Math 054 Partial Differential Equations - HW Assignment 2
1. Explain in a short paragraph where the uxx(x, y) or ∇2u comes from in the heat equation. This should be understandable to someone outside the class. You can use terminology like heat flux...
2. How is the heat equation related to the Laplace equation?
Problems to think about-
1. In class we discussed the discrete Laplace equation example where
This can be more generally written as
[LN]UN = fN
where N is the number of interior nodes, UN is the vector of unknowns, fN is a known vector, and [LN] is the N x N coefficient matrix. [LN] is always a symmetric matrix with the largest value on the diagonals. Convince yourself that the following are true
(a) fN = 0 implies UN = 0.
(b) For each fN there exists a unique UN.
(c) [LN] has real eigenvalues, all of one sign.
(d) [LN] has a complete set of orthogonal eigenvectors.
(e) UN satisfies the discrete max-min principle.
2. One numerical technique used for approximating a solution for differential equations is the finite difference method. It is based on approximating a derivative as a difference quotient. Specifically, as long as is small,
U'(x) ≈ (u(x + ∈) - u(x)/∈), u''(x) ≈ (u(x) - 2u(x) + u(x - ∈)/∈2).
Use the finite difference method to discretize the heat equation, ut = kuxx for the interior nodes (notice that we are going in the opposite direction than we did in class).