Differential Equations Lecture Work Solutions 95

Differential Equations Lecture Work Solutions 95 - 2 ut =...

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Unformatted text preview: 2. ut = k (uxx + uyy + uzz ) ˙ T XY Z = kT (X Y Z + XY Z + XY Z ) ˙ X Y Z T = + + = −λ kT X Y Z X Y Z =− − − λ = −µ X Y Z ˙ T + λkT = 0 X + µX = 0 X (0) = X (L) = 0 Y Z =− − λ + µ = −ν Y Z Y + νY = 0 Y (0) = Y (H ) = 0 Z + (λ − µ − ν ) Z = 0 Z (0) = Z (W ) = 0 nπ L Xn = sin mπ H Ym = sin y π W Znm = sin ∞ x νm = z ∞ ∞ n=1 m=1 =1 L 0 H 0 L 0 W 0 H 0 mπ H λnm = u(x, y, z, t) = anm = nπ L µn = 2 2 nπ L n = 1, 2, · · · m = 1, 2, · · · 2 + anm e−kλnm t sin mπ H 2 + π W 2 mπ π nπ x sin y sin z L H W π f (x, y, z ) sin nLπ x sin m π y sin W z dz dy dx H W 2 nπ π x sin2 m π y sin2 W z dz dy dx 0 sin L H 95 = 1, 2, · · · ...
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