# 1 the three most important linear partial

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Unformatted text preview: resented on [a, b] as a convergent sum of eigenfunctions. Suppose that f1 (x), f2 (x), . . . , fn (x), . . . are eigenfunctions corresponding to the eigenvalues λ1 , λ2 , . . . , λn , . . . . Then the general solution to the heat equation (4.28) together with the boundary conditions u(a, t) = u(b, t) = 0 is ∞ u(x, t) = cn fn (x)e−λn t , n=0 4 For further discussion, see Boyce and DiPrima, Elementary diﬀerential equations and boundary value problems , Seventh edition, Wiley, New York, 2001. 108 where the cn ’s are arbitrary constants. To determine the cn ’s in terms of the initial temperature h(x), we need a generalization of the theory of Fourier series. The key idea here is that the eigenspaces should be orthogonal with respect to an appropriate inner product. The inner product should be one which makes L like a symmetric matrix. To arrange this, the inner product that we need to use on V0 is the one deﬁned by the formula b f, g = ρ(x)σ (x)f (x)g (x)dx. a Lemma. With respect to this inner product, eigenfunctions corresponding to distinct eigenvalues are perpendicular. The proof hinges...
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