Quiz6 - ditions, to find the eigenvalues and eigenfunctions...

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MAE105 Quiz #6 (closed book, closed notes) Name:_________________________ Time: 3:35 to 3:55pm Date: May 31, 2007 Consider the following non-homogeneous PDE: u t = 2 u x 2 + sin x , t >0 ,0 < x <1, (1) with the following non-homogeneous boundary conditions: u (0, t ) = 1, u (1 , t ) = 2. (2) (a) (1.5 Point) First consider the following related ODE: d 2 w ( x ) dx 2 =- sin x , w (0) = 1, w (1 ) = 2, (3) and find an explicit expression for w ( x ). (b) (1.5 Point) Nowset u ( x , t ) = v ( x , t ) + u E ( x ), substitute into PDE (1), and, in light of your results above ,find an ODE with suitable boudary conditions for u E ( x )such that the resulting PDE for v ( x , t )ishomogeneous, having homoge- neous boundary conditions. Neatly write down the PDE and the boundary condi- tions for v ( x , t ). (c) (1 Point) Use the separation of variable, and the homogeneous boundary con-
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Unformatted text preview: ditions, to find the eigenvalues and eigenfunctions necessary to obtain the general series solution for v ( x , t ). Write down this solution. (d) (1 Point) If the initial condition for the ORIGINAL problem [i.e., for u ( x , t ) which satisfies PDE (1) and boundary conditions (2)], is given by u ( x , 0) = x , find the initial condition for v ( x , t ). (e) (1 Point) Explain how you would use this initial condition for v ( x , t ) to find all the constants in the corresponding series solution of v ( x , t ). For an extra point, find all the constants and write down the complete solution for u ( x , t ); note that sin sin = 1 2 [ cos ( + )-cos(-)]....
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