Differential Equations Lecture Work Solutions 164

Differential Equations Lecture Work Solutions 164 - c. 2 u...

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c. 2 u = S ( x, y ) u x (0 ,y )=0 u x ( L, y )=0 cos L x u y ( x, 0) = u y ( x, H )=0 cos H y Use a double Fourier cosine series u ( x, y )= X n =0 X m =0 u nm cos L x cos H y S ( x, y )= X n =0 X m =0 s nm cos L x cos H y s nm = R H 0 R L 0 S ( x, y )cos H y cos L xdxdy R H 0 R L 0 cos 2 H y cos 2 L xdxdy X n =0 X m =0 ( u nm ) " L ± 2 + H ± 2 # cos L x cos H y = S ( x, y ) Thus u nm ² L 2 + H 2 = s nm Substituing for s nm , we get the unknowns u nm u nm = R H 0 R L 0 S ( x, y )cos H y cos L xdxdy ² L 2 +
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