Differential Equations Lecture Work Solutions 295

Differential Equations Lecture Work Solutions 295 - 3...

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3. Develop first order approximation for u yy i j using the points u i j , u i j +1 , u i j +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 i j + Bu i j +1 + Cu i j 1 with A, B, C to be determined. Now take Taylor series expansions u i j +1 = u i j + h 1 u y i j + h 2 1 2 u yy i j + h 3 1 6 u yyy i j + · · · u i j 1 = u i j h 2 u y i j + h 2 2 2 u yy i j h 3 2 6 u yyy i j + · · · So Au i j + Bu i j +1 + Cu i j 1 = ( A + B + C ) u i j + ( Bh 1 Ch 2 ) u y i j + ( 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 Bh 1 Ch 2 = 0 B h 2 1 2 + C h 2 2 2 = 1 This system of 3 equations can be solved for
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