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2D Heat equation with mixed boundary conditions

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ut = k1 uxx + k2 uyy

on a rectangle (0<x<L, 0<y<H) Subject to

u(0 , y, t) = 0 uy = (x, 0, t) = 0
u(x, y, 0) = f(x,y)
u(L, y, t) = 0 uy = (x, H, t) = 0

Left and Right sides are kept at zero temp top and bottom are insulated

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Solution Preview

The equation is:
With boundary conditions:
While the initial condition is:
We use separation of variables:
The boundary conditions become:


Substituting (1.5) into (1.1) we get:

The left hand side is a function of t while the right hand sides is s function of x and y.
Since it must be true for any both sides must be equal a constant:

The left hand side of (1.10) is a function of x while the right hand side is a function of y, therefore both sides must be equal a constant:

And by the same token we can write:

so again the equation is separated and both sides equal a different constant:
And from (1.9) we see that :

So we start with equation (1.11):
with the boundary conditions

Solution Summary

The 12-pages solution contains detailed explanation how to solve the homogeneous heat equation on a 2D rectangle with mixed boundary conditions.