# A solution of the 2d Laplace equation

Solve Laplace of u = 0 subject to the conditions:

u(x,0) = f1(x)

u(0,y) = 0

u(x,b) = 0

u(a,y) = 0

0<x<a

0<y<b

(The question attachment contains a slightly different question. The question is restated correctly in the solution attachment)

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#### Solution Preview

Problem #2

Solve subject to

Boundary conditions:

u(x,0) = f1(x)

u(0,y) = 0

u(x,b) = 0

u(a,y) = 0

0<x<a

0<y<b

Use the method of separation of variables to derive u1 and An where

u1(x,y) = ∑ Ansin(nπx/a) sinh(nπ(b-y)/a)

and

An = 2/a sinh(nπb/a) ∫ f1(x) sin(nπx/a) dx

Let's look for a solution of the form . The 2d Laplace equation is , which leads to

As usual with separation of variables, we can set both of these to a constant giving the two ODEs:

I recall (I think!) showing you previously how to ...

#### Solution Summary

Separation of variables is an commonly used technique for solving Laplace's equation and other PDEs. The solution illustrates an example of this technique comprising two pages written in Word with equations in Mathtype. Some familiarity with the technique of separation of variables is assumed (namely, that the solutions of the resulting ODEs split into 3 cases depending on the value of the constant) but the solution is otherwise explained step by step.