116 u dxdydz t 51 on the other hand heat ow can be

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Unformatted text preview: log x), where log denotes the natural or base e logarithm. The lemma implies that these eigenfunctions will be perpendicular with respect to the inner product ·, · , defined by eπ f, g = 1 1 f (x)g (x)dx. x Exercises: 4.7.1. Show by direct integration that if m = n, the functions fm (x) = sin(m log x) and fn (x) = sin(n log x) are perpendicular with respect to the inner product defined by (4.33). 110 (4.33) 4.7.2. Find the function u(x, t), defined for 1 ≤ x ≤ eπ and t ≥ 0, which satisfies the following initial-value problem for a heat equation with variable coefficients: ∂u ∂ =x ∂t ∂x x ∂u ∂x u(1, t) = u(eπ , t) = 0, , u(x, 0) = 3 sin(log x) + 7 sin(2 log x) − 2 sin(3 log x). 4.7.3.a. Find the solution to the eigenvalue problem for the operator L=x d dx x d dx − 3, which acts on the space V0 of functions f : [1, eπ ] → R which vanish at the endpoints of the interval [1, eπ ]. b. Find the function u(x, t), defined for 1 ≤ x ≤ eπ and t ≥ 0, which satisfies the following initial-value problem for a heat equation...
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