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Unformatted text preview: r R is a Cauchy-Euler equidimensional equation
and we can ﬁnd a nontrivial solution by setting
R(r) = rm .
Then the equation becomes
dr r dm
dr = n2 r m , and carrying out the diﬀerentiation on the left-hand side yields the characteristic
m2 − n2 = 0,
which has the solutions m = ±n. In this case, the solution is
R(r) = Arn + Br−n .
Once again, in order for this equation to be well-behaved as r → 0 we must
have B = 0, so R(r) is a constant multiple of rn , and
un (r, θ) = an rn cos nθ + bn rn sin nθ.
139 The general solution to (5.25) which is well-behaved at r = 0 and satisﬁes
the periodicity condition u(r, θ + 2π ) = u(r, θ) is therefore
∞ u(r, θ) = a0
(an rn cos nθ + bn rn sin nθ),
n=1 where a0 , a1 , . . . , b1 , . . . are constants. To determine these constants we must
apply the boundary condition:
∞ h(θ) = u(1, θ) = a0
(an cos nθ + bn sin nθ).
n=1 We conclude that the constants a0 , a1 , . . . , b1 , . . . are simply the Fourier coeﬃcients of h.
5.6.1. Solve the following boundary value problem...
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This document was uploaded on 01/12/2014.
- Winter '14