Differential Equations Lecture Work Solutions 304

Differential Equations Lecture Work Solutions 304 - 1 1....

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1. Utilize Taylor series expansions about the point ( n + 1 2 ,j ) to determine the T.E. of the Crank Nicolson representation of the heat equation. u n +1 j u n j t = α 2(∆ x ) 2 h u n +1 j +1 + u n j +1 2( u n +1 j + u n j )+ u n +1 j 1 + u n j 1 i Expand about ( n +1 / 2 ) u n +1 j = u + t 2 u t + 1 2 ( t 2 ) 2 u tt + 1 6 ( t 2 ) 3 u ttt + 1 24 ( t 2 ) 4 u tttt + ··· All terms on the right are at ( n / 2 ) u n j = u t 2 u t + 1 2 ( t 2 ) 2 u tt 1 6 ( t 2 ) 3 u ttt + LHS = u t |{z} term from PDE + (∆ t ) 2 24 u ttt + u n +1 j +1 = u + t 2 u t +∆ xu x + 1 2 ( t 2 ) 2 u tt + t 2 xu tx + 1 2 (∆ x ) 2 u xx + 1 6 ( t 2 ) 3 u ttt +3( t 2 ) 2 x 6 u ttx +3 t 2 (∆ x ) 2 6 u txx + (∆ x ) 3 6 u xxx + 1 24 ( t 2 ) 4 u tttt +4( t 2 ) 3 x 24 u tttx +6( t 2 ) 2 (∆ x ) 2 24 u ttxx +4 t 2 (∆ x ) 3 24 u txxx + (∆ x ) 4 24 u xxxx + u n +1 j 1 = u + t 2 u t xu x + 1 2 ( t 2 ) 2 u tt
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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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