## Computational Differential Equations

### Instructors

Lectures: Martin Jagersand
Labs and Seminars: Karteek Popuri

### Communication

The course web page course material, dates etc: http://www.cs.ualberta.ca/~jag/courses/cde/
The course newsgroup is used for class discussion, questions etc: ualberta.courses.cmput.419
Course instructors and TA available for meetings Tue after class and lab respectively.

### Objectives

Learn how to model real world problems using differential equations, and solve these with numerical methods, in particular Galerkin FEM methods.

### Description

Since Newton's time differential equations have been the most successful modeling method for describing the world around us. Mathematical models using Ordinary and Partial Differential Equations (ODE and PDE) are fundamental to Science and Engineering. For hundreds of years the focus was on analytical solutions, often expressed in function series, of simple problems on simple geometries. This allowed for instance an understanding of the atom through solving the Schroedinger equation. However, we know an exact solution for only the simplest atom of the hundreds of elements, namely for the hydrogen atom. Even this solution requires a fair amount of knowledge of e.g. Bessel functions and spherical harmonics to develop and express analytically.

The focus in this course is to develop a theory and method of computationally solving general differential equations on general geometries. Instead of analytic closed form solutions using (combinations of) known functions, we will develop computational methods which numerically compute the solution using discrete basis functions (usually local polynomials -- "finite elements"). Unlike conventional numerical analysis courses, our main focus is more on formulating and computationally solving real world problems, and a bit less on e.g. error analysis, though this is brought up.

The course will start with simple ODE you may know from examples in calculus, e.g. population dynamics, 1D diffusion, and use these to introduce Galerkins method. Then we will move to solve several important PDE including the heat equation, wave equation, convection-diffusion problems, and elliptic eigenvalue problems. In the process we will cover basic concepts of function spaces as well as special numerical methods for solving the large equation systems resulting from Galerkin's finite element method (FEM). However, on these side topics the course is not as deep as a functional analysis course, nor as general as a typical introduction to numerical methods courses. Instead it is complementary to these courses.

The course examination will include homework problems throughout the term, some simpler programming/use of Matlab, and two exams.

### Schedule

Lectures: Tuesday, Thursday 11:00 - 12:20, ED 170, 01/05-04/12/2010
Labs: Lab H01 Mon 5:00PM - 7:50PM CSC 235
Lab H02 Tue 2:00PM - 4:50PM CSC 235