Mathematics 22: Lecture 10

Size: px
Start display at page:

Download "Mathematics 22: Lecture 10"

Transcription

1 Mathematics 22: Lecture 10 Euler s Method Dan Sloughter Furman University January 22, 2008 Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

2 Euler s method Consider the initial-value problem du dt = f (t, u), u(t 0) = u 0. Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

3 Euler s method Consider the initial-value problem For a small h > 0, let t 1 = t 0 + h. du dt = f (t, u), u(t 0) = u 0. Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

4 Euler s method Consider the initial-value problem For a small h > 0, let t 1 = t 0 + h. du dt = f (t, u), u(t 0) = u 0. To approximate u(t 1 ), first note that implies t1 t 0 u(t 1 ) u(t 0 ) = du t1 dt dt = f (s, u)ds t 0 t1 t 0 f (s, u)ds hf (t 0, u(t 0 )). Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

5 Euler s method Consider the initial-value problem For a small h > 0, let t 1 = t 0 + h. du dt = f (t, u), u(t 0) = u 0. To approximate u(t 1 ), first note that implies Hence t1 t 0 u(t 1 ) u(t 0 ) = du t1 dt dt = f (s, u)ds t 0 t1 t 0 f (s, u)ds hf (t 0, u(t 0 )). u(t 0 + h) u 0 + f (t 0, u 0 )h. Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

6 Euler s method (cont d) To approximate a solution to the initial-value problem on an interval [t 0, t 0 + T ]: du dt = f (t, u), u(t 0) = u 0, Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

7 Euler s method (cont d) To approximate a solution to the initial-value problem du dt = f (t, u), u(t 0) = u 0, on an interval [t 0, t 0 + T ]: divide the interval into N subintervals of equal length h = T N, Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

8 Euler s method (cont d) To approximate a solution to the initial-value problem du dt = f (t, u), u(t 0) = u 0, on an interval [t 0, t 0 + T ]: divide the interval into N subintervals of equal length let ti = t 0 + ih, i = 1, 2,..., N, h = T N, Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

9 Euler s method (cont d) To approximate a solution to the initial-value problem du dt = f (t, u), u(t 0) = u 0, on an interval [t 0, t 0 + T ]: divide the interval into N subintervals of equal length let ti = t 0 + ih, i = 1, 2,..., N, for i = 0, 1, 2,..., N 1, let h = T N, u i+1 = u i + hf (t i, u i ). Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

10 Euler s method (cont d) To approximate a solution to the initial-value problem du dt = f (t, u), u(t 0) = u 0, on an interval [t 0, t 0 + T ]: divide the interval into N subintervals of equal length let ti = t 0 + ih, i = 1, 2,..., N, for i = 0, 1, 2,..., N 1, let h = T N, u i+1 = u i + hf (t i, u i ). Then u i u(t i ). Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

11 Example Consider the initial-value problem du dt = u cos(t), u(0) = 1. Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

12 Example Consider the initial-value problem du dt = u cos(t), u(0) = 1. To approximate u on [0, 6], we will take h = 0.1, that is, N = 60. Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

13 Example Consider the initial-value problem du dt = u cos(t), u(0) = 1. To approximate u on [0, 6], we will take h = 0.1, that is, N = 60. We have u 0 = 1.0, u 1 = (0.1)(1.0) cos(0) = 1.1, u 2 = (0.1)(1.1) cos(0.1) = , u 3 = (0.1)( ) cos(0.2) = Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

14 Example Consider the initial-value problem du dt = u cos(t), u(0) = 1. To approximate u on [0, 6], we will take h = 0.1, that is, N = 60. We have u 0 = 1.0, u 1 = (0.1)(1.0) cos(0) = 1.1, u 2 = (0.1)(1.1) cos(0.1) = , u 3 = (0.1)( ) cos(0.2) = Hence, for example, u(0.3) Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

15 Example (cont d) Note: the exact solution of the equation is u(t) = e sin(t), so u(0.3) = e sin(0.3) = Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

16 Using Octave Solution in Octave: octave:1> t=[0:0.1:6]; octave:2> u(1)=1; octave:3> for i = 1:60 > u(i+1) = u(i) + 0.1*u(i)*cos(t(i)); > endfor Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

17 Using Octave Solution in Octave: octave:1> t=[0:0.1:6]; octave:2> u(1)=1; octave:3> for i = 1:60 > u(i+1) = u(i) + 0.1*u(i)*cos(t(i)); > endfor Note: indices begin with 1 in Octave, so u starts with u(1) and ends with u(61). The same comment holds for t. Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

18 Using Octave Solution in Octave: octave:1> t=[0:0.1:6]; octave:2> u(1)=1; octave:3> for i = 1:60 > u(i+1) = u(i) + 0.1*u(i)*cos(t(i)); > endfor Note: indices begin with 1 in Octave, so u starts with u(1) and ends with u(61). The same comment holds for t. In particular, u 60 is u(61) in Octave. Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

19 Using Octave Solution in Octave: octave:1> t=[0:0.1:6]; octave:2> u(1)=1; octave:3> for i = 1:60 > u(i+1) = u(i) + 0.1*u(i)*cos(t(i)); > endfor Note: indices begin with 1 in Octave, so u starts with u(1) and ends with u(61). The same comment holds for t. In particular, u 60 is u(61) in Octave. For example, we now see that u(6) u 60 = Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

20 Using Octave Solution in Octave: octave:1> t=[0:0.1:6]; octave:2> u(1)=1; octave:3> for i = 1:60 > u(i+1) = u(i) + 0.1*u(i)*cos(t(i)); > endfor Note: indices begin with 1 in Octave, so u starts with u(1) and ends with u(61). The same comment holds for t. In particular, u 60 is u(61) in Octave. For example, we now see that u(6) u 60 = Exact value: u(6) = Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

21 Octave output The command octave:4> [t,u ] will print out the values of t and u in side-by-side columns. Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

22 Octave output The command octave:4> [t,u ] will print out the values of t and u in side-by-side columns. Note: We need t, and u, because t and u are row vectors. Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

23 Octave output The command octave:4> [t,u ] will print out the values of t and u in side-by-side columns. Note: We need t, and u, because t and u are row vectors. Note: In some versions of Octave, u is not a row vector. Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

24 Octave output The command octave:4> [t,u ] will print out the values of t and u in side-by-side columns. Note: We need t, and u, because t and u are row vectors. Note: In some versions of Octave, u is not a row vector. If we define octave:5> z=exp(sin(t)); then octave:6> [t,u,z ] will print out the values of t, u, and z (the exact solution) in side-by-side columns. Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

25 Octave output (cont d) The command octave:7> plot(t,u) will plot the approximate solution, and the command octave:8> plot(t,u,t,z) will plot a comparison of the approximate solution with the exact solution. Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

26 Octave output (cont d) Comparison of exact (green) and approximate (red) solutions: Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

27 Modified Euler s method Again consider the initial-value problem du dt = f (t, u), u(t 0) = u 0. Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

28 Modified Euler s method Again consider the initial-value problem du dt = f (t, u), u(t 0) = u 0. Let T, h, N, and t i, i = 0, 1, 2,..., N be as before. Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

29 Modified Euler s method Again consider the initial-value problem du dt = f (t, u), u(t 0) = u 0. Let T, h, N, and t i, i = 0, 1, 2,..., N be as before. Assuming we have computed u 1, u 2,..., u i, let be the predicted value for u(t i+1 ). ũ i+1 = u i + hf (t i, u i ). Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

30 Modified Euler s method Again consider the initial-value problem du dt = f (t, u), u(t 0) = u 0. Let T, h, N, and t i, i = 0, 1, 2,..., N be as before. Assuming we have computed u 1, u 2,..., u i, let be the predicted value for u(t i+1 ). Now let ũ i+1 = u i + hf (t i, u i ). u i+1 = u i + f (t i, u i ) + f (t i+1, ũ i+1 ) h. 2 Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

31 Modified Euler s method Again consider the initial-value problem du dt = f (t, u), u(t 0) = u 0. Let T, h, N, and t i, i = 0, 1, 2,..., N be as before. Assuming we have computed u 1, u 2,..., u i, let be the predicted value for u(t i+1 ). Now let ũ i+1 = u i + hf (t i, u i ). u i+1 = u i + f (t i, u i ) + f (t i+1, ũ i+1 ) h. 2 This is an example of a predictor-corrector method. Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

32 Example For our previous example, we would have ũ 1 = (0.1)(1.0) cos(0) = 1.1, and so u 1 = cos(0) cos(0.1) (0.1) = Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

33 Example For our previous example, we would have ũ 1 = (0.1)(1.0) cos(0) = 1.1, and so Next u 1 = cos(0) cos(0.1) (0.1) = and ũ 2 = (0.1)( ) cos(0.1) = u 2 = = cos(0.1) cos(0.2) (0.1) 2 Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

34 Using Octave Note: octave:9> clear u will clear the old values of u. Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

35 Using Octave Note: octave:9> clear u will clear the old values of u. To define f (t, u) in Octave: octave:10> function w = f(t, u) > w = u*cos(t); > endfunction Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

36 Using Octave Note: octave:9> clear u will clear the old values of u. To define f (t, u) in Octave: octave:10> function w = f(t, u) > w = u*cos(t); > endfunction Modified Euler s method in Octave: octave:11> t = [0:0.1:6]; octave:12> u(1) = 1.0; octave:13> for i=1:60 > p = u(i) + 0.1*f(t(i),u(i)); > u(i+1) = u(i) + 0.1*(f(t(i),u(i)) + f(t(i+1),p))/2; > endfor Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

37 Using Octave (cont d) This time we have u(6) u 60 = Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

38 Using Octave (cont d) This time we have u(6) u 60 = Note: again, this is u(61) in Octave. Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

39 Using Octave (cont d) This time we have u(6) u 60 = Note: again, this is u(61) in Octave. Recall: exact value is u(6) = Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

40 Using Octave (cont d) This time we have u(6) u 60 = Note: again, this is u(61) in Octave. Recall: exact value is u(6) = Recall: the approximation from Euler s method was Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

41 Using Octave (cont d) Comparison of exact (green) and approximate (red) solutions: Dan Sloughter (Furman University) Mathematics 22: Lecture 10 January 22, / 14

Mathematics 22: Lecture 11

Mathematics 22: Lecture 11 Mathematics 22: Lecture 11 Runge-Kutta Dan Sloughter Furman University January 25, 2008 Dan Sloughter (Furman University) Mathematics 22: Lecture 11 January 25, 2008 1 / 11 Order of approximations One

More information

Mathematics 22: Lecture 7

Mathematics 22: Lecture 7 Mathematics 22: Lecture 7 Separation of Variables Dan Sloughter Furman University January 15, 2008 Dan Sloughter (Furman University) Mathematics 22: Lecture 7 January 15, 2008 1 / 8 Separable equations

More information

Mathematics 22: Lecture 5

Mathematics 22: Lecture 5 Mathematics 22: Lecture 5 Autonomous Equations Dan Sloughter Furman University January 11, 2008 Dan Sloughter (Furman University) Mathematics 22: Lecture 5 January 11, 2008 1 / 11 Solving the logistics

More information

Mathematics 13: Lecture 4

Mathematics 13: Lecture 4 Mathematics 13: Lecture Planes Dan Sloughter Furman University January 10, 2008 Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, 2008 1 / 10 Planes in R n Suppose v and w are nonzero

More information

Mathematics 22: Lecture 19

Mathematics 22: Lecture 19 Mathematics 22: Lecture 19 Legendre s Equation Dan Sloughter Furman University February 5, 2008 Dan Sloughter (Furman University) Mathematics 22: Lecture 19 February 5, 2008 1 / 11 Example: Legendre s

More information

Mathematics 22: Lecture 12

Mathematics 22: Lecture 12 Mathematics 22: Lecture 12 Second-order Linear Equations Dan Sloughter Furman University January 28, 2008 Dan Sloughter (Furman University) Mathematics 22: Lecture 12 January 28, 2008 1 / 14 Definition

More information

Mathematics 13: Lecture 10

Mathematics 13: Lecture 10 Mathematics 13: Lecture 10 Matrices Dan Sloughter Furman University January 25, 2008 Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, 2008 1 / 19 Matrices Recall: A matrix is a

More information

Mathematics 22: Lecture 4

Mathematics 22: Lecture 4 Mathematics 22: Lecture 4 Population Models Dan Sloughter Furman University January 10, 2008 Dan Sloughter (Furman University) Mathematics 22: Lecture 4 January 10, 2008 1 / 6 Malthusian growth model Let

More information

The Chain Rule. Mathematics 11: Lecture 18. Dan Sloughter. Furman University. October 10, 2007

The Chain Rule. Mathematics 11: Lecture 18. Dan Sloughter. Furman University. October 10, 2007 The Chain Rule Mathematics 11: Lecture 18 Dan Sloughter Furman University October 10, 2007 Dan Sloughter (Furman University) The Chain Rule October 10, 2007 1 / 15 Example Suppose that a pebble is dropped

More information

Antiderivatives. Mathematics 11: Lecture 30. Dan Sloughter. Furman University. November 7, 2007

Antiderivatives. Mathematics 11: Lecture 30. Dan Sloughter. Furman University. November 7, 2007 Antiderivatives Mathematics 11: Lecture 30 Dan Sloughter Furman University November 7, 2007 Dan Sloughter (Furman University) Antiderivatives November 7, 2007 1 / 9 Definition Recall: Suppose F and f are

More information

Change of Variables: Indefinite Integrals

Change of Variables: Indefinite Integrals Change of Variables: Indefinite Integrals Mathematics 11: Lecture 39 Dan Sloughter Furman University November 29, 2007 Dan Sloughter (Furman University) Change of Variables: Indefinite Integrals November

More information

Pivotal Quantities. Mathematics 47: Lecture 16. Dan Sloughter. Furman University. March 30, 2006

Pivotal Quantities. Mathematics 47: Lecture 16. Dan Sloughter. Furman University. March 30, 2006 Pivotal Quantities Mathematics 47: Lecture 16 Dan Sloughter Furman University March 30, 2006 Dan Sloughter (Furman University) Pivotal Quantities March 30, 2006 1 / 10 Pivotal quantities Definition Suppose

More information

Some Trigonometric Limits

Some Trigonometric Limits Some Trigonometric Limits Mathematics 11: Lecture 7 Dan Sloughter Furman University September 20, 2007 Dan Sloughter (Furman University) Some Trigonometric Limits September 20, 2007 1 / 14 The squeeze

More information

Nonparametric Tests. Mathematics 47: Lecture 25. Dan Sloughter. Furman University. April 20, 2006

Nonparametric Tests. Mathematics 47: Lecture 25. Dan Sloughter. Furman University. April 20, 2006 Nonparametric Tests Mathematics 47: Lecture 25 Dan Sloughter Furman University April 20, 2006 Dan Sloughter (Furman University) Nonparametric Tests April 20, 2006 1 / 14 The sign test Suppose X 1, X 2,...,

More information

Sampling Distributions

Sampling Distributions Sampling Distributions Mathematics 47: Lecture 9 Dan Sloughter Furman University March 16, 2006 Dan Sloughter (Furman University) Sampling Distributions March 16, 2006 1 / 10 Definition We call the probability

More information

Calculus: Area. Mathematics 15: Lecture 22. Dan Sloughter. Furman University. November 12, 2006

Calculus: Area. Mathematics 15: Lecture 22. Dan Sloughter. Furman University. November 12, 2006 Calculus: Area Mathematics 15: Lecture 22 Dan Sloughter Furman University November 12, 2006 Dan Sloughter (Furman University) Calculus: Area November 12, 2006 1 / 7 Area Note: formulas for the areas of

More information

Example. Mathematics 255: Lecture 17. Example. Example (cont d) Consider the equation. d 2 y dt 2 + dy

Example. Mathematics 255: Lecture 17. Example. Example (cont d) Consider the equation. d 2 y dt 2 + dy Mathematics 255: Lecture 17 Undetermined Coefficients Dan Sloughter Furman University October 10, 2008 6y = 5e 4t. so the general solution of 0 = r 2 + r 6 = (r + 3)(r 2), 6y = 0 y(t) = c 1 e 3t + c 2

More information

Math 128A Spring 2003 Week 12 Solutions

Math 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 information

Solving Ordinary Differential Equations

Solving 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 Runge-Kutta Methods Multistep Methods

More information

Goodness of Fit Tests: Homogeneity

Goodness of Fit Tests: Homogeneity Goodness of Fit Tests: Homogeneity Mathematics 47: Lecture 35 Dan Sloughter Furman University May 11, 2006 Dan Sloughter (Furman University) Goodness of Fit Tests: Homogeneity May 11, 2006 1 / 13 Testing

More information

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

What we ll do: Lecture 21. Ordinary Differential Equations (ODEs) Differential Equations. Ordinary Differential Equations

What we ll do: Lecture 21. Ordinary Differential Equations (ODEs) Differential Equations. Ordinary Differential Equations What we ll do: Lecture Ordinary Differential Equations J. Chaudhry Department of Mathematics and Statistics University of New Mexico Review ODEs Single Step Methods Euler s method (st order accurate) Runge-Kutta

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

The Geometry. Mathematics 15: Lecture 20. Dan Sloughter. Furman University. November 6, 2006

The Geometry. Mathematics 15: Lecture 20. Dan Sloughter. Furman University. November 6, 2006 The Geometry Mathematics 15: Lecture 20 Dan Sloughter Furman University November 6, 2006 Dan Sloughter (Furman University) The Geometry November 6, 2006 1 / 18 René Descartes 1596-1650 Dan Sloughter (Furman

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

3.5: Euler s Method (cont d) and 3.7: Population Modeling

3.5: Euler s Method (cont d) and 3.7: Population Modeling 3.5: Euler s Method (cont d) and 3.7: Population Modeling Mathematics 3 Lecture 19 Dartmouth College February 15, 2010 Typeset by FoilTEX Example 1 Let s consider a very simple first-order IVP: dy dx =

More information

Chap. 20: Initial-Value Problems

Chap. 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 information

Part: Frequency and Time Domain

Part: Frequency and Time Domain Numerical Methods Fourier Transform Pair Part: Frequency and Time Domain For more details on this topic Go to Clic on eyword Clic on Fourier Transform Pair You are free to Share to copy, distribute, display

More information

Numerical Analysis MTH603. dy dt = = (0) , y n+1. We obtain yn. Therefore. and. Copyright Virtual University of Pakistan 1

Numerical Analysis MTH603. dy dt = = (0) , y n+1. We obtain yn. Therefore. and. Copyright Virtual University of Pakistan 1 Numerical Analysis MTH60 PREDICTOR CORRECTOR METHOD Te metods presented so far are called single-step metods, were we ave seen tat te computation of y at t n+ tat is y n+ requires te knowledge of y n only.

More information

Ex. 1. Find the general solution for each of the following differential equations:

Ex. 1. Find the general solution for each of the following differential equations: MATH 261.007 Instr. K. Ciesielski Spring 2010 NAME (print): SAMPLE TEST # 2 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1.

More information

ODE Background: Differential (1A) Young Won Lim 12/29/15

ODE Background: Differential (1A) Young Won Lim 12/29/15 ODE Background: Differential (1A Copyright (c 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version

More information

MTH 452/552 Homework 3

MTH 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 m-files odesample.m and odesampletest.m available from the author s webpage.

More information

Section 5.8. Taylor Series

Section 5.8. Taylor Series Difference Equations to Differential Equations Section 5.8 Taylor Series In this section we will put together much of the work of Sections 5.-5.7 in the context of a discussion of Taylor series. We begin

More information

Problem 1 In each of the following problems find the general solution of the given differential

Problem 1 In each of the following problems find the general solution of the given differential VI Problem 1 dt + 2dy 3y = 0; dt 9dy + 9y = 0. Problem 2 dt + dy 2y = 0, y(0) = 1, y (0) = 1; dt 2 y = 0, y( 2) = 1, y ( 2) = Problem 3 Find the solution of the initial value problem 2 d2 y dt 2 3dy dt

More information

Numerical methods for solving ODEs

Numerical 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 information

DFT Frequency (9A) Each Row of the DFT Matrix. Young Won Lim 7/31/10

DFT Frequency (9A) Each Row of the DFT Matrix. Young Won Lim 7/31/10 DFT Frequency (9A) Each ow of the DFT Matrix Copyright (c) 2009, 2010 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GU Free Documentation License,

More information

Initial value problems for ordinary differential equations

Initial 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 information

Numerical Methods - Initial Value Problems for ODEs

Numerical 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 information

MATH 1242 FINAL EXAM Spring,

MATH 1242 FINAL EXAM Spring, MATH 242 FINAL EXAM Spring, 200 Part I (MULTIPLE CHOICE, NO CALCULATORS).. Find 2 4x3 dx. (a) 28 (b) 5 (c) 0 (d) 36 (e) 7 2. Find 2 cos t dt. (a) 2 sin t + C (b) 2 sin t + C (c) 2 cos t + C (d) 2 cos t

More information

CHAPTER 4 FOURIER SERIES S A B A R I N A I S M A I L

CHAPTER 4 FOURIER SERIES S A B A R I N A I S M A I L CHAPTER 4 FOURIER SERIES 1 S A B A R I N A I S M A I L Outline Introduction of the Fourier series. The properties of the Fourier series. Symmetry consideration Application of the Fourier series to circuit

More information

5 Numerical Integration & Dierentiation

5 Numerical Integration & Dierentiation 5 Numerical Integration & Dierentiation Department of Mathematics & Statistics ASU Outline of Chapter 5 1 The Trapezoidal and Simpson Rules 2 Error Formulas 3 Gaussian Numerical Integration 4 Numerical

More information

CHEE 319 Tutorial 3 Solutions. 1. Using partial fraction expansions, find the causal function f whose Laplace transform. F (s) F (s) = C 1 s + C 2

CHEE 319 Tutorial 3 Solutions. 1. Using partial fraction expansions, find the causal function f whose Laplace transform. F (s) F (s) = C 1 s + C 2 CHEE 39 Tutorial 3 Solutions. Using partial fraction expansions, find the causal function f whose Laplace transform is given by: F (s) 0 f(t)e st dt (.) F (s) = s(s+) ; Solution: Note that the polynomial

More information

MATH 2413 TEST ON CHAPTER 4 ANSWER ALL QUESTIONS. TIME 1.5 HRS.

MATH 2413 TEST ON CHAPTER 4 ANSWER ALL QUESTIONS. TIME 1.5 HRS. MATH 1 TEST ON CHAPTER ANSWER ALL QUESTIONS. TIME 1. HRS. M1c Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Use the summation formulas to rewrite the

More information

Optimal Control Strategies in a Two Dimensional Differential Game Using Linear Equation under a Perturbed System

Optimal Control Strategies in a Two Dimensional Differential Game Using Linear Equation under a Perturbed System Leonardo Journal of Sciences ISSN 1583-33 Issue 1, January-June 8 p. 135-14 Optimal Control Strategies in a wo Dimensional Differential Game Using Linear Equation under a Perturbed System Department of

More information

Solutions to Homework 3

Solutions to Homework 3 Solutions to Homework 3 Section 3.4, Repeated Roots; Reduction of Order Q 1). Find the general solution to 2y + y = 0. Answer: The charactertic equation : r 2 2r + 1 = 0, solving it we get r = 1 as a repeated

More information

Audio /Video Signal Processing. Lecture 2, Quantization, SNR Gerald Schuller, TU Ilmenau

Audio /Video Signal Processing. Lecture 2, Quantization, SNR Gerald Schuller, TU Ilmenau Audio /Video Signal Processing Lecture 2, Quantization, SNR Gerald Schuller, TU Ilmenau Quantization Signal to Noise Ratio (SNR). Assume we have a A/D converter with a quantizer with a certain number of

More information

SCIE1000 Theory and Practice in Science Final Examination, Semester One 2011

SCIE1000 Theory and Practice in Science Final Examination, Semester One 2011 1. Researchers propose two models for estimating the weight of accumulated flammable material in a forest, where weight is measured in appropriate units and t is the number of years since the most recent

More information

Speed and Velocity: Recall from Calc 1: If f (t) gives the position of an object at time t, then. velocity at time t = f (t) speed at time t = f (t)

Speed and Velocity: Recall from Calc 1: If f (t) gives the position of an object at time t, then. velocity at time t = f (t) speed at time t = f (t) Speed and Velocity: Recall from Calc 1: If f (t) gives the position of an object at time t, then velocity at time t = f (t) speed at time t = f (t) Math 36-Multi (Sklensky) In-Class Work January 8, 013

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

Math 3191 Applied Linear Algebra

Math 3191 Applied Linear Algebra Math 9 Applied Linear Algebra Lecture : Orthogonal Projections, Gram-Schmidt Stephen Billups University of Colorado at Denver Math 9Applied Linear Algebra p./ Orthonormal Sets A set of vectors {u, u,...,

More information

Chapter 8. Numerical Solution of Ordinary Differential Equations. Module No. 2. Predictor-Corrector Methods

Chapter 8. Numerical Solution of Ordinary Differential Equations. Module No. 2. Predictor-Corrector Methods Numerical Analysis by Dr. Anita Pal Assistant Professor Department of Matematics National Institute of Tecnology Durgapur Durgapur-7109 email: anita.buie@gmail.com 1 . Capter 8 Numerical Solution of Ordinary

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

Lecture 3: Initial and boundary value problems in 1 dimension

Lecture 3: Initial and boundary value problems in 1 dimension Lecture 3: Initial and boundary value problems in 1 dimension Gantumur Tsogtgerel Assistant professor of Mathematics Math 319: Introduction to PDEs McGill University, Montréal January 10, 2011 Euler s

More information

Lectures 9-10: Polynomial and piecewise polynomial interpolation

Lectures 9-10: Polynomial and piecewise polynomial interpolation Lectures 9-1: Polynomial and piecewise polynomial interpolation Let f be a function, which is only known at the nodes x 1, x,, x n, ie, all we know about the function f are its values y j = f(x j ), j

More information

Math 2214 Solution Test 1D Spring 2015

Math 2214 Solution Test 1D Spring 2015 Math 2214 Solution Test 1D Spring 2015 Problem 1: A 600 gallon open top tank initially holds 300 gallons of fresh water. At t = 0, a brine solution containing 3 lbs of salt per gallon is poured into the

More information

Finite difference methods for the diffusion equation

Finite difference methods for the diffusion equation Finite difference methods for the diffusion equation D150, Tillämpade numeriska metoder II Olof Runborg May 0, 003 These notes summarize a part of the material in Chapter 13 of Iserles. They are based

More information

Numerical Methods for the Solution of Differential Equations

Numerical 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 information

Honors Differential Equations

Honors Differential Equations MIT OpenCourseWare http://ocw.mit.edu 18.034 Honors Differential Equations Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. UNIT III: HIGHER-ORDER

More information

Applied Calculus I. Lecture 29

Applied Calculus I. Lecture 29 Applied Calculus I Lecture 29 Integrals of trigonometric functions We shall continue learning substitutions by considering integrals involving trigonometric functions. Integrals of trigonometric functions

More information

MA2264 -NUMERICAL METHODS UNIT V : INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL. By Dr.T.Kulandaivel Department of Applied Mathematics SVCE

MA2264 -NUMERICAL METHODS UNIT V : INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL. By Dr.T.Kulandaivel Department of Applied Mathematics SVCE MA64 -NUMERICAL METHODS UNIT V : INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS B Dr.T.Kulandaivel Department of Applied Matematics SVCE Numerical ordinar differential equations is te part

More information

Solutions to Assignment 4

Solutions to Assignment 4 EE35 Spectrum Analysis and Discrete Time Systems (Fall 5) Solutions to Assignment. Consider the continuous-time periodic signal: x(t) = sin(t 3) + sin(6t) (8) [] (a) Obviously, the fundamental frequency

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

Physics 170 Week 10, Lecture 1

Physics 170 Week 10, Lecture 1 Physics 170 Week 10, Lecture 1 http://www.phas.ubc.ca/ gordonws/170 Physics 170 Week 10, Lecture 1 1 Textbook Chapter 14: Section 14.1-3 Physics 170 Week 10, Lecture 1 2 Learning Goals: We will define

More information

Lecture Notes on Numerical Differential Equations: IVP

Lecture 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 information

Second Printing Errata for Advanced Engineering Mathematics, by Dennis G. Zill and Michael R. Cullen

Second Printing Errata for Advanced Engineering Mathematics, by Dennis G. Zill and Michael R. Cullen 59X_errataSecondPrint.qxd /7/7 9:9 AM Page Second Printing Errata for Advanced Engineering Mathematics, by Dennis G. Zill and Michael R. Cullen (Yellow highlighting indicates corrected material) Page 7

More information

Riemann Sums. Outline. James K. Peterson. September 15, Riemann Sums. Riemann Sums In MatLab

Riemann Sums. Outline. James K. Peterson. September 15, Riemann Sums. Riemann Sums In MatLab Riemann Sums James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 15, 2013 Outline Riemann Sums Riemann Sums In MatLab Abstract This

More information

Calculus of Variations Summer Term 2016

Calculus of Variations Summer Term 2016 Calculus of Variations Summer Term 2016 Lecture 14 Universität des Saarlandes 28. Juni 2016 c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 28. Juni 2016 1 / 31 Purpose of Lesson Purpose

More information

ζ (s) = s 1 s {u} [u] ζ (s) = s 0 u 1+sdu, {u} Note how the integral runs from 0 and not 1.

ζ (s) = s 1 s {u} [u] ζ (s) = s 0 u 1+sdu, {u} Note how the integral runs from 0 and not 1. Problem Sheet 3. From Theorem 3. we have ζ (s) = + s s {u} u+sdu, (45) valid for Res > 0. i) Deduce that for Res >. [u] ζ (s) = s u +sdu ote the integral contains [u] in place of {u}. ii) Deduce that for

More information

Euler s Method, cont d

Euler s Method, cont d Jim Lambers MAT 461/561 Spring Semester 009-10 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 information

Later in this chapter, we are going to use vector functions to describe the motion of planets and other objects through space.

Later in this chapter, we are going to use vector functions to describe the motion of planets and other objects through space. 10 VECTOR FUNCTIONS VECTOR FUNCTIONS Later in this chapter, we are going to use vector functions to describe the motion of planets and other objects through space. Here, we prepare the way by developing

More information

A MATH 1225 Practice Test 4 NAME: SOLUTIONS CRN:

A MATH 1225 Practice Test 4 NAME: SOLUTIONS CRN: A MATH 5 Practice Test 4 NAME: SOLUTIONS CRN: Multiple Choice No partial credit will be given. Clearly circle one answer. No calculator!. Which of the following must be true (you may select more than one

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, Matematics and Computer Science. ANSWERS OF THE TEST NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS (WI3097 TU) Tuesday January 9 008, 9:00-:00

More information

Find the equation of a plane perpendicular to the line x = 2t + 1, y = 3t + 4, z = t 1 and passing through the point (2, 1, 3).

Find the equation of a plane perpendicular to the line x = 2t + 1, y = 3t + 4, z = t 1 and passing through the point (2, 1, 3). CME 100 Midterm Solutions - Fall 004 1 CME 100 - Midterm Solutions - Fall 004 Problem 1 Find the equation of a lane erendicular to the line x = t + 1, y = 3t + 4, z = t 1 and assing through the oint (,

More information

Math 3C Lecture 25. John Douglas Moore

Math 3C Lecture 25. John Douglas Moore Math 3C Lecture 25 John Douglas Moore June 1, 2009 Let V be a vector space. A basis for V is a collection of vectors {v 1,..., v k } such that 1. V = Span{v 1,..., v k }, and 2. {v 1,..., v k } are linearly

More information

Integration by Parts Logarithms and More Riemann Sums!

Integration by Parts Logarithms and More Riemann Sums! Integration by Parts Logarithms and More Riemann Sums! James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 16, 2013 Outline 1 IbyP with

More information

A Brief Introduction to Numerical Methods for Differential Equations

A Brief Introduction to Numerical Methods for Differential Equations A Brief Introduction to Numerical Methods for Differential Equations January 10, 2011 This tutorial introduces some basic numerical computation techniques that are useful for the simulation and analysis

More information

First In-Class Exam Solutions Math 246, Professor David Levermore Thursday, 20 September 2018

First In-Class Exam Solutions Math 246, Professor David Levermore Thursday, 20 September 2018 First In-Class Exam Solutions Math 246, Professor David Levermore Thursday, 20 September 208 () [6] In the absence of predators the population of mosquitoes in a certain area would increase at a rate proportional

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

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. 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 Adams-Bashforth

More information

Vector fields Lecture 2

Vector fields Lecture 2 Vector fields Lecture 2 Let U be an open subset of R n and v a vector field on U. We ll say that v is complete if, for every p U, there exists an integral curve, γ : R U with γ(0) = p, i.e., for every

More information

ES.1803 Topic 19 Notes Jeremy Orloff. 19 Variation of parameters; exponential inputs; Euler s method

ES.1803 Topic 19 Notes Jeremy Orloff. 19 Variation of parameters; exponential inputs; Euler s method ES83 Topic 9 Notes Jeremy Orloff 9 Variation of parameters; eponential inputs; Euler s method 9 Goals Be able to derive and apply the eponential response formula for constant coefficient linear systems

More information

MA 201, Mathematics III, July-November 2018, Laplace Transform (Contd.)

MA 201, Mathematics III, July-November 2018, Laplace Transform (Contd.) MA 201, Mathematics III, July-November 2018, Laplace Transform (Contd.) Lecture 19 Lecture 19 MA 201, PDE (2018) 1 / 24 Application of Laplace transform in solving ODEs ODEs with constant coefficients

More information

Displacement and Total Distance Traveled

Displacement and Total Distance Traveled Displacement and Total Distance Traveled We have gone over these concepts before. Displacement: This is the distance a particle has moved within a certain time - To find this you simply subtract its position

More information

Math 408 Advanced Linear Algebra

Math 408 Advanced Linear Algebra Math 408 Advanced Linear Algebra Chi-Kwong Li Chapter 4 Hermitian and symmetric matrices Basic properties Theorem Let A M n. The following are equivalent. Remark (a) A is Hermitian, i.e., A = A. (b) x

More information

Lecture 8: The First Fundamental Form Table of contents

Lecture 8: The First Fundamental Form Table of contents Math 38 Fall 2016 Lecture 8: The First Fundamental Form Disclaimer. As we have a textbook, this lecture note is for guidance and sulement only. It should not be relied on when rearing for exams. In this

More information

Math 307 Lecture 19. Laplace Transforms of Discontinuous Functions. W.R. Casper. Department of Mathematics University of Washington.

Math 307 Lecture 19. Laplace Transforms of Discontinuous Functions. W.R. Casper. Department of Mathematics University of Washington. Math 307 Lecture 19 Laplace Transforms of Discontinuous Functions W.R. Casper Department of Mathematics University of Washington November 26, 2014 Today! Last time: Step Functions This time: Laplace Transforms

More information

Calculus of Variations Summer Term 2015

Calculus of Variations Summer Term 2015 Calculus of Variations Summer Term 2015 Lecture 14 Universität des Saarlandes 24. Juni 2015 c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 1 / 20 Purpose of Lesson Purpose

More information

ECE257 Numerical Methods and Scientific Computing. Ordinary Differential Equations

ECE257 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 information

Math 497C Mar 3, Curves and Surfaces Fall 2004, PSU

Math 497C Mar 3, Curves and Surfaces Fall 2004, PSU Math 497C Mar 3, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 10 2.3 Meaning of Gaussian Curvature In the previous lecture we gave a formal definition for Gaussian curvature K in terms of the

More information

Math 361: Homework 1 Solutions

Math 361: Homework 1 Solutions January 3, 4 Math 36: Homework Solutions. We say that two norms and on a vector space V are equivalent or comparable if the topology they define on V are the same, i.e., for any sequence of vectors {x

More information

Section Vector Functions and Space Curves

Section Vector Functions and Space Curves Section 13.1 Section 13.1 Goals: Graph certain plane curves. Compute limits and verify the continuity of vector functions. Multivariable Calculus 1 / 32 Section 13.1 Equation of a Line The equation of

More information

MATLAB Project 2: MATH240, Spring 2013

MATLAB Project 2: MATH240, Spring 2013 1. Method MATLAB Project 2: MATH240, Spring 2013 This page is more information which can be helpful for your MATLAB work, including some new commands. You re responsible for knowing what s been done before.

More information

Differential Equations: Homework 8

Differential 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 information

Chapter 3 Convolution Representation

Chapter 3 Convolution Representation Chapter 3 Convolution Representation DT Unit-Impulse Response Consider the DT SISO system: xn [ ] System yn [ ] xn [ ] = δ[ n] If the input signal is and the system has no energy at n = 0, the output yn

More information

Computer Graphics: 2D Transformations. Course Website:

Computer Graphics: 2D Transformations. Course Website: Computer Graphics: D Transformations Course Website: http://www.comp.dit.ie/bmacnamee 5 Contents Wh transformations Transformations Translation Scaling Rotation Homogeneous coordinates Matri multiplications

More information

Inner Product Spaces 6.1 Length and Dot Product in R n

Inner Product Spaces 6.1 Length and Dot Product in R n Inner Product Spaces 6.1 Length and Dot Product in R n Summer 2017 Goals We imitate the concept of length and angle between two vectors in R 2, R 3 to define the same in the n space R n. Main topics are:

More information

Section x7 +

Section 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 information

Final Exam Sample Problems, Math 246, Spring 2018

Final Exam Sample Problems, Math 246, Spring 2018 Final Exam Sample Problems, Math 246, Spring 2018 1) Consider the differential equation dy dt = 9 y2 )y 2. a) Find all of its stationary points and classify their stability. b) Sketch its phase-line portrait

More information

PDEs in Image Processing, Tutorials

PDEs in Image Processing, Tutorials PDEs in Image Processing, Tutorials Markus Grasmair Vienna, Winter Term 2010 2011 Direct Methods Let X be a topological space and R: X R {+ } some functional. following definitions: The mapping R is lower

More information

Chapter 7. Tridiagonal linear systems. Solving tridiagonal systems of equations. and subdiagonal. E.g. a 21 a 22 a A =

Chapter 7. Tridiagonal linear systems. Solving tridiagonal systems of equations. and subdiagonal. E.g. a 21 a 22 a A = Chapter 7 Tridiagonal linear systems The solution of linear systems of equations is one of the most important areas of computational mathematics. A complete treatment is impossible here but we will discuss

More information