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# 2D Heat equation with mixed boundary conditions

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Solve:

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

https://brainmass.com/math/fourier-analysis/heat-equation-mixed-boundary-conditions-625953

#### Solution Preview

The equation is:
(1.1)
With boundary conditions:
(1.2)
And
(1.3)
While the initial condition is:
(1.4)
We use separation of variables:
(1.5)
The boundary conditions become:

(1.6)
And:

(1.7)
Substituting (1.5) into (1.1) we get:

(1.8)
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:
(1.9)
Furthermore,

(1.10)
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:
(1.11)

And by the same token we can write:

(1.12)
so again the equation is separated and both sides equal a different constant:
(1.13)
And from (1.9) we see that :

(1.14)