## Math 269A, Section 1, Fall 2018

### Numerical solution for systems of ordinary differential equations; initial and boundary value problems

Lecture Meeting Time: MWF 1.00PM - 1:50PM.
Lecture Location: MS 5138

Instructor: Luminita A. Vese
Office: MS 7620-D
Office hours: Wednesday 2-2.45pm, or by appointment.

E-mail: lveseATmath.ucla.edu

Discussion Section: Thursday, 1.00PM - 1.50PM, MS 5117

Teaching Assistant: Jean-Michel Maldague
Office: MS 2963
Office hours: Thursday 2-4pm.
E-mail: jmmaldagueATmath.ucla.edu

Recommended Textbooks:
• U.M. Ascher and L.R. Petzold, "Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations", SIAM 1998.
• A. Iserles, "A first course in the Numerical Analysis of Differential Equations", Cambridge Texts in Applied Mathematics, Cambridge University Press, 1998.
• J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis", Third Edition, Springer 2002. (Chapter 7).

Topics:
- Notations and terminology for ODE's and systems of ODE's; reduction of higher order ODE's to 1st order systems of ODE's; the fundamental existence and uniqueness thm. for ODE's (Lipschitz condition).
- Introduction of Euler's method, order of Euler's method, one step methods (introduction, definition, consistency, local truncation error).
- Explicit Runge-Kutta (ERK) methods (introduction of the method in the general case, notations in the general case, derivation of ERK of second order); Runge-Kutta method of fourth order.
- Examples of implicit methods: trapezoidal rule, midpoint rule, the theta method, and the implicit Euler's method; computation of orders for these methods.
- Convergence of one-step methods (the general case; see also convergence for Euler's method, etc).
- Asymptotic expansions for the global discretization error for one step methods, and applications to error estimate.
- Practical implementation of one step methods
- Linear Multistep methods: examples, derivation using the Lagrange interpolation polynomial
- Linear multistep methods: definition and computations of the local truncation error, order of the method, consistency.
- Implicit and explicit linear multistep methods; predictor-corrector methods.
- Examples of consistent multistep methods which diverge.
- Linear difference equations: stability (root) condition, general solution.
- Convergence Thm. for linear multistep methods
- Order and consistency for linear multistep methods
- Adaptive methods for one-step and multi-step methods, error control, Milne device, extrapolation
- Stiff differential equations, stability and intervals (regions) of absolute stability, A-stable methods, BDF methods
- Numerical methods and stability for systems of ODE's
- Finite difference methods for linear BVP
- Functional (fixed point) iteration and Newton's iteration for solving systems of ODE's using an implicit method

Course Requisites: Math 115A, Math 151A, and Math 151B or equivalent.

• PIC Lab: Boelter Hall 2817 and Mathematical Sciences 3970
http://www.pic.ucla.edu/piclab/
• MATLAB Documentation (thanks to Prof. C. Anderson, UCLA)
• Class Web Page: http://www.math.ucla.edu/~lvese/269a.1.18f/
• Numerical Analysis Qualifying Exam
• Numerical Recipes in C
• Getting started with MATLAB

Homework Policy: There will be several assignments involving both theoretical and computational exercises. These will be collected on Friday in class.

Examinations: A one-hour final exam.
Final Exam: Friday, December 7, 1-2pm. There will be a one-hour final exam during the last lecture.
The examinations are closed-book and closed-note.
No exams at a time other than the designated ones will be allowed (exceptions for illness with document proof or emergency).

Grading Policy: HW 70%, Final 30%

Weekly Homework Assignments:

HW #1 (due on Friday, October 5)
HW1.pdf

HW #2 (due on Monday, October 15)
HW2.pdf

HW #3 (due on Friday, October 26)
HW3.pdf HW3.tex

HW #4 (due on Monday, November 5)
HW4.pdf HW4.pdf

Sample practice problems for the midterm

HW #5 (due on Friday, November 16)
HW5.pdf HW5.tex

HW #6 (due on Wednesday, November 28, or on Friday, November 30)
HW6.pdf HW6.tex

HW #7 (due on Friday, December 7)
HW7.pdf HW7.tex

Sample practice problems for the final and announcements