Unformatted text preview: scribed boundary conditions. Once v is found, write u ( x, t ) = v ( x ) + w ( x, t ) where w ( x, t ) is the transient response. Substitue this u back into the PDE to produce a new PDE for w which now has homogeneous boundary conditions. 18. Solve Laplace’s equation, ∇ 2 u = 0 , ≤ x ≤ π, ≤ y ≤ π subject to the boundary conditions u ( x, 0) = sin x + 2 sin 2 x u ( π, y ) = 0 u ( x, π ) = 0 u (0 , y ) = 0 19. Repeat the above problem with u ( x, 0) = − π 2 x 2 + 2 πx 3 − x 4 50...
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 Fall '08
 BELL,D
 Thermodynamics, Laplace, Boundary conditions, initial temperature distribution, uxxx

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