Question

1. Consider a thin rectangular plate in the ry-plane, the figure. The PDE describing the temperature of the plate is the heat equation shown in as 0 xa, 0< y < b, t>0. D + at where u(x, y, t) is the temperature at point (x, y) diffusivity at time t andD> 0 is the thermal (a) Suppose that the solution to the PDE (once we impose initial and boundary con ditions) reaches equilibrium when t o, that is there is a function ueg(, y) such that (3 points) lim u(r, y, t) = ueq(x, y), Vre [0, a], Vy[0, b) t00 Prove that u Solves the 2D Laplace's equation, that is 0. + (b) Conversely, show that any solution v(x, y) of Laplace's equation is also a solution of the heat equation. These are called steady state solutions, since they don't depend on time. (3 points) Total for Question 1: 6 points

Answer 1

(1) Here we see that if u(x,y,t) is a solution of the heat equation; then

e(x, y)lim u(x, y, t) to

which gives us that

bane at 0

and since D>0, we see that

0 leq ueg

thus the result follows.

(2) Since any solution of the form v(x,y) of the laplace equation is independent of time; we see that the partial derivative with respect to time is zero and hence the since D>0 the heat equation follows immediately.

i.e.

- 0 = 0, D> 0; at

hence the LHS and RHS of the heat equation are satisfied.