# Darned heat conduction problem

1. Jun 1, 2010

### Jamin2112

1. The problem statement, all variables and given/known data

In each of Problems 1 through 8 find the steady-state solution of the heat conduction equation ∂2uxx=ut that satisfieds that given set of boundary problems.

.....

3. ux(0,t)=0, u(L,t)=0

2. Relevant equations

Assume u(x,t)=X(x)T(t)

3. The attempt at a solution

2X'' T = X T'

X'' / X = (1/∂2) T' / T = -ß

X'' + ßX = 0,
T + ∂2ßT' = 0.

Since ux(0,t)=0 and u(x,t)=X(x)T(t),

X'(0)T(t)=0 ---> X'(0)=0.

Solving X'' + ßX = 0 like a I would any differential equation,

X(x) = C1cos(√(ß)x) + C2sin(√(ß)x)

--->

X'(x) = -√(ß)C1sin(√(ß)x) + √(ß)C2(cos(√(ß)x)

--->

X'(0)=0= 0 + √(ß)C2 ---> C2=0

--->

X(x) = C1cos(√(ß)x).

Also, u(L,t)=0 ---> X(L)T(t)=0 ---> X(L)=0.

I'm getting somewhere, right?

Since we're talking about a steady-state solution being reached, some function of t and possibly x will disappear, leaving us with just a function v(x) that shows the temperature at any place in the rod.

u(x,t) = v(x) + w(x,t),

limt-->∞ u(x,t) = v(x)

...............

Anyhow, I'm sort of stuck. Problem 1 was was easy because it gave me u(0,t)=T1, u(L,t)=T2, just like the examples in the chapter; but this one is throwing me off with the ux(0,t). I'm having trouble putting the pieces together.

2. Jun 1, 2010

### Mapes

At steady state, $\partial T/\partial t=0$. Why not just integrate $\partial^2 T/\partial x^2=0$ twice and apply the boundary conditions? Or are you required to use separation of variables?

3. Jun 1, 2010

### Jamin2112

Hmmmmm ..... In the textbook example it never has us integrate T''(t)=0 when solving these problems. It does say, however:

Since v(x) must satisfy the equation of the heat equation ß2uxx=ut, we have

v''(x)=0, 0<x<L.