# Implementing a weird-looking boundary condition (PDE/FDM)

• A
So I have this problem, taken from Kraus's heat transfer book.

So deriving the computational molecule, the conditions for (3.251a), (3.251b) is a bit of a no brainer. The issue I am having is about the boundaries for (3.251c) and (3.251d). This is actually the first time I have seen this kind of boundary condition.

How do I deal with this? My hypothesis is to integrate the equations (lol) but the constants of integration stay around... I have no idea what to do at all. Bi and γ are constants. Thanks!

## Answers and Replies

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Oh, to add.

There's an analytical expression for the solution; and I am able to graph the resulting multivariate graph. I intend to do the numerical analysis as another way of solving it. The problem is, even if I implement the 'integrating the boundary condition' part or if I use the ghost point strategy, both of it does not work - the graph from the numerical solution is way too far from the analytical expression's graph.

Chestermiller
Mentor
You're solving this by finite difference, correct?

Yes sir. Finite difference.

Chestermiller
Mentor
$$\theta (1+\Delta R, Z)-\theta (1-\Delta R, Z)=-2(\Delta R) Bi\ \theta (1,Z)$$
so
$$\frac{\partial ^2 \theta}{\partial R^2}=2\frac{(\theta (1-\Delta R, Z)-\theta (1,Z))-(\Delta R) Bi\ \theta (1,Z)}{(\Delta R)^2}$$
or
$$\frac{\partial ^2 \theta}{\partial R^2}=2\frac{(\theta (1-\Delta R, Z)-\theta (1,Z))}{(\Delta R)^2}-2\frac{Bi\ \theta(1,Z)}{\Delta R}$$
or
$$\frac{\partial ^2 \theta}{\partial R^2}=2\frac{(\theta (1-\Delta R, Z)-[1+(\Delta R) Bi]\theta (1,Z))}{(\Delta R)^2}$$

Thanks for replying sir. Actually I did it already, but I seem to be getting erroneous results. Could you have my computational molecules for each boundary checked first?

I intend to implement a solution similar to Gauss-Seidel iteration; thus I kept on factoring out the 'center' molecule.

Last edited:
Chestermiller
Mentor
Sorry. I'll help with the finite differencing, but, as far as the debugging is concerned, you're on your own.

Oh, it's alright sir.

Apparently my computational molecules for the boundaries are correct; but the numerical solution is still given different values. I'm starting to think that the problem comes from the corners - problem is what should I do with the corners (0,1) and (1,1). What boundary should I invoke? Say for the upper left corner (0,1); is it 3.251b or 3.251d?

Chestermiller
Mentor
Oh, it's alright sir.

Apparently my computational molecules for the boundaries are correct; but the numerical solution is still given different values. I'm starting to think that the problem comes from the corners - problem is what should I do with the corners (0,1) and (1,1). What boundary should I invoke? Say for the upper left corner (0,1); is it 3.251b or 3.251d?
Both