Mathematics Homework Solutions
Problem
#170570

A solution of the 2d Laplace equation using separation of variables.

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

00
(The question attachment contains a slightly different question.  The question is restated correctly in the solution attachment)

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prob#2.doc
Problem #2

Boundary conditions:

u(x,0) = f1(x)

u(0,y) = g1(y)

u(x,b) = f2(x)

u(a,y) = g2(x)

0
0
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

h



h



h



h



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

**This problem consisting of Laplace’s equation on a region in the
plane together with specified boundary values is called a Dirichlet
problem.



n=1

0

a

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.

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