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Midterm1

# Midterm1 - MAE105 First Midterm Exam(open book closed notes...

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MAE105 First Midterm Exam (open book; closed notes; no computers, no calculators, no cell phones) Name:_________________________ Time: 3:50 to 4:50pm Date: April 17, 2008 Problem 1: Consider the following diffusion PDE: u t 2 u x 2 = 0, t >0, 0< x < π , (1) with the following boundary conditions: u x (0, t ) = 0, u ( π , t ) = 0. (2) (a) (0.5 Point) Based on the method of "separation of variables", set u ( x , t ) = φ ( x ) G ( t ), differentiate as necessary, and substitute into the PDE (1). (b) (0.5 Point) Collect terms such that a function of only x is set equal to a function of only t , and use this fact to find two ODE’s, one for φ ( x ), and the other for G ( t ), in such a way that the solution for φ ( x ) would be periodic. (c) (0.5 Point) Write down the general solution of the ODE for φ ( x ). Is your solution periodic? (d) (0.5 Point) Write down the general solution of the ODE for G ( t ). Does your solution decay exponentially in time? (e) (1 Point) Use the boundary conditions (2) to obtain the values of φ ( x ) at x = 0 and φ ( x ) at x = π .

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Midterm1 - MAE105 First Midterm Exam(open book closed notes...

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