pracex1

# pracex1 - u(0,t = 0 u x(3,t = 0 for all t> 0(Hint Look...

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Math 425 Dr. DeTurck Practice Midterm 1 February 2010 1. Suppose f is a function of one variable that has a continuous second derivative. Show that for any constants a and b , the function u ( x, y ) = f ( ax + by ) is a solution of the PDE u xx u yy - u 2 xy = 0 . 2. Give an example that shows why solutions of the wave equation u tt = u xx do not necessarily satisfy the maximum principle (i.e., give an example of an explicit solution of the equation for which the maximum principle does not hold). 3. Find the function u ( x, t ) that satisfies u t = 2 u xx for ( x, t ) (0 , 3) × (0 , ), together with the initial condition u ( x, 0) = sin πx 6 + 4 sin 5 πx 6 for x [0 , 3], and the boundary conditions:
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Unformatted text preview: u (0 ,t ) = 0 u x (3 ,t ) = 0 for all t > 0. (Hint: Look for “separated” solutions.) 4. Find the closed form (similar to d’Alembert’s formula) of the solution u ( x,t ) of the initial-boundary value problem for the semi-inﬁnite string: u tt-c 2 u xx = 0 for x,t > where u ( x, 0) = f ( x ) for x > 0, and u t ( x, 0) = 0 for x > 0, and u (0 ,t ) = α ( t ) for t ≥ 0, where f and α are C 2 functions and satisfy f (0) = α (0), α (0) = 0 and α 00 (0) = c 2 f 00 (0). Verify that the solution is C 2 for all x,t > 0....
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