Unformatted text preview: h the theory of Fourier
series, allows us to solve the Dirichlet problem for Laplace’s equation in a disk.
Indeed, we can now formulate the Dirichlet problem as follows: Find u(r, θ), for
0 < r ≤ 1 and θ ∈ mathbbR, such that
1. u satisﬁes Laplace’s equation,
1∂
r ∂r r ∂u
∂r + 1 ∂2u
= 0,
r2 ∂θ2 (5.25) 2. u satisﬁes the periodicity condition u(r, θ + 2π ) = u(r, θ),
3. u is wellbehaved near r = 0,
4. u satisﬁes the boundary condition u(1, θ) = h(θ), where h(θ) is a given
wellbehaved function satisfying the periodicity condition h(θ +2π ) = h(θ).
The ﬁrst three of these conditions are homogeneous linear. To treat these
conditions via the method of separation of variables, we set
u(r, θ) = R(r)Θ(θ), where Θ(θ + 2π ) = Θ(θ). Substitution into (5.25) yields
1d
r dr R d2 Θ
= 0.
r2 dθ2 r dR
dr Θ+ r dR
dr Θ+R We multiply through by r2 ,
r d
dr d2 Θ
= 0,
dθ2 and divide by −RΘ to obtain
− rd
R dr r dR
dr = 1 d2 Θ
.
Θ dθ2 The lefthand side of this equation does not depend on θ while the righthand
side d...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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