# 21 the reason for this terminology will become

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.

Ask a homework question - tutors are online