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## Homework Statement

Solve ∇

^{2}T(x, y) = 0

with boundary conditions

T(0, y) = T(L, y) = T

_{0}

T(x, L/2) = T(x, -L/2) = T

_{0}+ T

_{1}sin(πx/L)

## Homework Equations

## The Attempt at a Solution

Set T(x, y) = X(x)Y(y)

Then ∇

^{2}T(x,y) = (∂

^{2}X/∂x

^{2}) Y + (∂

^{2}Y/∂y

^{2}) X = 0

Rearrange to find two separate ODEs:

d

^{2}X/dx

^{2}= k

^{2}X

d

^{2}Y/dy

^{2}= -k

^{2}Y

Two solutions:

X(x) = Ae

^{kx}+ Be

^{-kx}

Y(x) = Ce

^{iky}+ De

^{-iky}

Thus

T(x, y) = (Ae

^{kx}+ Be

^{-kx})(Ce

^{iky}+ De

^{-iky})

Boundary conditions:

T(0, y) = (A+B)(Ce

^{iky}+ De

^{-iky}) = T

_{0}

T(L, y) = (Ae

^{kL}+ Be

^{-kL})(Ce

^{iky}+ De

^{-iky}) = T

_{0}

Equate the two and rearrange to find:

B = Ae

^{kL}

Next boundary condition:

T(x, L/2) = (Ae

^{kx}+ Be

^{-kx})(Ce

^{ikL/2}+ De

^{-ikL/2}) = T

_{0}+ T

_{1}sin(πx/L)

T(x, -L/2) = (Ae

^{kx}+ Be

^{-kx})(Ce

^{-ikL/2}+ De

^{ikL/2}) = T

_{0}+ T

_{1}sin(πx/L)

Equate the two and rearrange to find:

C=D

Thus:

T(x, y) = AC(e

^{kx}+ e

^{kL}e

^{-kx})(e

^{iky}+ e

^{-iky})

Returning to first boundary conditions:

T(0, y) = AC(1 + e

^{kL})(e

^{iky}+ e

^{-iky}) = T

_{0}

True for all y, so set y=0, then can find:

AC = T

_{0}/2(1+e

^{kL})

So now:

T(x, y) = (T

_{0}/2(1+e

^{kL}))(e

^{kx}+ e

^{kL}e

^{-kx})(e

^{iky}+ e

^{-iky})

Returning to second boundary condition:

T(x, L/2) = (T

_{0}/2(1+e

^{kL}))(e

^{kx}+ e

^{kL}e

^{-kx})(e

^{ikL/2}+ e

^{-ikL/2}) = T

_{0}+ T

_{1}sin(πx/L)

True for all x, so set x=0:

T(0, L/2) = (T

_{0}/2(1+e

^{kL}))(1 + e

^{kL})(e

^{ikL/2}+ e

^{-ikL/2}) = T

_{0}

Reduces to:

(e

^{ikL/2}+ e

^{-ikL/2}) = 2

Which solves to give:

k = 2πn/L , for n∈ℤ

Thus:

T(x, y) = (T

_{0}/2(1+e

^{2πn}))(e

^{2πnx/L}+ e

^{2πn}e

^{-2πnx/L})(e

^{i2πny/L}+ e

^{-i2πny/L})

But when I check this back for ∇

^{2}T(x, y) = 0 it doesn't work.

I think the problem is where I make the "same for all x so set x=0" assumptions..?

But if I don't make those assumptions I can't find AC or k.

It's all rather confusing..!