# Heat equation w/ Newton's Law

1. Sep 8, 2007

### Mindscrape

The equation given is

$$\frac{\partial T}{\partial t} = \frac{\partial}{\partial x} \left( K(x) \frac{\partial T}{\partial x} \right), 0<x<L$$

where T is temperature, t is time, and x is the 1-d spatial coordinate, also

$$K(x) = \kappa e^{-x}, \kappa > 0$$

and the boundary conditions are that T(x=0, t) = A = constant, and we have a Newton law of cooling condition at x = L : $$\frac{\partial T}{\partial x} = - \alpha T$$ with alpha a positive constant. Find the equilibrium temperature.

I'm fine with the PDE part, which actually reduces to the ODE because we want the equilibrium temperature, but the Newton's Law part is sort of confusing for me. Someone needs to point me in the right direction. So...

$$0 = \frac{d}{dx} (K(x) \frac{dT}{dx})$$

$$c_1 = K(x) \frac{dT}{dx}$$

Now according to Newton's law of cooling

$$K(L) \frac{dT}{dx}(L,t) = H[T(L,t) - T_e (t)]$$

where T_e would be the external temperature/bath.

Does that mean that c1 is H[T(L,t) - T_e (t)]? Then continue to solve the ODE to find that

$$limit t->\infty T(x,t) = T(x) = \frac{c_1}{\kappa}e^x + c_2$$

and use the other boundary value to find that c_2 = A - c_1?

Last edited: Sep 8, 2007