Math 170C (Introduction to Numerical Analyis: ODEs)

Course Topics: Introduction to Numerical Analysis: Ordinary Differential Equations
Instructor: Prof. Michael Holst (5739 AP&M, mholst@math.ucsd.edu; Office Hours: MW 1-2pm)
Term: Spring 2010
Lecture: 11:00a-11:50a MWF, AP&M B412
TA: Jonny Serencsa (5720 AP&M, jserencs@math.ucsd.edu; Office Hours: See the TA)
Discussion: See the TA
Main Class Webpage: http://ccom.ucsd.edu/~mholst/teaching/ucsd/170c_s10/index.html
Textbook(s): Numerical Analysis, by D. Kincaid and W. Cheney.
Printable Syllabus: Can be found [ here ].

CATALOG DESCRIPTION: 170C. INTRODUCTION TO NUMERICAL ANALYSIS: ORDINARY DIFFERENTIAL EQUATIONS (4)
Numerical differentiation and integration. Ordinary differential equations and their numerical solution. Basic existence and stability theory. Difference equations. Boundary value problems. Three lectures, one recitation. Prerequisite: Math. 170B and Math. 20D or consent of instructor.

COURSE INFORMATION: Many of the advances of modern science have been made possible only through the sophisticated use of computer modeling. The mathematical foundation of the computer modeling techniques now used in all areas of mathematics, engineering, and science is known as numerical analysis. The Math 170ABC series at UCSD provides an introduction to the exciting field of numerical analysis, which is also sometimes referred to as computational mathematics or scientific computing. Professor Holst has a passion for this particular area of mathematics, and much of his published research is in this area.

Math 170C deals primarily with the numerical solution of ordinary differential equations (ODEs). In 170C we will also study related numerical techniques for approximating the derivatives and integrals of functions in a stable and accurate way. Previous quarters dealt with numerical linear algebra (170A) and nonlinear equations and numerical approximation theory (170B), and we will have occasion to use these ideas to develop new techniques in 170C. The textbook for the course will be as listed above.

GRADES, HOMEWORKS, EXAMS, AND IMPORTANT DATES: Course information, such as homework assignments, due dates, and exam dates, will be maintained on the class webpage. Note that I sometimes make minor changes to the homework assignments as the quarter progresses, based on how much I am able to cover in the lectures. Therefore, CHECK THE WEBPAGE FREQUENTLY. The course will be graded on the homework assignments, two midterm examinations and a final examination, according to the following guidelines:

 Written and Computer HW (five homeworks): 30% of grade Midterm #1 (In class on Friday April 23): 15% of grade Midterm #2 (In class on Friday May 21): 15% of grade Final (11:30am-2:29pm, Friday June 11, APM B412): 40% of grade

There will be five homework assignments throughout the quarter. The first midterm will be based on homeworks 1 and 2, and the second midterm will be based on homeworks 3 and 4. The final will be cummulative and based on homeworks 1-4, as well as a small amount of new material from homework 5. The following policies regarding homeworks and exams will be applied:
1. All HW assignments will count towards the final grade (i.e., none can be dropped). Late HW will not be accepted.
2. In order to receive credit on a homework, you must at least attempt the computer parts of the homework assignments (if there are any).
3. There will be no make-up exams. If you miss a midterm with an excused absence (i.e., illness with a note from a doctor), the other midterm and the final exam will be weighted accordingly.
4. You are not allowed to use a calculator on midterms or finals.
5. You are allowed to bring a single 8x11 sheet of paper containing notes on both sides (formulas, whatever you find useful) to each midterm and to the final. My view is that this allows you to focus on learning how to do the problems and understanding the material, rather than on memorizing formulas.

LECTURES: The lectures will follow the textbook quite closely; in particular, we will cover Chapters 7 and 8, in that order. (Chapters 2, 4, and 5 were covered in 170A, and Chapters 3 and 6 were covered in 170B. Chapter 10 is covered in 171AB, and Chapter 9 is covered in 175.) Polynomial interpolation (Chapter 6, covered in 170B) will be reviewed briefly in 170C due the central role it plays in numerical differential equations. Homework assignments will be a combination of theoretical and computer problems; this will require some computer programming using MATLAB. The TA will be able to assist you in accessing your computer accounts as well as MATLAB.

 Week Topics Covered Week 1(3/29-4/2) Review, 6.1-6.2: Review of mathematical background and notation, differentiation of multivariate functions, Taylor series, polynomial interpolation, and error in interpolation. Week 2(4/5-4/9) 7.1-7.2: Numerical differentiation, Richardson extrapolation, numerical integration (quadrature) based on interpolation. Week 3(4/12-4/16) 7.3-7.5: Gaussian quadrature, Romberg integration, adaptive quadrature. Homework 1 due in TA section on Monday April 12. Homework 1 solutions may be found [ here ]. Week 4(4/19-4/23) 8.1: Basic theory of ordinary differential equations; existence and uniqueness of solutions. Homework 2 due Wednesday April 21. Homework 2 solutions may be found [ here ]. Midterm 1 given in class on Friday April 23. Covers: 6.1-6.2, 7.1-7.5 Midterm 1 solutions may be found [ here ]. Week 5(4/26-4/30) 8.2-8.3: Taylor-Series methods for ODE, Runge-Kutta Methods. Week 6(5/3-5/7) 8.4-8.5: Multi-step methods, local and global errors, stability. Homework 3 due Wednesday May 5. Homework 3 solutions may be found [ here ]. Week 7(5/10-5/14) 8.6: Systems and higher-order ODEs. Week 8(5/17-5/21) 8.7-8.8: Basic theory of boundary-value problems in ODEs, shooting methods. Homework 4 due Wednesday May 19. Homework 4 solutions may be found [ here ]. Midterm 2 given in class on Friday May 21. Covers: 8.1-8.5 Midterm 2 solutions may be found [ here ]. Week 9(5/24-5/28) 8.9-8.10: Finite difference methods, collocation methods. Week 10(5/31-6/4) 8.11, Review: Linear systems of autonomous ODEs, Review for final. Homework 5 due Wednesday June 2. Homework 5 solutions may be found [ here ]. Final(6/7-6/11) Final Exam: Covers: 6.1-6.2, 7.1-7.5, 8.1-8.10 NOTE: To help you prepare for the final exam, Jonny will be holding special office hours during finals week: Monday and Tuesday, 12-2pm, and by appointment.

HOMEWORKS ASSIGNMENTS: The following are the five homework assignments:

Homework 1 (due week 3 in TA section on Monday April 12; Problems covering 6.1-6.2, 7.1-7.2):
1. Problems 6.1: #22
2. Problems 6.1: #26
3. Problems 7.1: #14
4. Problems 7.1: #15
5. Problems 7.2: #10

Homework 2 (due week 4 at end of day on Wednesday April 21; Problems covering 7.2-7.4):
1. Problems 7.2: #18-#19
2. Problems 7.3: #7
3. Problems 7.3: #20
4. Problems 7.4: #7
5. Computer Problems 7.4: #1

Homework 3 (due week 6 at end of day on Wednesday May 5; Problems covering 8.1-8.3):
1. Problems 8.1: #12
2. Problems 8.1: #18
3. Problems 8.2: #2
4. Problems 8.2: #8
5. Problems 8.3: #1
6. Computer Problems 8.3: #1

Homework 4 (due week 8 at end of day on Wednesdy May 19; Problems covering 8.4-8.6):
1. Problems 8.4: #1
2. Problems 8.4: #7
3. Problems 8.5: #1
4. Problems 8.6: #1
5. Problems 8.6: #3
6. Computer Problems 8.6: #4

Homework 5 (due week 10 at end of day on Wednesday June 2; Problems covering 8.7-8.10):
1. Problems 8.7: #1
2. Problems 8.7: #9
3. Problems 8.9: #3
4. Computer Problems 8.9: #1
5. Computer Problems 8.9: #2

Extra Credit Homework (NORWEGIAN EXCHANGE STUDENTS ONLY -- DOES NOT COUNT TOWARDS GRADE IN THE CLASS):
1. Problem 1: Read about the Laplace Transform for solving IVP in ODE on your own; here are two reasonable sources:

Wikipedia
Wolfram MathWorld

The Wikipedia source has an example showing how to solve an IVP using Laplace Transform. Both sources have the table of Laplace Transforms you will need for the problems below.

2. Problem 2: Find the Laplace Transform of the following functions (t^p means t raised to the power p):

3 cos 5t
(t^2 + 1)^2
3 cosh 5t - 4 sinh 5t
(5 e^(2t) - 3)^2
t^2 sin t

3. Problem 3: Solve the following IVP for y(t) using the Laplace Transform:

y'' + 6 y' + 34 y = 0, t > 0,
y(0) = 3,
y'(0) = 1.