Heat Equation and Boundary Value Problems : Steady-State Solution, Neumann Boundary Conditions

Solution Preview

The following is a syllabus I found on the net concerning everything you were afraid to ask concerning heat equations. I thought it might help you see what you need to know as the semester progresses. Also I found your homework problem published on the net and I also found http://online.redwoods.cc.ca.us/instruct/darnold/deproj/sp02/AbeRichards/paper.pdf

which pretty much discusses your homework set as I believe it is written by your professor.

2) Heat equation on finite intervals:
- the heat equation is derived using the law of conservation of (heat) energy, plus experimental data. Typical extra data: initial data at time t=0, plus boundary data at positions x=0, x=a: Dirichlet type, Neumann type, etc.
- homogeneous problems are solved via the method of separation of variables. This breaks the problem into 3 "subproblems": 1) an eigenvalue problem for x, yielding solutions Phi_n(x) determined by constants lambda_n; 2) an exponential-type equation for t which we solve for those lambda_n, yielding solutions T_n(t); 3) an initial data problem for the series built using Phi_n(x)T_n(t), which we solve via orthogonality arguments (if we're lucky, this is just Fourier theory; otherwise, it's Sturm-Liouville theory: see below).
- non-homogeneous problems are solved in two steps: 1) we solve the corresponding "steady-state" problem, finding a solution v(x); 2) we solve the corresponding homogeneous problem, finding a solution w(x,t). The solution to the initial problem is then u(x,t):=w(x,t)+v(x).

Week 5: Continue heat equation; HOMEWORK ON CHAPTER 1 DUE ON WEDNESDAY!
1) Comments regarding the steady-state equation:
- the steady-state equation sometimes leads to an incomplete answer (so: to fully determine the steady-state temperature, we need to rely on additional info hidden somewhere else in the problem), and sometimes cannot be solved (ie: there is no steady-state temperature). The latter case implies that our separation of variables method will not work, so we must try another method ...