Differential Equations Lecture Work Solutions 50

Differential Equations Lecture Work Solutions 50 - scribed...

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(a) u xx + u yy =1 (b ) u xy + u yy = u (c) x 2 yu xx + y 4 u yy =4 u (d) u t + uu x =0 (e) u tt + f ( t ) u t = u xx (f) x 2 y 2 u xxx = u y 16. (a) Solve the one dimensional heat equation in a bar u t = ku xx 0 <x<L which is insulated at either end, given the initial temperature distribution u ( x, 0) = f ( x ) (b) What is the equilibrium temperature of the bar? and explain physically why your answer makes sense. 17. Solve the 1-D heat equation u t = ku xx 0 <x<L subject to the nonhomogeneous boundary conditions u (0) = 1 u x ( L )=1 with an initial temperature distribution u ( x, 0) = 0. (Hint: First solve for the equilibrium temperature distribution v ( x
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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 Laplaces equation, 2 u = 0 , x , y subject to the boundary conditions u ( x, 0) = sin x + 2 sin 2 x u ( , y ) = 0 u ( x, ) = 0 u (0 , y ) = 0 19. Repeat the above problem with u ( x, 0) = 2 x 2 + 2 x 3 x 4 50...
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This note was uploaded on 12/22/2011 for the course MAP 2302 taught by Professor Bell,d during the Fall '08 term at UNF.

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