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Quiz4

# Quiz4 - boundary conditions for φ x and G y(d(1 Point Find...

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MAE105 Quiz #4 (closed book, closed notes, no computers or calculators) Name:_________________________ Time: 3:35 to 3:55pm Date : May 8 , 200 7 (a) (1 Point) Use the separation of variables, u ( x , t ) = φ ( x ) G ( y ), to find two ODE’s, one defining the x-variation and the other the y-variation of the general solution of the following PDE: x 2 2 u + y 2 2 u = 0 0 < x < 1 , 0 < y < H , in such a way that the solution for φ ( x ) is periodic. (b) (1 Point) Solve the two ODE’s to find the general expressions for φ ( x ) and G ( y ). Consider the following boundary conditions: u ( x ,0) =3 sin( 5 π x ) +7 sin( 3 π x ) , u ( x , H ) = 0 , u (0, y ) = 0 , u (1, y ) = 0. (c) (1 Point) Apply the last three boundary conditions to find the corresponding

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Unformatted text preview: boundary conditions for φ ( x ) and G ( y ). (d) (1 Point) Find the eigenvalues and eigenfunctions corresponding to φ ( x ). (e) (1 Point) Write down the general inﬁnite series solution for u ( x , y ). For an extra point, apply the ﬁrst boundary condition above, and obtain the com-plete solution for u ( x , y ). Note: To receive full credit, all steps must be neatly shown. Writing down the ﬁnal results will receive no credit.- 2 -...
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Quiz4 - boundary conditions for φ x and G y(d(1 Point Find...

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