Course Description - Math 338 - Topics in Applied Mathematics

In this course we are interested in generating and understanding various solutions to certain "classical" partial differential equations such as the heat, wave, and Laplace equations with various boundary conditions. These classical equations arise from modeling simple physical problems. For example the wave equation can be used to model a vibrating string with fixed end points or in 2-dimensions the vibrations of a drum.

	A vibrating string Boundary condtions: fixed ends Initial displacement f(x)=x(sin(πx/10))+cos(4πx/10) Initial velocity g(x)=0
	A vibrating drum Boundary condtions: fixed edge Initial displacement f(r,θ)=-1/2(exp(-5r²cos²(θ)+5rcos(θ)-5/4-5r²sin²(θ))) Initial velocity g(r,θ)=0

The actual mechanics, a method called "separation of variables", for solving such equations are in fact deceivingly easy whereas the underlying mathematics is not. The method of "separation of variables" employs orthogonal (Fourier, Bessel, eigenfunction) series expansions to construct a formal power series solution and it is here in which the mathematical subtleties are contained. We will first consider Fourier series and although developing Fourier Series in a truly rigorous way is outside the scope of this course, we will make an effort to understand "where Fourier Series takes place" and what steps of "separation of variables" require justification.

The material may be divided into 5 groupings, each of which should take 2 or 3 weeks to cover. The relevant sections from the text are in parentheses.

1 Introduction to PDEs (1.1, 1.2); brief intro to the "method of separation of variables" (3.3); Fourier series (most of Ch. 2).

2 Separation of variables applied to Wave, Heat, Laplace equations in rectangular coordinates. Applications of Fourier series (most of Ch. 3).

3
Applications of our new techniques to PDEs in polar and cylindrical coordinates. The ideas are mostly the same, but require some more subtle analysis (most of Ch. 4).

4
Sturm-Liouville problems; beautifully unifies a lot of the theory used in the previous sections (most of Ch. 6).

5
Fourier transform; used for solving PDEs on infinite domains, as opposed to the finite domains of the previous sections. (most of Ch. 7).

1	Introduction to PDEs (1.1, 1.2); brief intro to the "method of separation of variables" (3.3); Fourier series (most of Ch. 2).
2	Separation of variables applied to Wave, Heat, Laplace equations in rectangular coordinates. Applications of Fourier series (most of Ch. 3).
3	Applications of our new techniques to PDEs in polar and cylindrical coordinates. The ideas are mostly the same, but require some more subtle analysis (most of Ch. 4).
4	Sturm-Liouville problems; beautifully unifies a lot of the theory used in the previous sections (most of Ch. 6).
5	Fourier transform; used for solving PDEs on infinite domains, as opposed to the finite domains of the previous sections. (most of Ch. 7).