MATHS 473: Boundary Value Problems (3)

Syllabus

1.        Prerequisite:  MATHS 374

2.        Catalog Description:  Fourier series and integrals, heat and wave equations in one dimension, Laplace's equation in two dimensions, problems in higher dimensions, numerical methods of solving boundary value problems.

3.        Course Objectives:  The goal of this course is to provide a solid foundation in the study of classical boundary value problems that arise in engineering and physics.   In addition, the student will learn how to utilize a computer algebra package such as Mathematica in problem solving a crucial skill for today’s applied mathematician.  This course is designed to complement the department’s course on Partial Differential Equations (MATHS 475).

   

4.        Course Rationale:  Many real-world problems in engineering or mathematical physics can be modeled with the heat, wave, or potential equations.  Students who wish to work as an applied mathematician must be familiar with these equations and their standard solution techniques.   

5.        Course Content:  The core of the course should include the following topics:

·         Fourier Series and Transform:  Periodic functions and Fourier series, arbitrary period and half-range expansions, convergence of Fourier series, superposition of solutions, orthogonal sets of functions, Fourier transform.

·         The Heat Equation: Derivation of heat equation and boundary conditions for heat flow in a rod, steady-state temperatures, separation of variables, solution of basic examples:  fixed end temperatures, insulated rod, different boundary conditions at each end.

·         The Wave Equation: Derivation of wave equation and boundary conditions for vibrating string, solution of vibrating string problem, D’Alembert’s solution.

·         The Potential Equation:  The potential equation (Laplace’s equation) in rectangular, polar, and cylindrical coordinate systems, solution of the potential equation in a rectangle, solution of the potential equation in a disk.

·         Problems in Several Dimensions:  Derivation and solution of the wave and heat equations in two dimensions

Note that it is a good idea to include a quick review of ordinary differential equations, including the idea of Green’s functions.  Illustrations of how computer algebra systems, such as Mathematica or Maple, can be applied to solve or explore the core ideas should be included in the course. Additional topics are at the discretion of the instructor.  For a more theoretical flavor, more time can be spent on the theory of Fourier series, including uniform convergence, mean error and convergence in mean, Stürm-Liouville problems, and Bessel functions and Legendre polynomials.  For a more applied flavor, more time can be spent on numerical methods and the Laplace Transform.

6.        Course Format:  This course is usually taught in a lecture or lecture/discussion format.  Although not required, computer labs, group discussion, projects, and technology such as graphing calculators or computer algebra systems may be used to help illustrate ideas.   

7.        Methods of Evaluating Student Performance:  Course grades are determined primarily by student performance on tests, quizzes, and projects, as well as possibly homework, labs, or class participation. The evaluation and weight of these various components are at the discretion of the individual instructor.

8.        Evaluation of the Course:  The instruction of the course is evaluated by departmental student evaluations and peer evaluation.  The course is reviewed and revised periodically by the Departmental Undergraduate Programs Committee.

M. Karls, M. Toda, December, 2000

M. Karls, A. Mohammed, J.P. Liamba 4/2004,  Ahmed Mohammed, Fall 2005