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Unformatted text preview: xpect U to also be an element of C 2(0 1). Mathematically, we regard U as belonging to a nite-dimensional function space that is a subspace of C 2 (0 1).
We express this condition by writing U 2 S N (0 1) C 2(0 1). (The restriction of these
functions to the interval 0 < x < 1 will, henceforth, be understood and we will no longer
write the (0 1).) With this interpretation, we'll call S N the trial space and regard the
preselected functions j (x), j = 1 2 : : : N , as forming a basis for S N .
Likewise, since v 2 L2, we'll regard V as belonging to another nite-dimensional
function space S N called the test space. Thus, V 2 S N L2 and j (x), j = 1 2 : : : N ,
provide a basis for S N .
Now, replacing v and u in (1.2.2c) by their approximations V and U , we have
(V L U] ; f) = 0 ^
8V 2 S N : (1.2.4a) The residual r(x) := L U ] ; f (x) (1.2.4b) 1.2. Weighted Residual Methods 5 is apparent and clari es the name \method of weighted residuals." The vanishing of the
inner product (1.2.4a) implies that the...
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- Spring '14