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Unformatted text preview: Math 228A Homework 1 Due Tuesday, 10/12/10 1. Let L be the linear operator Lu = u xx , u x (0) = u x (1) = 0. (a) Find the eigenfunctions and corresponding eigenvalues of L . (b) Show that the eigenfunctions are orthogonal in the L 2 [0 , 1] inner product: ( u,v ) = integraldisplay 1 uv dx. (c) It can be shown that the eigenfunctions, j ( x ), form a complete set in L 2 [0 , 1]. This means that for any f L 2 [0 , 1], f ( x ) = j j j ( x ). Express the solution to u xx = f, u x (0) = u x (1) = 0 , (1) as a series solution of the eigenfunctions. (d) Note that equation (1) does not have a solution for all f . Express the condition for existence of a solution in terms of the eigenfunctions of L . 2. Define the functional F : X by F ( u ) = integraldisplay 1 1 2 ( u x ) 2 + fudx, where X is the space of real valued functions on [0 , 1] that have at least one continuous derivative and are zero at x = 0 and x = 1. The Frechet derivative of F at a point u is defined to be the linear operator...
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