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ex1 - u satisFes the Dirichlet boundary conditions show...

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Math 425 Dr. DeTurck Midterm 1 February 8, 2007 1. Solve u x + yu y + u = 0, u (0 , y ) = y . In what domain in the plane is your solution valid? 2. Let u ( x, t ) be the temperature in a rod of length L that satisfies the partial differential equation: u t = ku xx for ( x, t ) (0 , L ) × (0 , ), where k is a positive constant, together with the initial condition u ( x, 0) = φ ( x ) for x [0 , L ], where φ satisfies φ (0) = φ ( L ) = 0 and φ ( x ) > 0 for x (0 , L ). (a) If u also satisfies the Neumann boundary conditions u x (0 , t ) = 0 , u x ( L, t ) = 0 , show that the average temperature in the rod at time t , which is given by A ( t ) = 1 L Z L 0 u ( x, t ) dx is a constant (independent of t ). (b) On the other hand, if u satisfies the Dirichlet boundary conditions u (0 , t ) = 0 , u ( L, t ) = 0 , show that it must be the case the u ( x, t ) 0 for all ( x, t ) satisfying 0 < x < L and t > 0. (c) Still under the assumption that
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Unformatted text preview: u satisFes the Dirichlet boundary conditions, show that A ( t ) is a non-increasing function of t . (Hint for (a) and (c): Use an argument similar to an energy argument). 3. (a) Solve the wave equation with friction: u xx = u tt + 2 u t for 0 < x < π and t > with the initial conditions u ( x, 0) = sin x , u t ( x, 0) = 0, and the boundary conditions u (0) = u ( π ) = 0. (Hint: Look for “separated solutions”) (b) If E ( t ) = 1 2 Z π u 2 t + u 2 x dx, show that lim t →∞ E ( t ) = 0 . (Hint: To do this, you can calculate E ( t ) explicitly). 4. ²ind as general a solution u ( x, y, z ) as you can to the third-order equation u xyz = 0...
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