Homework_3_Spring_2010

# Homework_3_Spring_2010 - T(x 0 = 100 Given that the...

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1 MAE 105: Introduction to Mathematical Physics Homework 3 Posted: April 20, 2010 , DUE: April 30, 2010 (Friday, 1 pm), 1. Consider the equation, . Solve for the eigenvalues of this equation, when: (i) h > 0, with the boundary conditions (BCs), X(0) = 0, X(L) = 0 (ii) h < 0, with the boundary conditions (BCs), X(0) = 0, X(L) = 0 (iii) h = 0, with the boundary conditions (BCs), X(0) = 0, X(L) = 0 (iv) h > 0, with the boundary conditions (BCs), X(0) = 0, Briefly explain the practical plausibility of the solutions in each case. 2. Consider the heat conduction equation, for a one-dimensional rod of length L . Solve the above equation, subject to the following boundary and initial conditions: (a) Both ends of the rod at T=0, with an initial temperature, i.e.,

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Unformatted text preview: T (x, 0) = 100 . Given that the material of the rod is Silicon (with thermal conductivity, k = 130 W/mK, density ( ) = 2330 kg/m 3 , and the specific heat ( c ) = 700 J/kgK) calculate the thermal diffusivity ( ) and the ratio of the second and the first term of the T(x,t) expansion, at x= L/2 , assuming L= 1 m, and t = 1000 seconds. What does this imply? (b) One end of the rod at T=0 and the other end at T=100, with an initial temperature, T (x, 0) = x (c) Both ends of the rod at T=0, with an initial temperature, T (x, 0) = 6 sin + 4 sin (d) Both ends of the rod insulated, with an initial temperature, T (x, 0) = sin (e) One end of the rod at T=0 and the other end at T=100, with an initial temperature, T (x, 0) = cos 2...
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