Differential Equations Lecture Work Solutions 92

Differential Equations Lecture Work Solutions 92 - 1. ut =...

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1. u t = k ( u xx + u yy ) u ( x, y, 0) = f ( x, y ) u = X ( x ) Y ( y ) T ( t ) xY ˙ T = kY X 0 T + kXTY 0 ˙ T kT = X 0 X + Y 0 Y = λ ˙ T + λkT =0 X 0 X = Y 0 Y λ = µ X 0 + µX =0 Y 0 +( λ µ ) Y =0 a. X (0) = X 0 ( L )=0 Y (0) = Y ( H )=0 X n =s in n 1 2 π L n = n 1 2 π L 2 n =1 , 2 , ··· Y nm =s in H nm µ n = H ± 2 n =1 , 2 , ··· m =1 , 2 , ··· λ nm = n 1 2 π L 2 + H ± 2 n, m =1 , 2 , ··· T nm = e λ nm kt u ( x, y, t )= X n =1 X n =1 a nm e λ nm kt sin n 1 2 ± π L x sin H y f ( x, y )= u ( x, y, 0) =
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