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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 solution is attached below in two files. the files are identical in content, only differ in format. The first is in MS Word format, while the other is in Adobe pdf format. Therefore you can choose the format that is most suitable to you.
The differential equation is:
With the boundary conditions:
We start by writing the function as a product of two independent functions:
Then, the partial differentials turn into full differentials:
The boundary conditions now become:

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

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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