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# 28 where d dx the treatment of the last boundary

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Unformatted text preview: t of the last (boundary) term will need greater attention. For the moment, let v satisfy the same trivial boundary conditions (1.2.1b) as 0 6 Introduction u. In this case, the boundary term vanishes and (1.2.8) becomes A(v u) ; (v f ) = 0 where A(v u) = Z 1 (v pu + vqu)dx: 0 0 0 (1.2.9a) (1.2.9b) The integration by parts has eliminated second derivative terms from the formulation. Thus, solutions of (1.2.9) might have less continuity than those satisfying either (1.2.1) or (1.2.2). For this reason, they are called weak solutions in contrast to the strong solutions of (1.2.1) or (1.2.2). Weak solutions may lack the continuity to be strong solutions, but strong solutions are always weak solutions. In situations where weak and strong solutions di er, the weak solution is often the one of physical interest. Since we've added a derivative to v by the integration by parts, v must be restricted to a space where functions have more continuity than those in L2 . Having symmetry in mind, we will select functions...
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