Course Number: M233b (3 units)

Course Title: Applied Mathematics II

Prerequisite: Continuation of M233a

Catalog Description:

Continuation of M233a. Selected topics such as Green’s functions, eigenvalue problems, integral equations, or variational methods.

Texts:

Perturbation Methods, A. Nayfeh, Wiley Interscience 1973.

Partial differential equations of mathematical physics and integral equations, R. B. Guenther and J. W. Lee, Dover publications 1996.

An introduction to nonlinear partial differential equations, J. D. Logan, Wiley Interscience 1994.

Lab work:

Maple is used for displaying the solutions to equations covered in the course.

Project:

Each student has to hand in a written report on his research into a particular topic. The choice of topics can vary from mathematical modeling, to research on a specific equation and its solutions.

Students are strongly encouraged to choose their own topics. Recent research papers are used and a comprehensive bibliography is required.

Course Objectives:

To enable students to be able to use perturbation theory in analysing ordinary and partial differential equations. To be able to solve general nonlinear first order partial differential equations by the method of characteristics. To cover the basic theory of Sobelev spaces for pdes. To present the mathematical theory behind the solution of the classical heat, wave and Laplace equations. To cover the theory of Fredholm and Volterrs integral equations.

Student Outcomes:

The student should be able to:

1. Analyse singular and regular perturbation problems for ordinary and partial differential equations.

2. Solve first order nonlinear partial differential equations by the method of characteristics.

3. Demonstrate understanding of Sobelev function spaces and the use to which they are put in partial differential equations.

4. Demonstrate an understanding of the mathematical theory behind the separation of variables method of solution of the classical heat, wave and Laplace equations.

5. Derive the integral equation equivalent of boundary value problems for the classical partial differential equations in mathematical phusics.

6. Demonstrate an understanding of the theory of Fredholm and Volterra integral equations and their method of solution.

## Topics:

1. Regular and singular perturbation theory, multiple time scales, and the method of matched asymptotics. Examples taken from the literature.

2. The method of characteristics, envelopes, Monge cones, Cauchy data for first order partial differential equations.

3. Green’s functions, Sturm-Liouville problems and eigenfunction problems. Completeness in a function space. Application of the theory to the classical equations of mathematical physics.

4. Reformulation of boundary value problems for partial differential equations as integral equations.

5. Integral equations of Fredholm and Volterra, Neumann series and operator norms.

6. Potential theory, the maximum-minimum principle and its consequences.

7. Sobelev spaces and elementary discussion of their use in partial differential equation theory.

Possible course schedule:

Topics are listed by the number given in Topics.

Topics / No. of lectures
1. / 6
2. / 3
3. / 4
4. / 2
5. / 6
6. / 5
7. / 4
Total = / 30

Prepared by: R. K. Dodd, Spring 2000.

Updated: November 2003