Department of Mathematics & Computer Science

MATHEMATICS AND COMPUTER SCIENCE

Course: M233a(3 units) Applied Mathematics I

Semester: Fall 2000

Prerequisite: Math 133b or instructor consent

Time & location: T/Th 19.00-20.15 MH 235

Office hours: 12-13.30, 18.00-19.00

Catalog Description:

Derivation of the partial differential equations of classical mathematical physics. Existence and uniqueness of solutions of first order ordinary and partial differential equations. The classical theory of initial and boundary value problems for hyperbolic, parabolic and elliptic equations. Fourier series and transforms.

Texts:

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 the heat, wave and Laplace equations. A useful text specifically for this is: Partial equations and boundary value problems with Maple V, G. A. Articolo , Academic Press 1998.

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 cover the theory of the existence and uniqueness of solutions to first order systems of ordinary and partial differential equations. To introduce the elementary theory of function spaces including Banach and Hilbert spaces. To cover the theory of Fourier series and Fourier transforms. To discuss initial value-boundary value problems for the classical parabolic, hyperbolic and elliptic equations, and to give their solution by separation of variables. To present some basic types of nonlinear equation such as hyperbolic, diffusion and reaction-diffusion equations. To examine shock formation, weak solutions and travelling waves in representative equations derived from mathematical models taken from the literature.

Student Outcomes:

The student should be able to:

1. Solve linear first order systems of ordinary differential equations.

2. Solve first order linear, quasi-linear partial differential equations.

3. Demonstrate understanding of when solutions exist and are unique to systems of ordinary differential equations.

4. Solve the classical heat, wave, Laplace equations for a large variety of initial value-boundary value problems by separation of variables.

5. Derive Fourier representations of functions and manipulate Fourier transforms.

6. Investigate shock formation in hyperbolic equations.

7. Distinguish between hyperbolic, diffusion and reaction-diffusion equations.

8. Investigate the existence of travelling wave solutions in reaction-diffusion equations.

Topics:

1. Function spaces, normed spaces, Cauchy sequences, complete spaces (B and H-spaces).

2. Existence/Uniqueness of first order systems of odes, Lipschitz inequalities, Gromwall inequalities.

3. Linearisation of nonlinear odes, fundamental solutions of linear first order systems, examples of mathematical models from the literature which are represented as first order systems and their analysis.

4. Fourier series, pointwise and uniform convergence, Riemann-Lebesgue Lemma.

5. Fourier transforms, convolutions, functionals, Dirac delta-function, test functions, Galerkin approximations, generalised functions (distributions).

6. Solution of classical heat, wave, Laplace equations by separation of variables for a large variety of initial value-boundary value problems. This assumes students have taken M133b.

7. Solution of first order linear and quasi-linear pdes in 2 independent variables. Characteristics and shock formation. Weak solutions and their representation as functionals. Examples of models taken from the literature.

8. Nonlinear diffusion and reaction diffusion equations. Burger’s equation and Fisher’s equation and the analysis of their solutions.

Possible course schedule:

Topics are listed by the number given in Topics.

Updated: November 2003