Sample_Final_Exam_short_answers

Sample_Final_Exam_short_answers - where = 1 2 1 sin...

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ME 303 Sample Final Exam: Short Answers Question 1 a) The ODE can be solved in terms of Maclaurin series. b) y(x=0.1) = 1.0998. c) Shooting method Question 2 a) The provided expression satisfies the PDE. Thus, this is a valid solution. b) BCs: S(0,t)=0 and S(L,t)=0. c) ICs: ) sin( ) ( 0 L x D L x )=Cx S(x, π + - 0 0 = t S t = Question 3 a) 10 4 4 1 , 1 , , 1 , 1 , - + - + + + + j i j i j i j i j i T T T T = T b) (i) j j T T , 1 , 0 = , (ii) 9 , 10 , i i T T = , (iii) 5 . 1 1 , 0 , + = i i T T , (iv) 3 4 , 9 , 10 j j T T + = c) Have 117 unknowns. Part (a) leads to 81 equations. Part (b) leads to 36 equations. d) Yes, but we need to subdivide the solution to handle the non-homogeneous BCs. Question 4 a) The PDE separates into: 0 α = - τ k dt d and 0 2 2 = - φ k dx d BCs separate into: 0 ) 0 ( 0 = - = x dx d and 0 ) 1 ( 1 = + = x dx d b) t αλ 2 - = Ce - T decreases with time. Physically, this makes sense. c) 1 λ λ 2 λ tan 2 - = has infinite number of roots. The corresponding eigenfunctions are give by: [ ] ) sin( ) cos( ) ( x x B x n n n n n λ + = d) [ ] = - + = 1 t αλ ) sin( ) cos( ) , ( 2 n n n n n n x x e D t x T
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Unformatted text preview: , where [ ] [ ] ∫ ∫ + + = 1 2 1 ) sin( ) cos( ) sin( ) cos( 250 dx x x dx x x D n n n n n n n Question 5 a) The substitution leads to the same PDE for f(r,θ) and C=10. b) Eigenvalues are n = n λ , where n= 1,3,5, … and the eigenfunctions are ) cos( θ η n A n n = c) n n n n n r D r C-+ = φ d) ∑ ∞ = + = ... 5 , 3 , 1 ) cos( 10 ) , ( n n n n r c r T θ , where ∫ ∫ = 2 / 2 2 / ) ( cos ) cos( 90 π d n d n R c n n e) The same substitution as in part (a) will lead to an eigenvalue problem for that is not a Sturm-Liouville type of problem. Thus, cannot solve the PDE with such BCs using SofV method....
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This note was uploaded on 09/16/2011 for the course ME 303 taught by Professor Serhiyyarusevych during the Spring '10 term at Waterloo.

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Sample_Final_Exam_short_answers - where = 1 2 1 sin...

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