1. PDE help

1. Consider the heat equation in a two-dimensional region with an insulated boundary. An isotherm is a line of constant temperature, i.e., a level curve of the function u(x,t). Show that the isotherms are always perpendicular to the boundary.

I know that the level curve is going to be perpendicular to the gradient of the function, but I don't know where to go with that.

2. Consider the heat equation with internal heat source on a rod of length L:

u_t = u_xx + x - B
u(x,0) = f(x)
u_x(0,t) = u_x(L, t) = 0

For what values of the constant B does an equilibrium temperature distribution exist, and for each such B, what is the equilibrium distribution?

For 2, I got that the equilibrium temperature exists when B = .5L, but I don't know how to find the equilibrium distribution.

2. Originally Posted by Math Major
1. Consider the heat equation in a two-dimensional region with an insulated boundary. An isotherm is a line of constant temperature, i.e., a level curve of the function u(x,t). Show that the isotherms are always perpendicular to the boundary.

I know that the level curve is going to be perpendicular to the gradient of the function, but I don't know where to go with that.

2. Consider the heat equation with internal heat source on a rod of length L:

u_t = u_xx + x - B
u(x,0) = f(x)
u_x(0,t) = u_x(L, t) = 0

For what values of the constant B does an equilibrium temperature distribution exist, and for each such B, what is the equilibrium distribution?

For 2, I got that the equilibrium temperature exists when B = .5L, but I don't know how to find the equilibrium distribution.

For #2, at equilibrium $u_t = 0$ so
$
Now integrate twice (your condition $B = \frac{1}{2} L$ will satisfy the insulated BC's).