# PDE for temperature distribution in rectangle

1. Oct 26, 2014

### collectedsoul

1. The problem statement, all variables and given/known data
A rectangular chip of dimensions a by b is insulated on all sides and at t=o temperature u=0. The chip produces heat at a constant rate h. Find an expression for u(x,y,t)

2. Relevant equations
δu/δt = h + D(δ2u/δx2 + δ2u/δy2) x∈(0,a), y∈(0,b)

3. The attempt at a solution
I'm just wondering what effect the insulation on all sides will have on the equation. The temperature goes up by h per unit time throughout the chip but will the other term be affected?

2. Oct 27, 2014

### pasmith

"Insulated" means that there is no heat flux into or out of the rectangle. Here this reduces to the requirement that the component of $\nabla u$ normal to the boundary must vanish. This gives you the boundary condition you need in order to solve the PDE.

3. Oct 27, 2014

### collectedsoul

Why just the normal component? If its insulated won't the tangential component also vanish? So δu/δx=δu/δy=0 at x=0,a and y=0,b.

The question actually says that there is no need to solve the pde to get to u(x,y,t) - that there is a way to guess the answer without solving it. I can't think of any such way. Any ideas?

4. Oct 27, 2014

### pasmith

The physical constraint is "no heat flux across the boundary". The heat flux is $D\nabla u$, and the heat flux across the boundary is the component of $D\nabla u$ normal to the boundary. Flux parallel to the boundary is not a problem.

Is there any reason why the temperature should vary with position here, given that the heat source doesn't depend on position and the boundary conditions are consistent with a uniform temperature?

5. Oct 27, 2014

### collectedsoul

Alright...got it.

There's another related PDE: d2u/dx2 + d2u/dy2 = a*u with u = 0 on all edges for some eigenvalues a. I'm not sure how to go about this. I can start by solving the steady state equation but what do I do about the condition?

6. Oct 27, 2014

### pasmith

Seek a separable solution $u(x,y) = X(x)Y(y)$. You will find that there are particular values of $a$ (eigenvalues) for which $u(x,y) \equiv 0$ is not the only such solution.

7. Oct 27, 2014

### collectedsoul

I'm not clear what you mean by eigenvalues for which u=0 is not the only solution. Could you please elaborate?

8. Oct 28, 2014

### pasmith

Simple example: Let $X'' = kX$ subject to $X(0) = X(\pi) = 0$. The general solution of that ODE with satisfies $X(0) = 0$ is $$X(x) = \begin{cases} A\sin(\sqrt{|k|}x), & k < 0 \\ Ax, & k = 0 \\ A\sinh(\sqrt{k}x), & k > 0\end{cases}$$ In the second and third alternatives, the only way to satisfy $X(\pi) = 0$ is to take $A = 0$, which means that $X(x) = 0$. However in the first case we can set $k = -n^2 > 0$ for integer $n$ and then $X(x) = A \sin(nx)$ satisfies $X(\pi) = 0$ for any value of $A$. Thus $\{-n^2 : n \in \mathbb{N}\}$ are the eigenvalues, and $\sin(nx)$ is the corresponding eigenfunction. We don't need $n < 0$ because $\sin(-|n|x) = -\sin(|n|x)$.

9. Oct 30, 2014

### collectedsoul

Maybe I'm being dense but I'm not able to separate the variables, I get X''/X + Y''/Y = a. So how do I get separate equations for x and y?

10. Oct 30, 2014

### pasmith

X''/X is a function of x only. Y''/Y is a function of y only. Their sum is a constant. Therefore each of them is constant.

11. Oct 30, 2014

### collectedsoul

So X''/X=m (a cpnstant) and Y''/Y = n (another constant) and m + n = a?

12. Oct 30, 2014

### pasmith

Yes.

13. Oct 30, 2014

### collectedsoul

Ok...so do I break the equation down into 2 parts? A steady state one where d2u/dx2 + d2u/dy2 = 0 and another where it's = a*v and then add the 2 solutions? Or do I just do the latter equation?

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