Numerical Solutions of Differential Equations

AMS 213 – Numerical Solutions of Differential Equations. The course focuses on basic numerical linear algebra and numerical methods for ODEs and PDEs. Topics include LU, Cholesky and QR decompositions; iterative methods; Runge-Kutta methods; error estimation and error control; stability and convergence, etc. MATLAB is the chosen language for this course.

Instructor: Qi Gong (, Baskin Engineering 361A

Text Book

The lectures are based on notes extracted from various textbooks instead of using a single textbook. Listed in the following are some suggested references.

Numerical Linear Algebra, Trefethen & Bau, SIAM

Numerical Analysis of Differential Equations, Arieh Iserles, Cambridge Univ. Press

Numerical Recipes, Teukolsky, Vetterling & Flannery, Cambridge Univ. Press

Mastering MATLAB 6, Duane Hanselman and Bruce Littlefield, prentice Hall

Introduction to MATLAB, Ross L. Spencer and Michael Ware, available at

Lectures: Tuesday and Thursday, 10:00AM - 11:45AM, Crown Cirm 104

Office Hours: Monday 9:00AM - 11:30AM, BE 361A

Grading: Homework 50%, Final exam 50%.

Tentative Schedule

Week 1: Introduction, gaussian elimination, LU factorization and Cholesky factorization.

Week 2: Mathematical preliminaries on linear space, QR factorization by Gram-Schmidt and Householder transformation.

Week 3: Lest squares fitting, condition number and stability.

Week 4: Iterative methods for linear systems, conjugate gradient method.

Week 5: Eigenvalue and eigenvector problem.

Week 6: Initial value problem of ODEs. Eluer’s method, Explicit Runge-Kutta methods, convergence rate analysis, stiff equations and absolute stability.

Week 7: Quadrature integration and Implicit Runge-Kutta and multistep methods.  

Week 8: Two point boundary value problems, finite difference and shooting methods. PDE 1: Poisson equations.

Week 9: PDE 2: Diffusion equations, consistency, order and stability. Lax Equivalence Theorem.

Week 10: PDE 3: Stability analysis: eigenvalue approach and Fourier transform approach, advection equations.

Instructors and Assistants