Differential Equations Lecture Work Solutions 292

Differential Equations Lecture Work Solutions 292 - 1....

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1. Verify u xxx ij = 3 x u ij (∆ x ) 3 + O (∆ x ) Recall 3 x u =∆ x 2 x u x (∆ x ( u i +1 j u )) x ( u i +2 j u i +1 j u i +1 j + u ) x ( u i +2 j 2 u i +1 j + u ) = u i +3 j u i +2 j 2 u i +2 j +2 u i +1 j + u i +1 j u = u i +3 j 3 u i +2 j +3 u i +1 j u Now use Taylor series for each term u i +3 j = u +3∆ xu xij + (3∆ x ) 2 2 u xx i j + (3∆ x ) 3 6 u xxx i j + (3∆ x ) 4 24 u xxxx i j + ··· u i +2 j = u +2∆ xu + (2∆ x ) 2 2 u xx i j + (2∆ x ) 3 6 u xxx i j + (2∆ x ) 4 24 u xxxx i j + u i +1 j = u +∆ xu + (∆ x ) 2 2 u xx i j + (∆ x ) 3 6 u xxx i j + (∆ x ) 4 24 u xxxx i j + Combine these series we get 3 x u = u i +3 j 3 u i +2 j u i +1 j u =( 1 3+3 1) | {z } =0 u +(3∆ x 6∆ x x ) | {z } =0 u +( 9 2 x 2 12 2 x 2 + 3 2 x 2 ) | {z } =0 u xx i j 27 6 x 3 24 6 x 3 + 3 6 x 3 ) | {z } =∆ x 3 u
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This note was uploaded on 12/22/2011 for the course MAP 2302 taught by Professor Bell,d during the Fall '08 term at UNF.

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