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# Thus u is determined as the solution of a v u v f n 8v

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Unformatted text preview: 2 S0 : (1.2.12a) 1.2. Weighted Residual Methods 7 The substitution of (1.2.3b) with explicit form j replaced by A( j U ) = ( j f ) in (1.2.12a) again reveals the more j j = 1 2 : : : N: (1.2.12b) Finally, to make (1.2.12b) totally explicit, we eliminate U using (1.2.3a) and interchange a sum and integral to obtain N X k=1 ck A( j k) = ( j f) j = 1 2 : : : N: (1.2.12c) Thus, the coe cients ck , k = 1 2 : : : N , of the approximate solution (1.2.3a) are determined as the solution of the linear algebraic equation (1.2.12c). Di erent choices of the basis j , j = 1 2 : : : N , will make the integrals involved in the strain energy (1.2.9b) and L2 inner product (1.2.2b) easy or di cult to evaluate. They also a ect the accuracy of the approximate solution. An example using a nite element basis is presented in the next section. Problems 1. Consider the variational form (1.2.6) and select j (x) = (x ; xj ) j = 1 2 ::: N where (x) is the Dirac delta function satisfying (x) = 0 x 6= 0 Z 1 (x)dx = 1 ;1 and 0 < x1 < x2 < : : : < xN < 1: Show that this choice of test function leads to the collocation method L U ] ; f (x)jx=xj =0 j = 1 2 : : : N: Thus, the di erential equation (1.2.1) is satis ed exactly at N distinct points on (0 1). 2. The subdomain method uses p...
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