Differential Equations Lecture Work Solutions 299

# Differential Equations Lecture Work Solutions 299 - 1. Let...

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1. Let h 1 = y j +2 y j +1 and h 2 = y j +1 y j ,then T yij = AT ij + BT +1 + CT +2 + O (∆ y 2 ) with A, B, C to be determined. Now take Taylor series expansions T +1 = T + h 2 T + h 2 2 2 T yy i j + h 3 2 6 T yyy i j + ··· T +2 = T +( h 1 + h 2 ) T + ( h 1 + h 2 ) 2 2 T yy i j + ( h 1 + h 2 ) 3 6 T yyy i j + So AT + +1 + +2 =( A + B + C ) T Bh 2 + C ( h 1 + h 2 )) T ( B h 2 2 2 + C ( h 1 + h 2 ) 2 2 ) T yy i j B h 3 2 6 + C ( h 1 + h 2 ) 3 6 ) T yyy i j ±··· Compare coeﬃcients with T to get A + B + C =0 2 + C ( h 1 + h 2 )=1 B h 2 2 2 + C ( h 1 + h 2 ) 2 2 This system of 3 equations can be solved for A, B, C to get from the third B = C h 1 + h 2 h 2 ! 2 Plugging in the second and solve for C C h 1 + h 2 h 1 + h 2 h 2 ! 2 =1 C = 1 h 1 + h 2 h 1 + h 2 h 2 2 Thus
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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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