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Unformatted text preview: MTHSC 208 (Differential Equations) Dr. Matthew Macauley HW 23 Due Monday April 20th, 2009 (1) Let u ( x, t ) be the temperature of a bar of length 10, at position x and time t (in hours). Suppose that initially, the temperature increases linearly from 70 at the left endpoint to 80 at the other end. Furthermore, suppose that the temperature at the left end of the bar is held at a constant 70 degrees, and that the right end is insulated so no heat can escape. Finally, suppose that the interior of the bar is poorly insulated, so heat escapes from it, causing the temperature to decrease at a constant rate of 1 per hour. Write an initial value problem for u ( x, t ) that could model this situation. (2) Consider the following PDE: u t = 2 u x 2 , u (0 , t ) = 0 , u x ( , t ) + u ( , t ) = 0 , u ( x, 0) = h ( x ) , where is a non-negative constant. (a) Describe a physical situation that this models. Be sure to describe the impact of the initial condition, both boundary conditions and the constant . (b) What is the steady-state solution, and why?...
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