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Unformatted text preview: aplace’s equation
+ 2 = 0.
∂y (5.4) 2. u(x, y ) satisﬁes the homogeneous boundary conditions u(0, y ) = u(a, y ) =
u(x, 0) = 0.
3. u(x, y ) satisﬁes the nonhomogeneous boundary condition u(x, b) = f (x),
where f (x) is a given function.
Note that the Laplace equation itself and the homogeneous boundary conditions satisfy the superposition principle—this means that if u1 and u2 satisfy
these conditions, so does c1 u1 + c2 u2 , for any choice of constants c1 and c2 .
Our method for solving the Dirichlet problem consists of two steps:
Step I. We ﬁnd all of the solutions to Laplace’s equation together with the
homogeneous boundary conditions which are of the special form
u(x, y ) = X (x)Y (y ).
By the superposition principle, an arbitrary linear superposition of these solutions will still be a solution.
Step II. We ﬁnd the particular solution which satisﬁes the nonhomogeneous
boundary condition by Fourier analysis.
To carry out Step I, we substitute u(x, y ) = X (x)Y (y ) into Laplaces equation
(5.3) and obtain
X (x)Y (y ) + X (x)Y (y ) = 0.
Next we separate variables, puttin...
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This document was uploaded on 01/12/2014.
- Winter '14