Separation of variables, or how I’m learning to ignore the math, and just take the Fourier expansion

April 17th, 2007 ~ Posted in: Mathematics

So I was just talking to the new French exchange student about math. He does harmonic analysis — Littlewood-Paley theory– and other comparatively stratospheric stuff. Meanwhile I’m stuck trying to solve the following basic PDE using separation of variables:

u_t = k u_{xx} + Q(x,t); u(x,0)=f(x); u(0,t)=A(t); u_{x}(L,t)=0

I’d rather be doing the stratospheric stuff, but I suppose one must start with the basics (must one?)

I like how my poison of choice (i.e. my reference book to supplement what’s covered– or, more often, not– in class ) explains the use of eigenfunction expansions in the method of separation of variables:

In our case, the generalized superposition principle implies that the formal expression [solution as Fourier expansion] is a natural candidate for a generalized solution of [the heat equation]. By a ‘formal solution’, we mean that if we ignore questions concerning convergence, continuity, and smoothness, and carry out term-by-term differentiations and substitutions …

Nice, huh? Now I have to figure out how to use separation of variables to solve that problem above. Funny how most examples of separation of variables are done on problems with no source terms and homogeneous BCs. Well, not really that funny.

Update
By the grace of Kevorkian, I finally got a straight answer on how to solve nonhomogeneous PDEs (or at least the heat equation) with source terms, using separation of variables. In the case above, there’s an obvious (in hindsight) trick, but there’s a change of variables technique behind the trick that works in less obvious situations. I’ll post about it later. Got to go finish this problem (I’ve been working on it for about 5 hours now). Damn I feel stupid.

Mathematics ~

2 Responses to “Separation of variables, or how I’m learning to ignore the math, and just take the Fourier expansion”

  • 1. ObsessiveMathsFreak
    April 19th, 2007 at 4:55 am

    So what did you use? Green’s functions? Fourier basis functions? Are you supposed to assume that Q(x,t) is separable as well?

    Not sure how you go about getting a green’s function for inhomogeneous boundary conditions. The only ones I use are green’s function that simply must tend to zero as x goes to infinity.

  • 2. Alex
    April 19th, 2007 at 7:29 am

    I used Fourier basis functions: a change of variables to get the pde in terms of a function with homogeneous BCs, assumed the solution was a fourier series with time varying coefficients, and used the initial conditions and the source terms determine them. I don’t know anything much about green’s functions– I think we’ll cover those soon. But I remember Kevorkian’s analytic solutions book talking about the relationship b/w green’s functions and separation of variables– there might have been something on green’s functions for inhomogeneous BCs.

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