[Faller] 03FALL - Exam1Practice SOLN

# [Faller] 03FALL - Exam1Practice SOLN - ECH 159 Fall 2003...

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ECH 159, Fall 2003, Solution for Test-Midterm 1, October 23, 2003 Problem 1 Given the homogeneous 1 dimensional heat equation with a source t u = k∂ 2 x u + q in an interval 0 < x < l . (a) Solve the steady state equation for a constant source q = c . Boundary conditions: u (0) = 0, u ( l ) = 100 (b) Now we have a source proportional to the function q = cu . Apply separation of variables and derive two ordinary diﬀerential equations. Give the complete solutions of these ODE for any possible c . Do not apply boundary conditions yet. (c) Solve the space equation completely using boundary conditions u (0 ,t ) = u ( l,t ) = 0. (d) Give the full solution for c = 0 (no source) and the initial condition u ( x, 0) = 4sin πx L - 17sin 5 πx L Solution (a) Steady State: t u = 0 2 x u + c = 0 We integrate the space equation twice to yield u ( x ) = - c 2 x 2 + a 1 x + a 0 Applying the BC u (0) = 0 a 0 = 0, u ( l ) = 100 - c 2 l 2 + a 1 l = 100 leading to a 1 = 100 l + c 2 l So the ﬁnal steady state solution is u ( x ) = - c 2 2 x 2 + ( 100 l + cl 2 ) x (b) Separation of Variables u ( x,t ) = X ( x ) T ( t ) leads to X∂ t T = T∂ 2 x X + cXT t T T - c = 2 x X X = λ We can put the c on either side, even split it to c 2 on both sides. So above we

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[Faller] 03FALL - Exam1Practice SOLN - ECH 159 Fall 2003...

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