Math 128A Spring 2003 Week 12 Solutions

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Math 128A Spring 2003 Week 12 Solutions"

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. Higher-Order Equations and Systems of Differential Equations 1. Use the Runge-Kutta method for systems to approximate the solution of the following system of first-order 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 Runge-Kutta method, we obtain the following results. t i w i, 1 u 1 (t i ) w i, 2 u 2 (t i ) Use the Runge-Kutta 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 higher-order 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 Runge-Kutta to this system we get the t i w i, 1 y(t i ) w i, Change the Adams Fourth-Order Predictor-Corrector Algorithm to obtain approximate solutions to systems of first-order equations. Solution. The changed algorithm is as follows. Adams Fourth-Order Predictor Corrector for Systems To approximate the solution of the mth-order system of first-order initial-value 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 initial-value 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 initial-value problem using the Runge-Kutta method for systems is as follows. 3

4 t i w i, 1 w i, b. The solution for this initial-value problem using the Runge-Kutta 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 initial-value problem and , , and radians respectively for the second initial-value 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 Runge-Kutta 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 initial-value 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 Runge-Kutta fourth-order method. Solution. c. Applying the Runge-Kutta method to the previous differential equation, we get the following: t i w i y(t i ) Solve the following stiff initial-value problem using the Runge-Kutta fourth-order 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 Runge-Kutta fourth-order method gives: t i w 1, i u 1 (t i ) w 2, i u 2 (t i ) b. When h = 0.025, the Runge-Kutta fourth-order method gives: 10

11 t i w 1, i u 1 (t i ) w 2, i u 2 (t i ) Show that the fourth-order Runge-Kutta 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 one-step 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 )

Fourth Order RK-Method

Fourth 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 information

Ordinary differential equation II

Ordinary differential equation II Ordinary Differential Equations ISC-5315 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 information

Numerical Differential Equations: IVP

Numerical 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 information

Graded Project #1. Part 1. Explicit Runge Kutta methods. Goals Differential Equations FMN130 Gustaf Söderlind and Carmen Arévalo

Graded 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 information

DELFT UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering, Mathematics and Computer Science

DELFT 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 information

Jim Lambers MAT 772 Fall Semester Lecture 21 Notes

Jim Lambers MAT 772 Fall Semester Lecture 21 Notes Jim Lambers MAT 772 Fall Semester 21-11 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 information

Chapter 10. Initial value Ordinary Differential Equations

Chapter 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 information

9.6 Predictor-Corrector Methods

9.6 Predictor-Corrector Methods SEC. 9.6 PREDICTOR-CORRECTOR METHODS 505 Adams-Bashforth-Moulton Method 9.6 Predictor-Corrector Methods The methods of Euler, Heun, Taylor, and Runge-Kutta are called single-step methods because they use

More information

1 Error Analysis for Solving IVP

1 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 information

Math 308 Week 8 Solutions

Math 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 information

Ordinary Differential Equations: Initial Value problems (IVP)

Ordinary 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 information

Review Higher Order methods Multistep methods Summary HIGHER ORDER METHODS. P.V. Johnson. School of Mathematics. Semester

Review 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 information

PowerPoints organized by Dr. Michael R. Gustafson II, Duke University

PowerPoints organized by Dr. Michael R. Gustafson II, Duke University Part 6 Chapter 20 Initial-Value Problems PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction

More information

Numerical solution of ODEs

Numerical 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 information

Ordinary Differential Equations I

Ordinary 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 information

dt 2 The Order of a differential equation is the order of the highest derivative that occurs in the equation. Example The differential equation

dt 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 information

Stability Ordinates of Adams Predictor-Corrector Methods

Stability Ordinates of Adams Predictor-Corrector Methods BIT manuscript No. will be inserted by the editor) Stability Ordinates of Adams Predictor-Corrector Methods Michelle L. Ghrist Bengt Fornberg Jonah A. Reeger Received: date / Accepted: date Abstract How

More information

CHAPTER 5: Linear Multistep Methods

CHAPTER 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 information

1 Ordinary Differential Equations

1 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 information

Numerical Integration of Ordinary Differential Equations for Initial Value Problems

Numerical 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 information

Numerical Methods for Ordinary Differential Equations

Numerical 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 Runge-Kutta Method. 1. Numerical

More information

Runge-Kutta methods. With orders of Taylor methods yet without derivatives of f (t, y(t))

Runge-Kutta methods. With orders of Taylor methods yet without derivatives of f (t, y(t)) Runge-Kutta 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 information

Lecture V: The game-engine loop & Time Integration

Lecture V: The game-engine loop & Time Integration Lecture V: The game-engine loop & Time Integration The Basic Game-Engine Loop Previous state: " #, %(#) ( #, )(#) Forces -(#) Integrate velocities and positions Resolve Interpenetrations Per-body change

More information

Differential Equations

Differential Equations Differential Equations Definitions Finite Differences Taylor Series based Methods: Euler Method Runge-Kutta Methods Improved Euler, Midpoint methods Runge Kutta (2nd, 4th order) methods Predictor-Corrector

More information

Section 7.4 Runge-Kutta Methods

Section 7.4 Runge-Kutta Methods Section 7.4 Runge-Kutta Methods Key terms: Taylor methods Taylor series Runge-Kutta; methods linear combinations of function values at intermediate points Alternatives to second order Taylor methods Fourth

More information

COSC 3361 Numerical Analysis I Ordinary Differential Equations (II) - Multistep methods

COSC 3361 Numerical Analysis I Ordinary Differential Equations (II) - Multistep methods COSC 336 Numerical Analysis I Ordinary Differential Equations (II) - Multistep methods Fall 2005 Repetition from the last lecture (I) Initial value problems: dy = f ( t, y) dt y ( a) = y 0 a t b Goal:

More information

4 Stability analysis of finite-difference methods for ODEs

4 Stability analysis of finite-difference methods for ODEs MATH 337, by T. Lakoba, University of Vermont 36 4 Stability analysis of finite-difference methods for ODEs 4.1 Consistency, stability, and convergence of a numerical method; Main Theorem In this Lecture

More information

Solving scalar IVP s : Runge-Kutta Methods

Solving scalar IVP s : Runge-Kutta Methods Solving scalar IVP s : Runge-Kutta Methods Josh Engwer Texas Tech University March 7, NOTATION: h step size x n xt) t n+ t + h x n+ xt n+ ) xt + h) dx = ft, x) SCALAR IVP ASSUMED THROUGHOUT: dt xt ) =

More information

An Analysis of Numerical Methods on Traffic Flow Models

An Analysis of Numerical Methods on Traffic Flow Models Bridgewater State University Virtual Commons - Bridgewater State University Honors Program Theses and Projects Undergraduate Honors Program 5-12-2015 An Analysis of Numerical Methods on Traffic Flow Models

More information

2015 Holl ISU MSM Ames, Iowa. A Few Good ODEs: An Introduction to Modeling and Computation

2015 Holl ISU MSM Ames, Iowa. A Few Good ODEs: An Introduction to Modeling and Computation 2015 Holl Mini-Conference @ ISU MSM Ames, Iowa A Few Good ODEs: An Introduction to Modeling and Computation James A. Rossmanith Department of Mathematics Iowa State University June 20 th, 2015 J.A. Rossmanith

More information

Euler s Method, Taylor Series Method, Runge Kutta Methods, Multi-Step Methods and Stability.

Euler s Method, Taylor Series Method, Runge Kutta Methods, Multi-Step Methods and Stability. Euler s Method, Taylor Series Method, Runge Kutta Methods, Multi-Step Methods and Stability. REVIEW: We start with the differential equation dy(t) dt = f (t, y(t)) (1.1) y(0) = y 0 This equation can be

More information

Introductory Numerical Analysis

Introductory Numerical Analysis Introductory Numerical Analysis Lecture Notes December 16, 017 Contents 1 Introduction to 1 11 Floating Point Numbers 1 1 Computational Errors 13 Algorithm 3 14 Calculus Review 3 Root Finding 5 1 Bisection

More information

Differential Equations FMNN10 Graded Project #1 c G Söderlind 2017

Differential Equations FMNN10 Graded Project #1 c G Söderlind 2017 Differential Equations FMNN10 Graded Project #1 c G Söderlind 2017 Published 2017-10-30. Instruction in computer lab 2017-11-02/08/09. Project report due date: Monday 2017-11-13 at 10:00. Goals. The goal

More information

Introduction to Initial Value Problems

Introduction to Initial Value Problems Chapter 2 Introduction to Initial Value Problems The purpose of this chapter is to study the simplest numerical methods for approximating the solution to a first order initial value problem (IVP). Because

More information

1 Systems of First Order IVP

1 Systems of First Order IVP cs412: introduction to numerical analysis 12/09/10 Lecture 24: Systems of First Order Differential Equations Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mark Cowlishaw, Nathanael Fillmore 1 Systems

More information

2D-Volterra-Lotka Modeling For 2 Species

2D-Volterra-Lotka Modeling For 2 Species Majalat Al-Ulum Al-Insaniya wat - Tatbiqiya 2D-Volterra-Lotka Modeling For 2 Species Alhashmi Darah 1 University of Almergeb Department of Mathematics Faculty of Science Zliten Libya. Abstract The purpose

More information

Practice Problems For Test 3

Practice Problems For Test 3 Practice Problems For Test 3 Power Series Preliminary Material. Find the interval of convergence of the following. Be sure to determine the convergence at the endpoints. (a) ( ) k (x ) k (x 3) k= k (b)

More information

The Milne error estimator for stiff problems

The Milne error estimator for stiff problems 13 R. Tshelametse / SAJPAM. Volume 4 (2009) 13-28 The Milne error estimator for stiff problems Ronald Tshelametse Department of Mathematics University of Botswana Private Bag 0022 Gaborone, Botswana. E-mail

More information

Logistic Map, Euler & Runge-Kutta Method and Lotka-Volterra Equations

Logistic 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 information

4.4 Computing π, ln 2 and e

4.4 Computing π, ln 2 and e 252 4.4 Computing π, ln 2 and e The approximations π 3.1415927, ln 2 0.69314718, e 2.7182818 can be obtained by numerical methods applied to the following initial value problems: (1) y = 4, 1 + x2 y(0)

More information

Lecture 8: Calculus and Differential Equations

Lecture 8: Calculus and Differential Equations Lecture 8: Calculus and Differential Equations Dr. Mohammed Hawa Electrical Engineering Department University of Jordan EE201: Computer Applications. See Textbook Chapter 9. Numerical Methods MATLAB provides

More information

Computational Methods CMSC/AMSC/MAPL 460. Ordinary differential equations

Computational Methods CMSC/AMSC/MAPL 460. Ordinary differential equations Computational Methods CMSC/AMSC/MAPL 460 Ordinar differential equations Ramani Duraiswami, Dept. of Computer Science Several slides adapted from Prof. ERIC SANDT, TAMU ODE: Previous class Standard form

More information

TMA4215: Lecture notes on numerical solution of ordinary differential equations.

TMA4215: Lecture notes on numerical solution of ordinary differential equations. TMA45: Lecture notes on numerical solution of ordinary differential equations. Anne Kværnø November, 6 Contents Eulers method. 3 Some background on ODEs. 6 3 Numerical solution of ODEs. 8 3. Some examples

More information

Theory, Solution Techniques and Applications of Singular Boundary Value Problems

Theory, Solution Techniques and Applications of Singular Boundary Value Problems Theory, Solution Techniques and Applications of Singular Boundary Value Problems W. Auzinger O. Koch E. Weinmüller Vienna University of Technology, Austria Problem Class z (t) = M(t) z(t) + f(t, z(t)),

More information

Numerical Methods for ODEs. Lectures for PSU Summer Programs Xiantao Li

Numerical 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 information

Interacting Populations.

Interacting Populations. Chapter 2 Interacting Populations. 2.1 Predator/ Prey models Suppose we have an island where some rabbits and foxes live. Left alone the rabbits have a growth rate of 10 per 100 per month. Unfortunately

More information

2.29 Numerical Fluid Mechanics Fall 2011 Lecture 20

2.29 Numerical Fluid Mechanics Fall 2011 Lecture 20 2.29 Numerical Fluid Mechanics Fall 2011 Lecture 20 REVIEW Lecture 19: Finite Volume Methods Review: Basic elements of a FV scheme and steps to step-up a FV scheme One Dimensional examples d x j x j 1/2

More information

8.7 Taylor s Inequality Math 2300 Section 005 Calculus II. f(x) = ln(1 + x) f(0) = 0

8.7 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 information

Additional exercises with Numerieke Analyse

Additional exercises with Numerieke Analyse Additional exercises with Numerieke Analyse March 10, 017 1. (a) Given different points x 0, x 1, x [a, b] and scalars y 0, y 1, y, z 1, show that there exists at most one polynomial p P 3 with p(x i )

More information

The Initial Value Problem for Ordinary Differential Equations

The Initial Value Problem for Ordinary Differential Equations Chapter 5 The Initial Value Problem for Ordinary Differential Equations In this chapter we begin a study of time-dependent differential equations, beginning with the initial value problem (IVP) for a time-dependent

More information

Astrodynamics (AERO0024)

Astrodynamics (AERO0024) Astrodynamics (AERO0024) 5. Numerical Methods Gaëtan Kerschen Space Structures & Systems Lab (S3L) Why Different Propagators? Analytic propagation: Better understanding of the perturbing forces. Useful

More information

Stability Ordinates of Adams Predictor-Corrector Methods

Stability Ordinates of Adams Predictor-Corrector Methods BIT anuscript No. will be inserted by the editor Stability Ordinates of Adas Predictor-Corrector Methods Michelle L. Ghrist Jonah A. Reeger Bengt Fornberg Received: date / Accepted: date Abstract How far

More information

Robotics. Dynamics. Marc Toussaint U Stuttgart

Robotics. Dynamics. Marc Toussaint U Stuttgart Robotics Dynamics 1D point mass, damping & oscillation, PID, dynamics of mechanical systems, Euler-Lagrange equation, Newton-Euler recursion, general robot dynamics, joint space control, reference trajectory

More information

MATH 307 Introduction to Differential Equations Autumn 2017 Midterm Exam Monday November

MATH 307 Introduction to Differential Equations Autumn 2017 Midterm Exam Monday November MATH 307 Introduction to Differential Equations Autumn 2017 Midterm Exam Monday November 6 2017 Name: Student ID Number: I understand it is against the rules to cheat or engage in other academic misconduct

More information

Dynamic Systems. Simulation of. with MATLAB and Simulink. Harold Klee. Randal Allen SECOND EDITION. CRC Press. Taylor & Francis Group

Dynamic Systems. Simulation of. with MATLAB and Simulink. Harold Klee. Randal Allen SECOND EDITION. CRC Press. Taylor & Francis Group SECOND EDITION Simulation of Dynamic Systems with MATLAB and Simulink Harold Klee Randal Allen CRC Press Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis

More information

Numerical Solution of Hybrid Fuzzy Dierential Equation (IVP) by Improved Predictor-Corrector Method

Numerical Solution of Hybrid Fuzzy Dierential Equation (IVP) by Improved Predictor-Corrector Method Available online at http://ijim.srbiau.ac.ir Int. J. Industrial Mathematics Vol. 1, No. 2 (2009)147-161 Numerical Solution of Hybrid Fuzzy Dierential Equation (IVP) by Improved Predictor-Corrector Method

More information

DELFT UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering, Mathematics and Computer Science

DELFT 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 (WI097 TU) Thursday January 014, 18:0-1:0

More information

Ordinary Differential Equations II

Ordinary Differential Equations II Ordinary Differential Equations II CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Ordinary Differential Equations II 1 / 33 Almost Done! Last

More information

Solutions of Math 53 Midterm Exam I

Solutions of Math 53 Midterm Exam I Solutions of Math 53 Midterm Exam I Problem 1: (1) [8 points] Draw a direction field for the given differential equation y 0 = t + y. (2) [8 points] Based on the direction field, determine the behavior

More information

Name Class. 5. Find the particular solution to given the general solution y C cos x and the. x 2 y

Name Class. 5. Find the particular solution to given the general solution y C cos x and the. x 2 y 10 Differential Equations Test Form A 1. Find the general solution to the first order differential equation: y 1 yy 0. 1 (a) (b) ln y 1 y ln y 1 C y y C y 1 C y 1 y C. Find the general solution to the

More information

5.2 - Euler s Method

5.2 - Euler s Method 5. - Euler s Method Consider solving the initial-value problem for ordinary differential equation: (*) y t f t, y, a t b, y a. Let y t be the unique solution of the initial-value problem. In the previous

More information

Dynamical Analysis of a Harvested Predator-prey. Model with Ratio-dependent Response Function. and Prey Refuge

Dynamical Analysis of a Harvested Predator-prey. Model with Ratio-dependent Response Function. and Prey Refuge Applied Mathematical Sciences, Vol. 8, 214, no. 11, 527-537 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/12988/ams.214.4275 Dynamical Analysis of a Harvested Predator-prey Model with Ratio-dependent

More information

Continuous Block Hybrid-Predictor-Corrector method for the solution of y = f (x, y, y )

Continuous Block Hybrid-Predictor-Corrector method for the solution of y = f (x, y, y ) International Journal of Mathematics and Soft Computing Vol., No. 0), 5-4. ISSN 49-8 Continuous Block Hybrid-Predictor-Corrector method for the solution of y = f x, y, y ) A.O. Adesanya, M.R. Odekunle

More information

MATH115. Sequences and Infinite Series. Paolo Lorenzo Bautista. June 29, De La Salle University. PLBautista (DLSU) MATH115 June 29, / 16

MATH115. Sequences and Infinite Series. Paolo Lorenzo Bautista. June 29, De La Salle University. PLBautista (DLSU) MATH115 June 29, / 16 MATH115 Sequences and Infinite Series Paolo Lorenzo Bautista De La Salle University June 29, 2014 PLBautista (DLSU) MATH115 June 29, 2014 1 / 16 Definition A sequence function is a function whose domain

More information

Three-Tank Experiment

Three-Tank Experiment Three-Tank Experiment Overview The three-tank experiment focuses on application of the mechanical balance equation to a transient flow. Three tanks are interconnected by Schedule 40 pipes of nominal diameter

More information

Series Solutions Near an Ordinary Point

Series Solutions Near an Ordinary Point Series Solutions Near an Ordinary Point MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Ordinary Points (1 of 2) Consider the second order linear homogeneous

More information

A computationally effective predictor-corrector method for simulating fractional order dynamical control system

A computationally effective predictor-corrector method for simulating fractional order dynamical control system ANZIAM J. 47 (EMA25) pp.168 184, 26 168 A computationally effective predictor-corrector method for simulating fractional order dynamical control system. Yang F. Liu (Received 14 October 25; revised 24

More information

The Simple Double Pendulum

The Simple Double Pendulum The Simple Double Pendulum Austin Graf December 13, 2013 Abstract The double pendulum is a dynamic system that exhibits sensitive dependence upon initial conditions. This project explores the motion of

More information

Solution: (a) Before opening the parachute, the differential equation is given by: dv dt. = v. v(0) = 0

Solution: (a) Before opening the parachute, the differential equation is given by: dv dt. = v. v(0) = 0 Math 2250 Lab 4 Name/Unid: 1. (35 points) Leslie Leroy Irvin bails out of an airplane at the altitude of 16,000 ft, falls freely for 20 s, then opens his parachute. Assuming linear air resistance ρv ft/s

More information

MATHEMATICS & APPLIED MATHEMATICS II

MATHEMATICS & APPLIED MATHEMATICS II RHODES UNIVERSITY DEPARTMENT OF MATHEMATICS (Pure & Applied) EXAMINATION : NOVEMBER 2011 MATHEMATICS & APPLIED MATHEMATICS II Paper 2 Examiners : Dr J.V. van Zyl Ms H.C. Henninger FULL MARKS : 150 Dr C.C.

More information

Computational Physics (6810): Session 8

Computational Physics (6810): Session 8 Computational Physics (6810): Session 8 Dick Furnstahl Nuclear Theory Group OSU Physics Department February 24, 2014 Differential equation solving Session 7 Preview Session 8 Stuff Solving differential

More information

Math 115 HW #5 Solutions

Math 115 HW #5 Solutions Math 5 HW #5 Solutions From 29 4 Find the power series representation for the function and determine the interval of convergence Answer: Using the geometric series formula, f(x) = 3 x 4 3 x 4 = 3(x 4 )

More information

BIBO STABILITY AND ASYMPTOTIC STABILITY

BIBO STABILITY AND ASYMPTOTIC STABILITY BIBO STABILITY AND ASYMPTOTIC STABILITY FRANCESCO NORI Abstract. In this report with discuss the concepts of bounded-input boundedoutput stability (BIBO) and of Lyapunov stability. Examples are given to

More information

Dynamics Analysis of Anti-predator Model on Intermediate Predator With Ratio Dependent Functional Responses

Dynamics Analysis of Anti-predator Model on Intermediate Predator With Ratio Dependent Functional Responses Journal of Physics: Conference Series PAPER OPEN ACCESS Dynamics Analysis of Anti-predator Model on Intermediate Predator With Ratio Dependent Functional Responses To cite this article: D Savitri 2018

More information

Differential Equations. - Theory and Applications - Version: Fall András Domokos, PhD California State University, Sacramento

Differential Equations. - Theory and Applications - Version: Fall András Domokos, PhD California State University, Sacramento Differential Equations - Theory and Applications - Version: Fall 2017 András Domokos, PhD California State University, Sacramento Contents Chapter 0. Introduction 3 Chapter 1. Calculus review. Differentiation

More information

Ordinary Differential Equations

Ordinary Differential Equations Chapter 13 Ordinary Differential Equations We motivated the problem of interpolation in Chapter 11 by transitioning from analzying to finding functions. That is, in problems like interpolation and regression,

More information

Math 113 HW #10 Solutions

Math 113 HW #10 Solutions Math HW #0 Solutions 4.5 4. Use the guidelines of this section to sketch the curve Answer: Using the quotient rule, y = x x + 9. y = (x + 9)(x) x (x) (x + 9) = 8x (x + 9). Since the denominator is always

More information

Physics: spring-mass system, planet motion, pendulum. Biology: ecology problem, neural conduction, epidemics

Physics: spring-mass system, planet motion, pendulum. Biology: ecology problem, neural conduction, epidemics Applications of nonlinear ODE systems: Physics: spring-mass system, planet motion, pendulum Chemistry: mixing problems, chemical reactions Biology: ecology problem, neural conduction, epidemics Economy:

More information

Ordinary Differential Equations II: Runge-Kutta and Advanced Methods

Ordinary Differential Equations II: Runge-Kutta and Advanced Methods Ordinary Differential Equations II: Runge-Kutta and Advanced Methods Sam Sinayoko Numerical Methods 3 Contents 1 Learning Outcomes 2 2 Introduction 2 2.1 Note................................ 4 2.2 Limitations

More information

Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number.

Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number. 997 AP Calculus BC: Section I, Part A 5 Minutes No Calculator Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f () is a real number..

More information

Exact and Approximate Numbers:

Exact and Approximate Numbers: Eact and Approimate Numbers: The numbers that arise in technical applications are better described as eact numbers because there is not the sort of uncertainty in their values that was described above.

More information

Lab #6: Predator Prey Interactions

Lab #6: Predator Prey Interactions Lab #6: Predator Interactions This exercise illustrates how different populations interact within a community, and how this interaction can influence the process of evolution in both species. The relationship

More information

Finite Difference and Finite Element Methods

Finite Difference and Finite Element Methods Finite Difference and Finite Element Methods Georgy Gimel farb COMPSCI 369 Computational Science 1 / 39 1 Finite Differences Difference Equations 3 Finite Difference Methods: Euler FDMs 4 Finite Element

More information

Lecture 42 Determining Internal Node Values

Lecture 42 Determining Internal Node Values Lecture 42 Determining Internal Node Values As seen in the previous section, a finite element solution of a boundary value problem boils down to finding the best values of the constants {C j } n, which

More information

LAURENTIAN UNIVERSITY UNIVERSITÉ LAURENTIENNE

LAURENTIAN UNIVERSITY UNIVERSITÉ LAURENTIENNE Page 1 of 15 LAURENTIAN UNIVERSITY UNIVERSITÉ LAURENTIENNE Friday, December 14 th Course and No. 2007 MATH 2066 EL Date................................... Cours et no........................... Total no.

More information

Computational project: Modelling a simple quadrupole mass spectrometer

Computational project: Modelling a simple quadrupole mass spectrometer Computational project: Modelling a simple quadrupole mass spectrometer Martin Duy Tat a, Anders Hagen Jarmund a a Norges Teknisk-Naturvitenskapelige Universitet, Trondheim, Norway. Abstract In this project

More information

DIFFERENTIATION RULES

DIFFERENTIATION RULES 3 DIFFERENTIATION RULES DIFFERENTIATION RULES Before starting this section, you might need to review the trigonometric functions. DIFFERENTIATION RULES In particular, it is important to remember that,

More information

Introduction to Hamiltonian systems

Introduction 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 information

Solutions to the Mathematics Masters Examination

Solutions to the Mathematics Masters Examination Solutions to the Mathematics Masters Examination OPTION 4 Spring 2007 COMPUTER SCIENCE 2 5 PM NOTE: Any student whose answers require clarification may be required to submit to an oral examination. Each

More information

Calculus IV - HW 3. Due 7/ Give the general solution to the following differential equations: y = c 1 e 5t + c 2 e 5t. y = c 1 e 2t + c 2 e 4t.

Calculus IV - HW 3. Due 7/ Give the general solution to the following differential equations: y = c 1 e 5t + c 2 e 5t. y = c 1 e 2t + c 2 e 4t. Calculus IV - HW 3 Due 7/13 Section 3.1 1. Give the general solution to the following differential equations: a y 25y = 0 Solution: The characteristic equation is r 2 25 = r 5r + 5. It follows that the

More information

Physics 115/242 Comparison of methods for integrating the simple harmonic oscillator.

Physics 115/242 Comparison of methods for integrating the simple harmonic oscillator. Physics 115/4 Comparison of methods for integrating the simple harmonic oscillator. Peter Young I. THE SIMPLE HARMONIC OSCILLATOR The energy (sometimes called the Hamiltonian ) of the simple harmonic oscillator

More information

LAURENTIAN UNIVERSITY UNIVERSITÉ LAURENTIENNE

LAURENTIAN UNIVERSITY UNIVERSITÉ LAURENTIENNE Page 1 of 15 LAURENTIAN UNIVERSITY UNIVERSITÉ LAURENTIENNE Wednesday, December 13 th Course and No. 2006 MATH 2066 EL Date................................... Cours et no........................... Total

More information

369 Nigerian Research Journal of Engineering and Environmental Sciences 2(2) 2017 pp

369 Nigerian Research Journal of Engineering and Environmental Sciences 2(2) 2017 pp 369 Nigerian Research Journal of Engineering and Environmental Sciences 2(2) 27 pp. 369-374 Original Research Article THIRD DERIVATIVE MULTISTEP METHODS WITH OPTIMIZED REGIONS OF ABSOLUTE STABILITY FOR

More information

Numerical Solutions to Partial Differential Equations

Numerical Solutions to Partial Differential Equations Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University The Implicit Schemes for the Model Problem The Crank-Nicolson scheme and θ-scheme

More information

MATH 108 FALL 2013 FINAL EXAM REVIEW

MATH 108 FALL 2013 FINAL EXAM REVIEW MATH 08 FALL 203 FINAL EXAM REVIEW Definitions and theorems. The following definitions and theorems are fair game for you to have to state on the exam. Definitions: Limit (precise δ-ɛ version; 2.4, Def.

More information

Section 5.6. Integration By Parts. MATH 126 (Section 5.6) Integration By Parts The University of Kansas 1 / 10

Section 5.6. Integration By Parts. MATH 126 (Section 5.6) Integration By Parts The University of Kansas 1 / 10 Section 5.6 Integration By Parts MATH 126 (Section 5.6) Integration By Parts The University of Kansas 1 / 10 Integration By Parts Manipulating the Product Rule d dx (f (x) g(x)) = f (x) g (x) + f (x) g(x)

More information

Exponential Integrators

Exponential Integrators Exponential Integrators John C. Bowman (University of Alberta) May 22, 2007 www.math.ualberta.ca/ bowman/talks 1 Exponential Integrators Outline Exponential Euler History Generalizations Stationary Green

More information

Basic Theory of Differential Equations

Basic Theory of Differential Equations page 104 104 CHAPTER 1 First-Order Differential Equations 16. The following initial-value problem arises in the analysis of a cable suspended between two fixed points y = 1 a 1 + (y ) 2, y(0) = a, y (0)

More information