What Is the Steady-State Solution for This Heat Conduction Problem?

[Jamin2112]

Homework Statement

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

...

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

Homework Equations

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

The Attempt at a Solution

∂²X'' T = X T'

X'' / X = (1/∂²) T' / T = -ß

X'' + ßX = 0,
T + ∂²ßT' = 0.

Since u_x(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),

lim_t-->∞ 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 u_x(0,t). I'm having trouble putting the pieces together.

Please help, geniuses.

 

[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?

 

[Jamin2112]

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?

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.