Unformatted text preview: ponding solutions to (4.24)
g (t) = λc2 g (t),
for λ = 0, −1, −4, −9, . . . , −n2 , . . . . As before, we ﬁnd that the solution is
g (t) = (constant)e−n 22 ct , where c is a constant. Thus the product solutions to the homogeneous part of
the problem are
u0 (θ, t) = a0
,
2 un (θ, t) = [an cos(nθ) + bn sin(nθ)]e−n where n = 1, 2, 3, . . . .
105 22 ct , Now we apply the superposition principle—an arbitrary superposition of
these product solutions must again be a solution. Thus
u(θ, t) = 22
a0
+ Σ∞ [an cos(nθ) + bn sin(nθ)]e−n c t
n=1
2 (4.27) is a periodic solution of period 2π to the heat equation (4.23).
To ﬁnish the solution to our problem, we must impose the initial condition
u(θ, 0) = h(θ).
But setting t = 0 in (4.27) yields
a0
+ Σ∞ [an cos(nθ) + bn sin(nθ)] = h(θ),
n=1
2
so the constants a0 , a1 , . . . , b1 , . . . are just the Fourier coeﬃcients of h(θ). Thus
the solution to our initial value problem is just (4.27) in which the constants ak
and bk can be determined via the familiar formulae
ak = 1
π π h(θ) cos k...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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