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### th_final_s08

Course: MATH 531, Fall 2008
School: San Diego State
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Word Count: 696

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2008 Spring I, Math 531 Take-Home Final - Due May 15, 2008 (your name), pledge that this exam is completely my own work, and that I did not take, borrow or steal work from any other person, and that I did not allow any other person to use, have, borrow or steal portions of my work. I understand that if I violate this honesty pledge, I am subject to disciplinary action pursuant to the appropriate sections of the...

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2008 Spring I, Math 531 Take-Home Final - Due May 15, 2008 (your name), pledge that this exam is completely my own work, and that I did not take, borrow or steal work from any other person, and that I did not allow any other person to use, have, borrow or steal portions of my work. I understand that if I violate this honesty pledge, I am subject to disciplinary action pursuant to the appropriate sections of the San Diego State University Policies. For all of the problems below, perform all integrations that can be readily be done. Use orthogonality to eliminate any zero coefficients. 1. Find the steady-state temperature in a cube, which satisfies: 2 u(x, y, z) = 0, 0 < x < 2, 0 < y < 2, 0 < z < 2. The cube is insulated on the faces with x = 0 and y = 2. The cube is kept at 0 C on the faces with x = 2 and z = 0 and kept at T0 when y = 0. Finally, it satisfies Newton's law of cooling on the other face (z = 2) with -k u(x, y, 2) = hu(x, y, 2). z 2. A can of beer at room temperature (20 C) is submersed in ice water (0 C). It satisfies the heat equation: u =k t 1 u r r r r + 2u , z 2 0 < r < 1, 0 < z < 4, t > 0, with the boundary conditions: u(1, z, t) = 0, and initial condition: u(r, z, 0) = 20. Find the temperature of the beer, u(r, z, t) for all t > 0. What is the t u(r, 0, t) = 0, u(r, 4, t) = 0, lim u(r, z, t), assuming it is not drunk and that there is infinite ice? 3. Consider heat conduction in a sphere given by: u k u = 2 2 , t with the boundary and initial conditions: u(a, t) = 0, u(, 0) = T0 . 0 < < a, t > 0, Solve this equation noting any other boundary conditions you might need to apply. State clearly your Sturm-Liouville problem(s) and any orthogonality relationships. (Hint: You might want to try the change of variables given by u(, t) = v(, t)/.) 4. A better model for the string problem is given by the nonhomogeneous partial differential equation: utt + 2kut = c2 uxx - g, where 2k is a small positive constant, which accounts for air resistance, and g is the acceleration due to gravity on the string. Assume that the ends of string the are fixed with u(0, t) = 0 and u(1, t) = 0. a. Find the equilibrium position for the string. b. Suppose that the initial displacement is the same as the equilibrium position and the initial velocity is 1 at each point of the string, i.e., ut (x, 0) = 1. Find u(x, t) and determine the limit of u(x, t) as t . 5. Solve the initial value problem for the nonhomogeneous heat equation: u =k t 1 u r r r r + 1 2u r2 2 + q0 e-t r sin(5), 0 < r < 1, 0<< , 2 t > 0, with the boundary conditions: u(1, , t) = 0, and initial condition: u(r, , 0) = f0 r3 sin(3). u(r, 0, t) = 0, u (r, /2, t) = 0, 6. Find the solution for Laplace's equation in a semi-infinite strip; 2 u= 2u 2u + 2 =0 x2 y x > 0, 0 < y < b, with the boundary conditions: u(0, y) = T0 cos 2y , b u(x, 0) = 0, y u(x, b) = T0 , 0, 0<x<a . x>a 7. Solve the following problem using the method of characteristics: w w +t = w, t x w(x, 0) = tanh(2x - 5). 8. a. Consider the following simplified model for the flow of traffic. We assume that there are no entrance or exit ramps on the road, so the traffic flow satisfies the equation: + c() = 0, t x - < x < , t > 0, where the velocity of the traffic depends only on and satisfies the equation: u() = umax 1 - ma...

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San Diego State - MATH - 531
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