The lemma implies that these eigenfunctions will be

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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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