Math 128A Spring 2003 Week 12 Solutions


 Marsha Osborne
 1 years ago
 Views:
Transcription
1 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. HigherOrder Equations and Systems of Differential Equations 1. Use the RungeKutta method for systems to approximate the solution of the following system of firstorder differential equations, and compare the result to the actual solution. b. u 1 = 4u 1 2u 2 + cos t + 4 sin t, 0 t 2, u 1 (0) = 0; u 2 = 3u 1 + u 2 3 sin t, 0 t 2, u 2 (0) = 1; h = 0.1; actual solutions u 1 (t) = 2e t 2e 2t + sin t and u 2 (t) = 3e t + 2e 2t. Solution. b. Applying the RungeKutta method, we obtain the following results. t i w i, 1 u 1 (t i ) w i, 2 u 2 (t i ) Use the RungeKutta for Systems Algorithm to approximate the solution of the following higherorder differential equation, and compare the result to the actual solution. b. t 2 y 2ty + 2y = t 3 ln t, 1 t 2, y(1) = 1, y (1) = 0, with h = 0.1; actual solution y(t) = 7 4 t t3 ln t 3 4 t3. 1
2 Solution. b. Solving this higherorder differential equation is equivalent to solving the following system of differential equations. u 1 = u 2, 1 t 2, u 1 (1) = 1; u 2 = 2 t u 2 2 t 2 u 1 + t ln t, 1 t 2, u 2 (1) = 0; where y(t) = u 1 (t) and y (t) = u 2 (t). following. Applying RungeKutta to this system we get the t i w i, 1 y(t i ) w i, Change the Adams FourthOrder PredictorCorrector Algorithm to obtain approximate solutions to systems of firstorder equations. Solution. The changed algorithm is as follows. Adams FourthOrder Predictor Corrector for Systems To approximate the solution of the mthorder system of firstorder initialvalue problems u j = f j (t, u 1, u 2,..., u m ) a t b, with u j (a) = α j for j = 1, 2,..., m at (N + 1) equally spaced numbers in the interval [a, b]: INPUT endpoints a, b; number of equations m; integer N; initial conditions α 1,..., α m. OUTPUT approximations of w j to u j (t) at the (N + 1) values of t. Step 1 Set h = (b a)/n; t 0 = a. Step 2 For j = 1, 2,..., m set w j = α j. Step 3 OUTPUT (t, w 1, w 2,..., w m ). Step 4 For i = 1, 2, 3 do steps Step 5 For j = 1, 2,..., m set k 1, j = hf j (t, w 1, w 2,..., w m ). Step 6 For j = 1, 2,..., m set k 2, j = hf j (t + h 2, w k 1, 1, w k 1, 2,..., w m k 1, m). Step 7 For j = 1, 2,..., m set k 3, j = hf j (t + h 2, w k 2, 1, w k 2, 2,..., w m k 2, m). Step 8 For j = 1, 2,..., m set k 4, j = hf j (t + h, w 1 + k 3, 1, w 2 + k 3, 2,..., w m + k 3, m ). Step 9 For j = 1, 2,..., m set w i, j = w i 1, j + (k 1, j + 2k 2, j + 2k 3, j + k 4, j )/6. Step 10 Set t i = a + ih. Step 11 OUTPUT (t, w 1, w 2,..., w m ). Step 12 For i = 4,..., N do Steps Step 13 Set t i = a + ih. Step 14 For j = 1, 2,..., m set 2
3 Step 15 w i, j = w i 1, j + h[55f j (t i 1, w i 1, 1, w i 1, 2,..., w i 1, m ) 59f j (t i 2, w i 2, 1, w i 2, 2,..., w i 2, m ) + 37f j (t i 3, w i 3, 1, w i 3, 2,..., w i 3, m ) 9f j (t i 4, w i 4, 1, w i 4, 2,..., w i 4, m )]/24 For j = 1, 2,..., m set w i, j = w i 1, j + h[9f j (t i, w i, 1, w i, 2,..., w i, m ) + 19f j (t i 1, w i 1, 1, w i 1, 2,..., w i 1, m ) 5f j (t i 2, w i 2, 1, w i 2, 2,..., w i 2, m ) + f j (t i 3, w i 3, 1, w i 3, 2,..., w i 3, m )]/24 Step 16 OUTPUT (t, w i, 1, w i, 2,..., w i, m ). Step 17 STOP 5. Repeat Exercise 2 using the algorithm developed in Exercise 3. Solution. b. Applying the algorithm, we arrive at the following results: t i w i, 1 w i, 2 y(t i ) Suppose the swinging pendulum described in the lead example of this chapter is 2 ft long and that g = ft/s 2. With h = 0.1 s, compare the angle θ obtained for the following two initialvalue problems: a. b. at t = 0, 1, and 2 s. d 2 θ dt 2 + g L sin θ = 0, θ(0) = π 6, θ (0) = 0. d 2 θ dt 2 + g L θ = 0, θ(0) = π 6, θ (0) = 0, Solution. a. The solution for this initialvalue problem using the RungeKutta method for systems is as follows. 3
4 t i w i, 1 w i, b. The solution for this initialvalue problem using the RungeKutta method for systems is as follows. t i w i, 1 w i, Thus we see that at 0, 1 and 2 s, we have , , and radians respectively for the first initialvalue problem and , , and radians respectively for the second initialvalue problem. 7. The study of mathematical models for predicting the population dynamics of competing species has its origin in independent works published in the early part of this century by A. J. Lotka and V. Volterra. Consider the problem of predicting the population of two species, one of which is 4
5 a predator, whose population at time t is x 2 (t), feeding on the other, which is the prey, whose population is x 1 (t). We will assume that the prey always has an adequate food supply and that its birth rate at any time is proportional to the number of prey alive at that time; that is, birth rate (prey) is k 1 x 1 (t). The death rate of the prey depends on both the number of prey and predators alive at that time. For simplicity, we assume death rate (prey) = k 2 x 1 (t)x 2 (t). The birth rate of the predator, on the other hand, depends on its food supply, x 1 (t), as well as on the number of predators available for reproductive purposes. For this reason, we assume that the birth (predator) is k 3 x 1 (t)x 2 (t). The death rate of the predator will be taken as simply proportional to the number of predators alive at the time; that is, death rate (predator) = k 4 x 2 (t). Since x 1(t) and x 2(t) represent the change in the prey and predator populations, respectively, with respect to time, the problem is expressed by the system of nonlinear differential equations x 1(t) = k 1 x 1 (t) k 2 x 1 (t)x 2 (t) and x 2(t) = k 3 x 1 (t)x 2 (t) k 4 x 2 (t). Solve this system for 0 t 4, assuming that the initial population of the prey is 1000 and of the predators is 500 and that the constants are k 1 = 3, k 2 = 0.002, k 3 = , and k 4 = 0.5. Sketch a graph of the solutions to this problem, plotting both populations with time, and describe the physical phenomena represented. Is there a stable solution to this population model? If so, for what values x 1 and x 2 is the solution stable? Solution. Running the RungeKutta for systems algorithm, we get that at time t = 4, the number of prey is approximately and the number of predators is approximately There are two stable solutions to the population model occurring when x 1 and x 2 are both 0. One occurs when both x 1 and x 2 are 0. The other occurs when both k 1 k 2 x 2 (t) = 0 and k 3 x 1 (t) k 4 = 0 implying that x 2 (t) = k 1 k 2 and x 1 (t) = k 4 k 3. With the current values of k 1, k 2, k 3, and k 4, this gives x 1 (t) = and x 2 (t) = which cannot happen since we cannot have one third of one prey. Burden & Faires Stability 4. Consider the differential equation a. Show that y = f(t, y), a t b, y(a) = α. y (t i ) = 3y(t i) + 4y(t i+1 ) y(t i+2 ) 2h for some ξ, where t i < ξ i < t i+2. b. Part (a) suggests the difference method use this method to solve + h2 3 y (ξ 1 ), w i+2 = 4w i+1 3w i 2hf(t i, w i ), for i = 0, 1,..., N 2. y = 1 y, 0 t 1, y(0) = 0, with h = 0.1. Use the starting values w 0 = 0 and w 1 = y(t 1 ) = 1 e 0.1. c. Repeat part (b) with h = 0.01 and w 1 = 1 e d. Analyze this method for consistency, stability, and convergence. Solution. a. Applying the three point method for approximating the derivative at t i given the values at t i, t i+1 = t i + h, and t i+2 = t i + 2h, we get the following equation: for some ξ i, where t i < ξ i < t i+2. y (t i ) = 3y(t i) + 4y(t i+1 ) y(t i+2 ) 2h + h2 3 y (ξ i ), 5
6 b. Applying this method with h = 0.1 we obtain the following results. t i w i c. Applying this method with h = 0.01 we obtain the following results
7
8 d. From part (a), we get the following: τ i+1 (h) = y(t i+1) 4y(t i ) + 3y(t i 1 ) h + 2f(t i 1, y(t i 1 )) = 2h2 3 y (ξ i 1 ). Thus this method is consistent. The method has characteristic equation given by: P (λ) = λ 2 4λ + 3. This equation has roots 1 and 3, but since 3 > 1, the method is unstable. Thus the method is not convergent. 5. Given the multistep method w i+1 = 3 2 w i + 3w i w i 2 + 3hf(t i, w i ), for i = 2,..., N 1, with starting values w 0, w 1, w 2 : a. Find the local truncation error. b. Comment on consistency, stability, and convergence. Solution. a. Using the polynomial approximation agreeing with y(t) at t i+1, t i, t i 1, and t i 2, we have: y(t) = P (t) + y(4) (ξ(t)) (t t i+1 )(t t i )(t t i 1 )(t t i 2 ). 24 Plugging P (t) into τ i+1 (h) = y(t i+1) y(t i) 3y(t i 1 ) y(t i 2) h we get 0 and plugging in the error term we get: 1 4 h3 y (4) (ξ(t i )). 3f(t i, y(t i )), b. Part (a) shows that the method is consistent. The characteristic equation is given by P (λ) = λ λ2 3λ = 0. Factoring, we get ( (λ 1) λ λ 1 ) =
9 The two roots other than 1 are given by but 5 ± 33, > 1, which implies that the method is unstable and hence not convergent. 8. Consider the problem y = 0, 0 t 10, y(0) = 0, which has the solution y = 0. If the difference method of Exercise 4 is applied to the problem, then w i+1 = 4w i 3w i 1, for i = 1, 2,..., N 1, w 0 = 0, and w 1 = α 1. Suppose w 1 = α 1 = ɛ, where ɛ is a small rounding error. Compute w i exactly for i = 2, 3, 4, 5, 6 to find how the error ɛ is propagated. Solution. Applying the method, we have w 0 = 0 w 1 = ε w 2 = 4w 1 3w 0 = 4ε w 3 = 13ε w 4 = 40ε w 5 = 121ε w 6 = 364ε Burden & Faires Stiff Differential Equations 1. Solve the following stiff initialvalue problem using Euler s method, and compare the result with the actual solution. c. Solution. y = 20y + 20 sin t + cos t, 0 t 2, y(0) = 1, with h = 0.25; actual solution y(t) = sin t + e 20t. c. Applying the Euler s method we get the following: t i w i y(t i )
10 2. Repeat Exercise 1 using the RungeKutta fourthorder method. Solution. c. Applying the RungeKutta method to the previous differential equation, we get the following: t i w i y(t i ) Solve the following stiff initialvalue problem using the RungeKutta fourthorder method with (a) h = 0.1 and (b) h = u 1 = 32u u t + 2 3, 0 t 0.5, u 1(0) = 1 3 ; u 2 = 66u 1 133u t 1 3, 0 t 0.5, u 2(0) = 1 3. Compare the results to the actual solution, u 1 (t) = 2 3 t e t 1 3 e 100t and u 2 (t) = 1 3 t 1 3 e t e 100t. Solution. a. When h = 0.1, the RungeKutta fourthorder method gives: t i w 1, i u 1 (t i ) w 2, i u 2 (t i ) b. When h = 0.025, the RungeKutta fourthorder method gives: 10
11 t i w 1, i u 1 (t i ) w 2, i u 2 (t i ) Show that the fourthorder RungeKutta method, k 1 = hf(t i, w i ), k 2 = hf(t i + h/2, w i + k 1 /2), k 3 = hf(t i + h/2, w i + k 2 /2), k 4 = hf(t i + h, w i + k 3 ), w i+1 = w i (k 1 + 2k 2 + 2k 3 + k 4 ), when applied to the differential equation y = λy, can be written in the form w i+1 = (1 + hλ + 12 (hλ) (hλ) ) (hλ)4 w i. 11
12 Solution. We calculate: k 1 = hλw ( i k 2 = hλ w i + k ) 1 2 = (hλ + 12 ) (hλ)2 w i ( k 3 = hλ w i + 1 ) 2 k 2 = (hλ + 12 (hλ) ) (hλ)3 w i k 4 = hλ(w i + k 3 ) = (hλ + (hλ) (hλ) ) (hλ)4 w i w i+1 = w i (k 1 + 2k 2 + 2k 3 + k 4 ) = w i + 1 ( hλ + 2hλ + (hλ) 2 + 2hλ + (hλ) (hλ)3 = + hλ + (hλ) (hλ)3 + 1 ) 4 (hλ)4 w i (1 + hλ + 12 (hλ) (hλ) ) (hλ)4 w i 8. The Backward Euler onestep method is defined by w i+1 = w i + hf(t i+1, w i+1 ), for i = 0,..., N 1. a. Show that Q(hλ) = 1/(1 hλ) for the Backward Euler method. b. Apply the Backward Euler method to the differential equation given in Exercise 1. Solution. a. Applying the Backward Euler method to the differential equation y = λy, we get: w i+1 = w i + hf(t i+1, w i+1 ) = w i + hλw i+1 (1 hλ)w i+1 = w i w i+1 = w i 1 hλ = Q(hλ)w i Q(hλ) = 1 1 hλ b. Applying the Backward Euler method to the differential equation from Exercise 1, we get: Thus the results are: w i+1 = w i + hf(t i+1, w i+1 ) = w i + h( 20w i sin(t i+1 ) + cos(t i+1 )) (1 + 20h)w i+1 = w i + 20h sin(t i+1 ) + h cos(t i+1 ) w i+1 = w i + 20h sin(t i+1 ) + h cos(t i+1 ) h 12
13 t i w i y(t i )
Math 128A Spring 2003 Week 11 Solutions Burden & Faires 5.6: 1b, 3b, 7, 9, 12 Burden & Faires 5.7: 1b, 3b, 5 Burden & Faires 5.
Math 128A Spring 2003 Week 11 Solutions Burden & Faires 5.6: 1b, 3b, 7, 9, 12 Burden & Faires 5.7: 1b, 3b, 5 Burden & Faires 5.8: 1b, 3b, 4 Burden & Faires 5.6. Multistep Methods 1. Use all the AdamsBashforth
More informationNumerical Methods  Initial Value Problems for ODEs
Numerical Methods  Initial Value Problems for ODEs Y. K. Goh Universiti Tunku Abdul Rahman 2013 Y. K. Goh (UTAR) Numerical Methods  Initial Value Problems for ODEs 2013 1 / 43 Outline 1 Initial Value
More informationInitial value problems for ordinary differential equations
Initial value problems for ordinary differential equations Xiaojing Ye, Math & Stat, Georgia State University Spring 2019 Numerical Analysis II Xiaojing Ye, Math & Stat, Georgia State University 1 IVP
More informationFourth Order RKMethod
Fourth Order RKMethod The most commonly used method is RungeKutta fourth order method. The fourth order RKmethod is y i+1 = y i + 1 6 (k 1 + 2k 2 + 2k 3 + k 4 ), Ordinary Differential Equations (ODE)
More informationLecture 5: Single Step Methods
Lecture 5: Single Step Methods J.K. Ryan@tudelft.nl WI3097TU Delft Institute of Applied Mathematics Delft University of Technology 1 October 2012 () Single Step Methods 1 October 2012 1 / 44 Outline 1
More informationConsistency and Convergence
Jim Lambers MAT 77 Fall Semester 01011 Lecture 0 Notes These notes correspond to Sections 1.3, 1.4 and 1.5 in the text. Consistency and Convergence We have learned that the numerical solution obtained
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 informationEuler s Method, cont d
Jim Lambers MAT 461/561 Spring Semester 00910 Lecture 3 Notes These notes correspond to Sections 5. and 5.4 in the text. Euler s Method, cont d We conclude our discussion of Euler s method with an example
More informationDo not turn over until you are told to do so by the Invigilator.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 216 17 INTRODUCTION TO NUMERICAL ANALYSIS MTHE612B Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions.
More informationmultistep methods Last modified: November 28, 2017 Recall that we are interested in the numerical solution of the initial value problem (IVP):
MATH 351 Fall 217 multistep methods http://www.phys.uconn.edu/ rozman/courses/m351_17f/ Last modified: November 28, 217 Recall that we are interested in the numerical solution of the initial value problem
More information8.1 Introduction. Consider the initial value problem (IVP):
8.1 Introduction Consider the initial value problem (IVP): y dy dt = f(t, y), y(t 0)=y 0, t 0 t T. Geometrically: solutions are a one parameter family of curves y = y(t) in(t, y)plane. Assume solution
More informationOrdinary differential equation II
Ordinary Differential Equations ISC5315 1 Ordinary differential equation II 1 Some Basic Methods 1.1 Backward Euler method (implicit method) The algorithm looks like this: y n = y n 1 + hf n (1) In contrast
More informationNumerical Differential Equations: IVP
Chapter 11 Numerical Differential Equations: IVP **** 4/16/13 EC (Incomplete) 11.1 Initial Value Problem for Ordinary Differential Equations We consider the problem of numerically solving a differential
More informationMTH 452/552 Homework 3
MTH 452/552 Homework 3 Do either 1 or 2. 1. (40 points) [Use of ode113 and ode45] This problem can be solved by a modifying the mfiles odesample.m and odesampletest.m available from the author s webpage.
More informationOrdinary differential equations  Initial value problems
Education has produced a vast population able to read but unable to distinguish what is worth reading. G.M. TREVELYAN Chapter 6 Ordinary differential equations  Initial value problems In this chapter
More informationInitial value problems for ordinary differential equations
AMSC/CMSC 660 Scientific Computing I Fall 2008 UNIT 5: Numerical Solution of Ordinary Differential Equations Part 1 Dianne P. O Leary c 2008 The Plan Initial value problems (ivps) for ordinary differential
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 informationMathematics for chemical engineers. Numerical solution of ordinary differential equations
Mathematics for chemical engineers Drahoslava Janovská Numerical solution of ordinary differential equations Initial value problem Winter Semester 20152016 Outline 1 Introduction 2 One step methods Euler
More informationScientific Computing with Case Studies SIAM Press, Lecture Notes for Unit V Solution of
Scientific Computing with Case Studies SIAM Press, 2009 http://www.cs.umd.edu/users/oleary/sccswebpage Lecture Notes for Unit V Solution of Differential Equations Part 1 Dianne P. O Leary c 2008 1 The
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 9 Initial Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign
More informationOrdinary Differential Equations
Chapter 7 Ordinary Differential Equations Differential equations are an extremely important tool in various science and engineering disciplines. Laws of nature are most often expressed as different equations.
More informationInitialValue Problems for ODEs. Introduction to Linear Multistep Methods
InitialValue Problems for ODEs Introduction to Linear Multistep Methods Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University
More informationGraded Project #1. Part 1. Explicit Runge Kutta methods. Goals Differential Equations FMN130 Gustaf Söderlind and Carmen Arévalo
20081107 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 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 informationLecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 9
Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 9 Initial Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction
More informationOrdinary Differential Equations
CHAPTER 8 Ordinary Differential Equations 8.1. Introduction My section 8.1 will cover the material in sections 8.1 and 8.2 in the book. Read the book sections on your own. I don t like the order of things
More informationDELFT UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering, Mathematics and Computer Science
DELFT UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering, Mathematics and Computer Science ANSWERS OF THE TEST NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS ( WI3097 TU AESB0 ) Thursday April 6
More informationChap. 20: InitialValue Problems
Chap. 20: InitialValue Problems Ordinary Differential Equations Goal: to solve differential equations of the form: dy dt f t, y The methods in this chapter are all onestep methods and have the general
More informationJim Lambers MAT 772 Fall Semester Lecture 21 Notes
Jim Lambers MAT 772 Fall Semester 2111 Lecture 21 Notes These notes correspond to Sections 12.6, 12.7 and 12.8 in the text. Multistep Methods All of the numerical methods that we have developed for solving
More informationPart IB Numerical Analysis
Part IB Numerical Analysis Definitions Based on lectures by G. Moore Notes taken by Dexter Chua Lent 206 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after
More informationECE257 Numerical Methods and Scientific Computing. Ordinary Differential Equations
ECE257 Numerical Methods and Scientific Computing Ordinary Differential Equations Today s s class: Stiffness Multistep Methods Stiff Equations Stiffness occurs in a problem where two or more independent
More informationNumerical Methods for the Solution of Differential Equations
Numerical Methods for the Solution of Differential Equations Markus Grasmair Vienna, winter term 2011 2012 Analytical Solutions of Ordinary Differential Equations 1. Find the general solution of the differential
More informationCS 450 Numerical Analysis. Chapter 9: Initial Value Problems for Ordinary Differential Equations
Lecture slides based on the textbook Scientific Computing: An Introductory Survey by Michael T. Heath, copyright c 2018 by the Society for Industrial and Applied Mathematics. http://www.siam.org/books/cl80
More informationChapter 10. Initial value Ordinary Differential Equations
Chapter 10 Initial value Ordinary Differential Equations Consider the problem of finding a function y(t) that satisfies the following ordinary differential equation (ODE): dy dt = f(t, y), a t b. The function
More informationMath Numerical Analysis Homework #4 Due End of term. y = 2y t 3y2 t 3, 1 t 2, y(1) = 1. n=(ba)/h+1; % the number of steps we need to take
Math 32  Numerical Analysis Homework #4 Due End of term Note: In the following y i is approximation of y(t i ) and f i is f(t i,y i ).. Consider the initial value problem, y = 2y t 3y2 t 3, t 2, y() =.
More informationBindel, Fall 2011 Intro to Scientific Computing (CS 3220) Week 12: Monday, Apr 18. HW 7 is posted, and will be due in class on 4/25.
Logistics Week 12: Monday, Apr 18 HW 6 is due at 11:59 tonight. HW 7 is posted, and will be due in class on 4/25. The prelim is graded. An analysis and rubric are on CMS. Problem du jour For implicit methods
More information9.6 PredictorCorrector Methods
SEC. 9.6 PREDICTORCORRECTOR METHODS 505 AdamsBashforthMoulton Method 9.6 PredictorCorrector Methods The methods of Euler, Heun, Taylor, and RungeKutta are called singlestep methods because they use
More informationAN OVERVIEW. Numerical Methods for ODE Initial Value Problems. 1. Onestep methods (Taylor series, RungeKutta)
AN OVERVIEW Numerical Methods for ODE Initial Value Problems 1. Onestep methods (Taylor series, RungeKutta) 2. Multistep methods (PredictorCorrector, Adams methods) Both of these types of methods are
More informationDifferential Equations: Homework 8
Differential Equations: Homework 8 Alvin Lin January 08  May 08 Section.6 Exercise Find a general solution to the differential equation using the method of variation of parameters. y + y = tan(t) r +
More informationDo not turn over until you are told to do so by the Invigilator.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 216 17 INTRODUCTION TO NUMERICAL ANALYSIS MTHE712B Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions.
More information1 Error Analysis for Solving IVP
cs412: introduction to numerical analysis 12/9/10 Lecture 25: Numerical Solution of Differential Equations Error Analysis Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mark Cowlishaw, Nathanael Fillmore
More informationPowerPoints organized by Dr. Michael R. Gustafson II, Duke University
Part 6 Chapter 20 InitialValue Problems PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright The McGrawHill Companies, Inc. Permission required for reproduction
More informationA STUDY OF GENERALIZED ADAMSMOULTON METHOD FOR THE SATELLITE ORBIT DETERMINATION PROBLEM
Korean J Math 2 (23), No 3, pp 27 283 http://dxdoiorg/568/kjm232327 A STUDY OF GENERALIZED ADAMSMOULTON METHOD FOR THE SATELLITE ORBIT DETERMINATION PROBLEM Bum Il Hong and Nahmwoo Hahm Abstract In this
More informationMath 308 Week 8 Solutions
Math 38 Week 8 Solutions There is a solution manual to Chapter 4 online: www.pearsoncustom.com/tamu math/. This online solutions manual contains solutions to some of the suggested problems. Here are solutions
More informationHigher Order Taylor Methods
Higher Order Taylor Methods Marcelo Julio Alvisio & Lisa Marie Danz May 6, 2007 Introduction Differential equations are one of the building blocks in science or engineering. Scientists aim to obtain numerical
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 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 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 informationNumerical methods for solving ODEs
Chapter 2 Numerical methods for solving ODEs We will study two methods for finding approximate solutions of ODEs. Such methods may be used for (at least) two reasons the ODE does not have an exact solution
More informationLecture Notes on Numerical Differential Equations: IVP
Lecture Notes on Numerical Differential Equations: IVP Professor Biswa Nath Datta Department of Mathematical Sciences Northern Illinois University DeKalb, IL. 60115 USA E mail: dattab@math.niu.edu URL:
More informationOrdinary Differential Equations (ODE)
Ordinary Differential Equations (ODE) Why study Differential Equations? Many physical phenomena are best formulated mathematically in terms of their rate of change. Motion of a swinging pendulum Bungeejumper
More informationMathematical Models. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Spring Department of Mathematics
Mathematical Models MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Ordinary Differential Equations The topic of ordinary differential equations (ODEs)
More informationMathematical Models. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics
Mathematical Models MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Ordinary Differential Equations The topic of ordinary differential equations (ODEs) is
More informationPowerPoints organized by Dr. Michael R. Gustafson II, Duke University
Part 6 Chapter 20 InitialValue Problems PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright The McGrawHill Companies, Inc. Permission required for reproduction
More informationOrdinary Differential Equations I
Ordinary Differential Equations I CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Ordinary Differential Equations I 1 / 32 Theme of Last Few
More informationOrdinary Differential Equations I
Ordinary Differential Equations I CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Ordinary Differential Equations I 1 / 27 Theme of Last Three
More informationVariable Step Size Differential Equation Solvers
Math55: Differential Equations 1/30 Variable Step Size Differential Equation Solvers Jason Brewer and George Little Introduction The purpose of developing numerical methods is to approximate the solution
More informationIntroduction to ODEs in Matlab. Jeremy L. Marzuola Charles Talbot Benjamin Wilson
Introduction to ODEs in Matlab Jeremy L. Marzuola Charles Talbot Benjamin Wilson Email address: marzuola@math.unc.edu Department of Mathematics, UNCChapel Hill, CB#3250 Phillips Hall, Chapel Hill, NC
More informationModule 4: Numerical Methods for ODE. Michael Bader. Winter 2007/2008
Outlines Module 4: for ODE Part I: Basic Part II: Advanced Lehrstuhl Informatik V Winter 2007/2008 Part I: Basic 1 Direction Fields 2 Euler s Method Outlines Part I: Basic Part II: Advanced 3 Discretized
More informationLinear Multistep Methods I: Adams and BDF Methods
Linear Multistep Methods I: Adams and BDF Methods Varun Shankar January 1, 016 1 Introduction In our review of 5610 material, we have discussed polynomial interpolation and its application to generating
More informationdt 2 The Order of a differential equation is the order of the highest derivative that occurs in the equation. Example The differential equation
Lecture 18 : Direction Fields and Euler s Method A Differential Equation is an equation relating an unknown function and one or more of its derivatives. Examples Population growth : dp dp = kp, or = kp
More informationLinear and Nonlinear Oscillators (Lecture 2)
Linear and Nonlinear Oscillators (Lecture 2) January 25, 2016 7/441 Lecture outline A simple model of a linear oscillator lies in the foundation of many physical phenomena in accelerator dynamics. A typical
More informationOn interval predictorcorrector methods
DOI 10.1007/s110750160220x ORIGINAL PAPER On interval predictorcorrector methods Andrzej Marcinia 1,2 Malgorzata A. Janowsa 3 Tomasz Hoffmann 4 Received: 26 March 2016 / Accepted: 3 October 2016 The
More informationStability Ordinates of Adams PredictorCorrector Methods
BIT manuscript No. will be inserted by the editor) Stability Ordinates of Adams PredictorCorrector Methods Michelle L. Ghrist Bengt Fornberg Jonah A. Reeger Received: date / Accepted: date Abstract How
More informationOutline. Calculus for the Life Sciences. What is a Differential Equation? Introduction. Lecture Notes Introduction to Differential Equa
Outline Calculus for the Life Sciences Lecture Notes to Differential Equations Joseph M. Mahaffy, jmahaffy@mail.sdsu.edu 1 Department of Mathematics and Statistics Dynamical Systems Group Computational
More informationSystems of Ordinary Differential Equations
Systems of Ordinary Differential Equations MATH 365 Ordinary Differential Equations J Robert Buchanan Department of Mathematics Fall 2018 Objectives Many physical problems involve a number of separate
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 informationCHAPTER 5: Linear Multistep Methods
CHAPTER 5: Linear Multistep Methods Multistep: use information from many steps Higher order possible with fewer function evaluations than with RK. Convenient error estimates. Changing stepsize or order
More informationMB4018 Differential equations
MB4018 Differential equations Part II http://www.staff.ul.ie/natalia/mb4018.html Prof. Natalia Kopteva Spring 2015 MB4018 (Spring 2015) Differential equations Part II 0 / 69 Section 1 SecondOrder Linear
More informationNumerical Methods for Differential Equations
Numerical Methods for Differential Equations Chapter 2: Runge Kutta and Linear Multistep methods Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the
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 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 information2tdt 1 y = t2 + C y = which implies C = 1 and the solution is y = 1
Lectures  Week 11 General First Order ODEs & Numerical Methods for IVPs In general, nonlinear problems are much more difficult to solve than linear ones. Unfortunately many phenomena exhibit nonlinear
More informationFORTRAN 77 Lesson 7. Reese Haywood Ayan Paul
FORTRAN 77 Lesson 7 Reese Haywood Ayan Paul 1 Numerical Integration A general all purpose Numerical Integration technique is the Trapezoidal Method. For most functions, this method is fast and accurate.
More informationLecture 4: Numerical solution of ordinary differential equations
Lecture 4: Numerical solution of ordinary differential equations Department of Mathematics, ETH Zürich General explicit onestep method: Consistency; Stability; Convergence. Highorder methods: Taylor
More informationMath 660 Lecture 4: FDM for evolutionary equations: ODE solvers
Math 660 Lecture 4: FDM for evolutionary equations: ODE solvers Consider the ODE u (t) = f(t, u(t)), u(0) = u 0, where u could be a vector valued function. Any ODE can be reduced to a first order system,
More informationLecture IV: Time Discretization
Lecture IV: Time Discretization Motivation Kinematics: continuous motion in continuous time Computer simulation: Discrete time steps t Discrete Space (mesh particles) Updating Position Force induces acceleration.
More information1 Ordinary Differential Equations
Ordinary Differential Equations.0 Mathematical Background.0. Smoothness Definition. A function f defined on [a, b] is continuous at ξ [a, b] if lim x ξ f(x) = f(ξ). Remark Note that this implies existence
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 informationElementary Differential Equations
Elementary Differential Equations George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 310 George Voutsadakis (LSSU) Differential Equations January 2014 1 /
More informationMathematics 22: Lecture 10
Mathematics 22: Lecture 10 Euler s Method Dan Sloughter Furman University January 22, 2008 Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, 2008 1 / 14 Euler s method Consider the
More informationRungeKutta methods. With orders of Taylor methods yet without derivatives of f (t, y(t))
RungeKutta metods Wit orders of Taylor metods yet witout derivatives of f (t, y(t)) First order Taylor expansion in two variables Teorem: Suppose tat f (t, y) and all its partial derivatives are continuous
More informationNumerical Integration of Ordinary Differential Equations for Initial Value Problems
Numerical Integration of Ordinary Differential Equations for Initial Value Problems Gerald Recktenwald Portland State University Department of Mechanical Engineering gerry@me.pdx.edu These slides are a
More informationMATH 250 Homework 4: Due May 4, 2017
Due May 4, 17 Answer the following questions to the best of your ability. Solutions should be typed. Any plots or graphs should be included with the question (please include the questions in your solutions).
More informationNotes for Numerical Analysis Math 5466 by S. Adjerid Virginia Polytechnic Institute and State University (A Rough Draft) Contents Numerical Methods for ODEs 5. Introduction............................
More informationSolution. Your sketch should show both x and y axes marked from 5 to 5 and circles of radius 1, 3, and 5 centered at the origin.
Solutions of the Sample Problems for the First InClass Exam Math 246, Fall 208, Professor David Levermore () (a) Sketch the graph that would be produced by the following Matlab command. fplot(@(t) 2/t,
More informationComputers, Lies and the Fishing Season
1/47 Computers, Lies and the Fishing Season Liz Arnold May 21, 23 Introduction Computers, lies and the fishing season takes a look at computer software programs. As mathematicians, we depend on computers
More informationSolutions of the Sample Problems for the First InClass Exam Math 246, Fall 2017, Professor David Levermore
Solutions of the Sample Problems for the First InClass Exam Math 246, Fall 207, Professor David Levermore () (a) Give the integral being evaluated by the following Matlab command. int( x/(+xˆ4), x,0,inf)
More informationNumerical Methods for Ordinary Differential Equations
CHAPTER 1 Numerical Methods for Ordinary Differential Equations In this chapter we discuss numerical method for ODE. We will discuss the two basic methods, Euler s Method and RungeKutta Method. 1. Numerical
More information4 Stability analysis of finitedifference methods for ODEs
MATH 337, by T. Lakoba, University of Vermont 36 4 Stability analysis of finitedifference methods for ODEs 4.1 Consistency, stability, and convergence of a numerical method; Main Theorem In this Lecture
More informationSolving Ordinary Differential Equations
Solving Ordinary Differential Equations Sanzheng Qiao Department of Computing and Software McMaster University March, 2014 Outline 1 Initial Value Problem Euler s Method RungeKutta Methods Multistep Methods
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 informationSection 7.2 Euler s Method
Section 7.2 Euler s Method Key terms Scalar first order IVP (one step method) Euler s Method Derivation Error analysis Computational procedure Difference equation Slope field or Direction field Error Euler's
More informationOn the stability regions of implicit linear multistep methods
On the stability regions of implicit linear multistep methods arxiv:1404.6934v1 [math.na] 8 Apr 014 Lajos Lóczi April 9, 014 Abstract If we apply the accepted definition to determine the stability region
More informationPhysics 584 Computational Methods
Physics 584 Computational Methods Introduction to Matlab and Numerical Solutions to Ordinary Differential Equations Ryan Ogliore April 18 th, 2016 Lecture Outline Introduction to Matlab Numerical Solutions
More informationChapter 7. Homogeneous equations with constant coefficients
Chapter 7. Homogeneous equations with constant coefficients It has already been remarked that we can write down a formula for the general solution of any linear second differential equation y + a(t)y +
More informationAIMS Exercise Set # 1
AIMS Exercise Set #. Determine the form of the single precision floating point arithmetic used in the computers at AIMS. What is the largest number that can be accurately represented? What is the smallest
More informationMultistep Methods for IVPs. t 0 < t < T
Multistep Methods for IVPs We are still considering the IVP dy dt = f(t,y) t 0 < t < T y(t 0 ) = y 0 So far we have looked at Euler s method, which was a first order method and Runge Kutta (RK) methods
More informationIntroduction to the Numerical Solution of IVP for ODE
Introduction to the Numerical Solution of IVP for ODE 45 Introduction to the Numerical Solution of IVP for ODE Consider the IVP: DE x = f(t, x), IC x(a) = x a. For simplicity, we will assume here that
More information