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### Midterm2

Course: MAE 105, Spring 2008
School: UCSD
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Word Count: 416

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Mid-term MAE105 2 (open book, closed notes) Name:_________________________ Time: 3:35 to 4:45pm Date: May 22, 2007 Problem 1 (a) (1 Point) Find a general expression x = x(t), for the characteristics of the following PDE: u u - x cos t = - u sin t . x t [Note that your expression must include a constant of integration, say, x(0) = x0 .] (b) (1 Point) In the x, t-plane, t &gt; 0, sketch a typical characteristic...

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Mid-term MAE105 2 (open book, closed notes) Name:_________________________ Time: 3:35 to 4:45pm Date: May 22, 2007 Problem 1 (a) (1 Point) Find a general expression x = x(t), for the characteristics of the following PDE: u u - x cos t = - u sin t . x t [Note that your expression must include a constant of integration, say, x(0) = x0 .] (b) (1 Point) In the x, t-plane, t > 0, sketch a typical characteristic curve. (c) (1 Point) Find the general solution of this PDE. 2 (d) (1 Point) Specialize the solution in (b) such that at t = 0, we have u(x0, 0) = u0 = (x0 + 1). Problem 2 Consider the wave equation 2 u 2 u - =0 t 2 x 2 in an infinite domain, - < x < , u(x, 0) = f (x) = (1) t > 0, with the initial conditions for for 0 < x <1 / 2 1 / 2 < x <1 , x 1- x u (x, 0) = g(x) = sin( x) , 0 < x < 1 , t with both f (x) and g(x) being zero for all other values of x. (a) (1 Point) Draw the (x, t)-plane, and below the x-axis, draw the initial conditions in two graphs, as discussed in the class and in your book. (b) (1.5 Points) In the x, t-plane, draw the characteristics that pass through the following points: [x = 0, t = 1 / 4] , [x = 1 / 2, t = 1 / 2] , [x = 1 / 4, t = 4 / 3] . (c) (1 Point) Write down the general solution of the wave equation an in infinite domain for PDE (1). -2- (d) (1.5 Points) Use your general solution and the characteristics to find the value of u(x, t) for the following indicated values of x and t: [x = 0, t = 1 / 4] , [x = 1 / 2, t = 1 / 2] , [x = 1 / 4, t = 4 / 3] . Problem 3 (a) (1 Point) Use the separation of variables, u(x,y) = (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: 2 u 2 u + =0 x 2 y2 0 < x <1 , 0< y< H , in such a way that the solution for (x) is periodic. (b) (1 Point) Solve the two ODE's to find the general expressions for (x) and G(y). Consider the following boundary conditions: u(x, 0) = 0 , u(0, y) = 0 , (x) and G(y). u(x, H) = 2 sin(5 x) + 5 sin( x) , u(1, y) = 0. (c) (1 Point) Apply the first and the last two boundary conditions to find the corresponding boundary conditions for (d) (1 Point) Find the eigenvalues and eigenfunctions corresponding to (x). (e) (1 Point) Write down the general infinite series solution for u(x, y). (f) (1 Point) Apply the last two boundary conditions above, and obtain the complete 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.
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