Using fourier analysis in order to solve differential equations.

Question

http://www.enm.bris.ac.uk/admin/courses/EMa2/Lecture%20Notes%2009-10/LSPDE5.pdf

The above PDF teaches us the separation of variables method. However, there are some things I dont understand, that I would like to ask about.

In the beginning of page $58$ , they set $b_n=AD$ , does this mean that for each choice of $n$ , we also make a specific choice of $A$ and $D$ ? If so, how come that the constants $A$ and $D$ depend upon $n$? Can anyone explain this?

Why do we write $u(x,t)$ to be an infinite sum of solutions, where each solution solves an unique system, determined by the choice of $n$ ? Is this actually what we are looking for? To find a $u(x,t)$ that solves the system for every possible choice of $n$ (and hence $k$ ?). Can anyone explain how and why we put $u(x,t)$ to be an infinite sum of different solutions for different systems?

Answer 1

It was Fourier who came up with this method for solving Partial Differential Equations. He found that, for many cases, he could determine all of the solutions of the special separated form X(x)Y(y)Z(z)T(t) and then form a sum of all of them with arbitrary multiplicative constants, and end up with enough possible solutions to solve the problem.

Fourier discovered "orthogonality conditions" (not his term) such that, when you integrate the solutions against each other, you get 0 unless they're the same solution. That allow him to isolate the coefficients required for a solution. It only took about a century to make sense of if all, and it led to a tremendous body of Mathematics that paved the way for modern Quantum Mechanics.

Fourier's idea that you could end up with enough possible separated solutions is really remarkable. Nearly everyone at the time believed that to be false, and you can see why. Fourier developed this technique to solve the new equations he had just developed to describe heat flow, which was brilliant enough on its own.

Answer 2

(1)For each choice of n, we can only determine the specific choice of AD (or you may prefer AD(n), as a function of n). The choice of individual A and D is arbitrary.

(2)One can imagine cos(nx) and sin(nx) as a set of vectors of infinite dimension (an individual x, serves as a dimension, so there are infinite number of the dimension) that add up (by summing over n) to be the vector f(x), where f(x) is any function of x. So AD(n) here just means the proportionality of each component and you have to determine it.

The above is of layman language. Professionals can improve the statement.