Homework 08. Due Wednesday, November 07
Exercise 8.1. Consider a solution u=u(x, y, t) of the 2D heat equation in
the unit square Ω = (0; 1)×(0; 1) satisfying the Newton’s cooling conditions
on the boundary. We assume that the thermal conductivity K0is a positive
function in Ω, and the heat transfer coefficient Htakes four different constant
values on four different sides of the rectangle.
•Using separation of variables, formulate the PDE for the eigenvalue
problem for the (x, y) part of the solution.
•Sketch the picture of the domain Ω and indicate the normal vector
non each piece of the boundary. Also, write down the boundary
conditions for the eigenvalue problem. Your answer should not contain
any gradients and normal vectors.
•For any function v=v(x, y) satisfying the above Newton’s cooling con-
ditions, write down the formula for RQ[v]. Choose a proper parametriza-
tion of each side of ∂Ω and show that the integral term I∂Ω
(...) ds can
be presented as a sum of four nonnegative terms.
•What can you say about the minimal eigenvalue of the problem?
Exercise 8.2. Let mbe a nonnegative integer, and λbe a positive real.
Consider the following ODE:
r2d2f
dr2+rdf
dr + (λr2−m2)f= 0
in the interval 0 ≤r < a.
•What is the origin of this equation?
•Show that the change of the variable z=√λr (ris outside of the square
root) leads to the Bessel ODE of order m. Write down this equation
in terms of the function F(z) = f(z/√λ).
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