# 2D Heat equation with mixed boundary conditions

please show all work in detail

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

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#### 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)

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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)

So we start with equation (1.11):

(1.15)

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.