21 the reason for this terminology will become

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Unformatted text preview: n for this terminology will become clearer as we develop the topic. Using the method of weighted residuals, we construct approximate solutions by replacing u and v by simpler functions U and V and solving (1.2.2c) relative to these choices. Speci cally, we'll consider approximations of the form u(x) U (x) = v(x) V (x) = N X j =1 N X j =1 cj j (x) (1.2.3a) dj j (x): (1.2.3b) The functions j (x) and j (x), j = 1 2 : : : N , are preselected and our goal is to determine the coe cients cj , j = 1 2 : : : N , so that U is a good approximation of u. For example, we might select j (x) = j (x) = sin j x j = 1 2 ::: N to obtain approximations in the form of discrete Fourier series. In this case, every function satis es the boundary conditions (1.2.1b), which seems like a good idea. The approximation U is called a trial function and, as noted, V is called a test function. Since the di erential operator L u] is second order, we might expect u 2 C 2 (0 1). (Actually, u can be slightly less smooth, but C 2 will su ce for the present discussion.) Thus, it's natural to e...
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