Differential Equations Lecture Work Solutions 295

Differential Equations Lecture Work Solutions 295 - 3...

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3. Develop frst order approximation For u yy i j using the points u ij , u +1 , u +2 nonuniFormly spaced. Let h 1 = y j +1 y j and h 2 = y j y j 1 ,then u yy i j = Au + Bu +1 + Cu 1 with A, B, C to be determined. Now take Taylor series expansions u +1 = u + h 1 u yij + h 2 1 2 u yy i j + h 3 1 6 u yyy i j + ··· u 1 = u h 2 u + h 2 2 2 u yy i j h 3 2 6 u yyy i j + So Au + +1 + 1 =( A + B + C ) u +( Bh 1 Ch 2 ) u B h 2 1 2 + C h 2 2 2 ) u yy i j B h 3 1 6 C h 3 2 6 ) u yyy i j ±··· Compare coefficients with u yy i j to get A + B + C =0 1 2 B h 2 1 2 + C h 2 2 2 =1 This system oF 3 equations can be solved For A, B, C to get From the second B = C h 2 h 1 Plugging in the third and solve For
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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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