Unformatted text preview: nd let φ : ∂D → R
be a continuous function. Find a harmonic function u : D → R such that
for (x, y ) ∈ ∂D. u(x, y ) = φ(x, y ), Our physical intuition suggests that the Dirichlet problem will always have a
unique solution. This is proven mathematically for many choices of boundary
in more advanced texts on complex variables and partial diﬀerential equations.
The mathematical proof that a unique solution exists provides evidence that
the mathematical model we have constructed for heat ﬂow may in fact be valid.
Our goal here is to ﬁnd the explicit solutions in the case where the region D
is suﬃciently simple. Suppose, for example, that
D = {(x, y ) ∈ R 2 : 0 ≤ x ≤ a, 0 ≤ y ≤ b}.
119 Suppose, moreover that the function φ : ∂D → R vanishes on three sides of ∂D,
so that
φ(0, y ) = φ(a, y ) = φ(x, 0) = 0,
while
φ(x, b) = f (x),
where f (x) is a given continuous function which vanishes when x = 0 and x = a.
In this case, we seek a function u(x, y ), deﬁned for 0 ≤ x ≤ a and 0 ≤ y ≤ b,
such that
1. u(x, y ) satisﬁes L...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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