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Cartesian Coordinates Problem

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Required to solve the attached Laplace Equation problems using separation of variables.

Let €:= {(x,y) : 0 < x < 1, 0 < y < 1}.

(1) Let u(x,y) satisfy the PDE
uxx + uyy = 0,
u(0, y) = 0, u(1, y) = 0, 0 < y < 1;
u(x, 0) = 0, u(x,1) = f(x), 0 < x <1.

(2) If f(x) = x, 0 < x < ½; and f(x) = ½, ½ < x <1; then what does u(x,y) from problem (1) look like?

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The differential equation is:
(1.1)
With the boundary conditions:
(1.2)
And:
(1.3)
We start by writing the function as a product of two independent functions:
(1.4)
Then, the partial differentials turn into full differentials:
(1.5)
(1.6)
The boundary conditions now become:
(1.7)

Plugging (1.5) and (1.6) back into (1.1) we get:

(1.8)
Equation (1.9) is separated. Each side of the equation is completely independent of the other side. Since it must be true for any , the two sides must be equal the same constant:
...

Solution Summary

Cartesian Coordinates are exemplified using separation of variables.

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