# 48 numerical solutions to the eigenvalue problem we

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Unformatted text preview: dθ, bk = −π 1 π π h(θ) sin kθdθ. −π Note that as t → ∞ the temperature in the circular wire approaches the constant value a0 /2. Exercises: 4.6.1. Find the function u(θ, t), deﬁned for 0 ≤ θ ≤ 2π and t ≥ 0, which satisﬁes the following conditions: ∂2u ∂u = , ∂t ∂θ2 u(θ + 2π, t) = u(θ, t), u(θ, 0) = 2 + sin θ − cos 3θ. You may assume that the nontrivial solutions to the eigenvalue problem f (θ) = λf (θ), f (θ + 2π ) = f (θ) are λ=0 and f (θ) = a0 , 2 and λ = −n2 and f (θ) = an cos nθ + bn sin nθ, for n = 1, 2, 3, . . . , where the an ’s and bn ’s are constants. 4.6.2. Find the function u(θ, t), deﬁned for 0 ≤ θ ≤ 2π and t ≥ 0, which satisﬁes the following conditions: ∂u ∂2u = , ∂t ∂θ2 u(θ + 2π, t) = u(θ, t), 106 u(θ, 0) = |θ|, for θ ∈ [−π, π ]. 4.6.3. Find the function u(θ, t), deﬁned for 0 ≤ θ ≤ 2π and t ≥ 0, which satisﬁes the following conditions: ∂2u ∂2u = , 2 ∂t ∂θ2 u(θ + 2π, t) = u(θ, t), ∂u (θ, 0) = 0. ∂t u(θ, 0) = 2 +...
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