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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 n π L x u y ( x, 0) = u y ( x, H ) = 0 cos m π H y Use a double Fourier cosine series u ( x, y ) = n =0 m =0 u nm cos n π L x cos m π H y S ( x, y ) = n =0 m =0 s nm cos n π L x cos m π H y s nm = H 0 L 0 S ( x, y ) cos m π H y cos n π L x dx dy H 0 L 0 cos 2 m π H y cos 2 n π L x dx dy n =0 m =0 ( u nm ) L 2 + H 2 cos L x cos H y = S ( x, y ) Thus u nm L 2 + H 2 = s nm Substituing for s nm , we get the unknowns u nm u nm = H 0 L 0 S ( x, y ) cos m π H y cos n π L x dx dy L 2 +
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