be sure to thoroughly study earlier material. The distribution
More information13 Implicit Differentiation
- 13 Implicit Differentiation This sections highlights the difference between explicit and implicit expressions, and focuses on the differentiation of the latter, which can be a very useful tool in mathematics.
More informationJim Lambers MAT 460/560 Fall Semester Practice Final Exam
Jim Lambers MAT 460/560 Fall Semester 2009-10 Practice Final Exam 1. Let f(x) = sin 2x + cos 2x. (a) Write down the 2nd Taylor polynomial P 2 (x) of f(x) centered around x 0 = 0. (b) Write down the corresponding
More informationx x2 2 + x3 3 x4 3. Use the divided-difference method to find a polynomial of least degree that fits the values shown: (b)
Numerical Methods - PROBLEMS. The Taylor series, about the origin, for log( + x) is x x2 2 + x3 3 x4 4 + Find an upper bound on the magnitude of the truncation error on the interval x.5 when log( + x)
More informationMath 106 Fall 2014 Exam 2.1 October 31, ln(x) x 3 dx = 1. 2 x 2 ln(x) + = 1 2 x 2 ln(x) + 1. = 1 2 x 2 ln(x) 1 4 x 2 + C
Math 6 Fall 4 Exam. October 3, 4. The following questions have to do with the integral (a) Evaluate dx. Use integration by parts (x 3 dx = ) ( dx = ) x3 x dx = x x () dx = x + x x dx = x + x 3 dx dx =
More information2003 Mathematics. Advanced Higher. Finalised Marking Instructions
2003 Mathematics Advanced Higher Finalised Marking Instructions 2003 Mathematics Advanced Higher Section A Finalised Marking Instructions Advanced Higher 2003: Section A Solutions and marks A. (a) Given
More information28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod)
28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod) θ + ω 2 sin θ = 0. Indicate the stable equilibrium points as well as the unstable equilibrium points.
More informationNon-Dimensionalization of Differential Equations. September 9, 2014
Non-Dimensionalization of Differential Equations September 9, 2014 1 Non-dimensionalization The most elementary idea of dimensional analysis is this: in any equation, the dimensions of the right hand side
More informationVirtual University of Pakistan
Virtual University of Pakistan File Version v.0.0 Prepared For: Final Term Note: Use Table Of Content to view the Topics, In PDF(Portable Document Format) format, you can check Bookmarks menu Disclaimer:
More informationNumerical solution of ODEs
Numerical solution of ODEs Arne Morten Kvarving Department of Mathematical Sciences Norwegian University of Science and Technology November 5 2007 Problem and solution strategy We want to find an approximation
More informationCHALLENGE! (0) = 5. Construct a polynomial with the following behavior at x = 0:
TAYLOR SERIES Construct a polynomial with the following behavior at x = 0: CHALLENGE! P( x) = a + ax+ ax + ax + ax 2 3 4 0 1 2 3 4 P(0) = 1 P (0) = 2 P (0) = 3 P (0) = 4 P (4) (0) = 5 Sounds hard right?
More informationNumerical Methods for Initial Value Problems; Harmonic Oscillators
1 Numerical Methods for Initial Value Problems; Harmonic Oscillators Lab Objective: Implement several basic numerical methods for initial value problems (IVPs), and use them to study harmonic oscillators.
More informationGraded Project #1. Part 1. Explicit Runge Kutta methods. Goals Differential Equations FMN130 Gustaf Söderlind and Carmen Arévalo
2008-11-07 Graded Project #1 Differential Equations FMN130 Gustaf Söderlind and Carmen Arévalo This homework is due to be handed in on Wednesday 12 November 2008 before 13:00 in the post box of the numerical
More informationFinite Difference Calculus
Chapter 3 Discretization of Computational Domain An Introduction to First Session Contents: 1) Approximation of Derivatives 2) Order Symbols 3) High-Order Derivatives 4) Richardson s Extrapolation (The
More informationVectors, matrices, eigenvalues and eigenvectors
Vectors, matrices, eigenvalues and eigenvectors 1 ( ) ( ) ( ) Scaling a vector: 0.5V 2 0.5 2 1 = 0.5 = = 1 0.5 1 0.5 ( ) ( ) ( ) ( ) Adding two vectors: V + W 2 1 2 + 1 3 = + = = 1 3 1 + 3 4 ( ) ( ) a
More informationBessel s Equation. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics
Bessel s Equation MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Background Bessel s equation of order ν has the form where ν is a constant. x 2 y + xy
More informationx 2 y = 1 2. Problem 2. Compute the Taylor series (at the base point 0) for the function 1 (1 x) 3.
MATH 8.0 - FINAL EXAM - SOME REVIEW PROBLEMS WITH SOLUTIONS 8.0 Calculus, Fall 207 Professor: Jared Speck Problem. Consider the following curve in the plane: x 2 y = 2. Let a be a number. The portion of
More informationContents. 2.1 Vectors in R n. Linear Algebra (part 2) : Vector Spaces (by Evan Dummit, 2017, v. 2.50) 2 Vector Spaces
Linear Algebra (part 2) : Vector Spaces (by Evan Dummit, 2017, v 250) Contents 2 Vector Spaces 1 21 Vectors in R n 1 22 The Formal Denition of a Vector Space 4 23 Subspaces 6 24 Linear Combinations and
More informationApplied Math for Engineers
Applied Math for Engineers Ming Zhong Lecture 15 March 28, 2018 Ming Zhong (JHU) AMS Spring 2018 1 / 28 Recap Table of Contents 1 Recap 2 Numerical ODEs: Single Step Methods 3 Multistep Methods 4 Method
More informationREVIEW OF DIFFERENTIAL CALCULUS
REVIEW OF DIFFERENTIAL CALCULUS DONU ARAPURA 1. Limits and continuity To simplify the statements, we will often stick to two variables, but everything holds with any number of variables. Let f(x, y) be
More informationIntroduction to Hamiltonian systems
Introduction to Hamiltonian systems Marlis Hochbruck Heinrich-Heine Universität Düsseldorf Oberwolfach Seminar, November 2008 Examples Mathematical biology: Lotka-Volterra model First numerical methods
More informationChapter 9 Global Nonlinear Techniques
Chapter 9 Global Nonlinear Techniques Consider nonlinear dynamical system 0 Nullcline X 0 = F (X) = B @ f 1 (X) f 2 (X). f n (X) x j nullcline = fx : f j (X) = 0g equilibrium solutions = intersection of
More informationMath Review Night: Work and the Dot Product
Math Review Night: Work and the Dot Product Dot Product A scalar quantity Magnitude: A B = A B cosθ The dot product can be positive, zero, or negative Two types of projections: the dot product is the parallel
More informationf (x) = k=0 f (0) = k=0 k=0 a k k(0) k 1 = a 1 a 1 = f (0). a k k(k 1)x k 2, k=2 a k k(k 1)(0) k 2 = 2a 2 a 2 = f (0) 2 a k k(k 1)(k 2)x k 3, k=3
1 M 13-Lecture Contents: 1) Taylor Polynomials 2) Taylor Series Centered at x a 3) Applications of Taylor Polynomials Taylor Series The previous section served as motivation and gave some useful expansion.
More information8 Numerical Integration of Ordinary Differential
8 Numerical Integration of Ordinary Differential Equations 8.1 Introduction Most ordinary differential equations of mathematical physics are secondorder equations. Examples include the equation of motion
More informationStudy # 1 11, 15, 19
Goals: 1. Recognize Taylor Series. 2. Recognize the Maclaurin Series. 3. Derive Taylor series and Maclaurin series representations for known functions. Study 11.10 # 1 11, 15, 19 f (n) (c)(x c) n f(c)+
More informationAnalysis of Dynamical Systems
2 YFX1520 Nonlinear phase portrait Mathematical Modelling and Nonlinear Dynamics Coursework Assignments 1 ϕ (t) 0-1 -2-6 -4-2 0 2 4 6 ϕ(t) Dmitri Kartofelev, PhD 2018 Variant 1 Part 1: Liénard type equation
More information1 Ordinary differential equations
Numerical Analysis Seminar Frühjahrssemester 08 Lecturers: Prof. M. Torrilhon, Prof. D. Kressner The Störmer-Verlet method F. Crivelli (flcrivel@student.ethz.ch May 8, 2008 Introduction During this talk
More informationFIXED POINT ITERATION
FIXED POINT ITERATION The idea of the fixed point iteration methods is to first reformulate a equation to an equivalent fixed point problem: f (x) = 0 x = g(x) and then to use the iteration: with an initial
More information4 Differential Equations
Advanced Calculus Chapter 4 Differential Equations 65 4 Differential Equations 4.1 Terminology Let U R n, and let y : U R. A differential equation in y is an equation involving y and its (partial) derivatives.
More informationMath Final Exam Review
Math - Final Exam Review. Find dx x + 6x +. Name: Solution: We complete the square to see if this function has a nice form. Note we have: x + 6x + (x + + dx x + 6x + dx (x + + Note that this looks a lot
More informationAstronomy 8824: Numerical Methods Notes 2 Ordinary Differential Equations
Astronomy 8824: Numerical Methods Notes 2 Ordinary Differential Equations Reading: Numerical Recipes, chapter on Integration of Ordinary Differential Equations (which is ch. 15, 16, or 17 depending on
More informationLecture Notes on Partial Dierential Equations (PDE)/ MaSc 221+MaSc 225
Lecture Notes on Partial Dierential Equations (PDE)/ MaSc 221+MaSc 225 Dr. Asmaa Al Themairi Assistant Professor a a Department of Mathematical sciences, University of Princess Nourah bint Abdulrahman,
More informationReview: Power series define functions. Functions define power series. Taylor series of a function. Taylor polynomials of a function.
Taylor Series (Sect. 10.8) Review: Power series define functions. Functions define power series. Taylor series of a function. Taylor polynomials of a function. Review: Power series define functions Remarks:
More information(f(x) P 3 (x)) dx. (a) The Lagrange formula for the error is given by
1. QUESTION (a) Given a nth degree Taylor polynomial P n (x) of a function f(x), expanded about x = x 0, write down the Lagrange formula for the truncation error, carefully defining all its elements. How
More information17.2 Nonhomogeneous Linear Equations. 27 September 2007
17.2 Nonhomogeneous Linear Equations 27 September 2007 Nonhomogeneous Linear Equations The differential equation to be studied is of the form ay (x) + by (x) + cy(x) = G(x) (1) where a 0, b, c are given
More informationSECTION A. f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes.
SECTION A 1. State the maximal domain and range of the function f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes. 2. By evaluating f(0),
More informationMA2AA1 (ODE s): The inverse and implicit function theorem
MA2AA1 (ODE s): The inverse and implicit function theorem Sebastian van Strien (Imperial College) February 3, 2013 Differential Equations MA2AA1 Sebastian van Strien (Imperial College) 0 Some of you did
More informationMath 5630: Iterative Methods for Systems of Equations Hung Phan, UMass Lowell March 22, 2018
1 Linear Systems Math 5630: Iterative Methods for Systems of Equations Hung Phan, UMass Lowell March, 018 Consider the system 4x y + z = 7 4x 8y + z = 1 x + y + 5z = 15. We then obtain x = 1 4 (7 + y z)
More information8.7 Taylor s Inequality Math 2300 Section 005 Calculus II. f(x) = ln(1 + x) f(0) = 0
8.7 Taylor s Inequality Math 00 Section 005 Calculus II Name: ANSWER KEY Taylor s Inequality: If f (n+) is continuous and f (n+) < M between the center a and some point x, then f(x) T n (x) M x a n+ (n
More informationSMA 208: Ordinary differential equations I
SMA 208: Ordinary differential equations I First Order differential equations Lecturer: Dr. Philip Ngare (Contacts: pngare@uonbi.ac.ke, Tue 12-2 PM) School of Mathematics, University of Nairobi Feb 26,
More informationTutorial-1, MA 108 (Linear Algebra)
Tutorial-1, MA 108 (Linear Algebra) 1. Verify that the function is a solution of the differential equation on some interval, for any choice of the arbitrary constants appearing in the function. (a) y =
More informationMathematical Economics: Lecture 2
Mathematical Economics: Lecture 2 Yu Ren WISE, Xiamen University September 25, 2012 Outline 1 Number Line The number line, origin (Figure 2.1 Page 11) Number Line Interval (a, b) = {x R 1 : a < x < b}
More informationINRIA Rocquencourt, Le Chesnay Cedex (France) y Dept. of Mathematics, North Carolina State University, Raleigh NC USA
Nonlinear Observer Design using Implicit System Descriptions D. von Wissel, R. Nikoukhah, S. L. Campbell y and F. Delebecque INRIA Rocquencourt, 78 Le Chesnay Cedex (France) y Dept. of Mathematics, North
More information