hw5 - ary conditions y = y = y 1 = y 1 Why is there no loss...

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MAE294B/SIO203B: Methods in Applied Mechanics Winter Quarter 2010 http://maecourses.ucsd.edu/mae294b Homework V Due February 11, 2010. 1 Find approximations to the large eigenvalues of the problem y 00 + EQ ( x ) y = 0 , a 0 y 0 ( 0 )+ b 0 y ( 0 ) = 0 , a 1 y 0 ( 1 )+ b 1 y ( 1 ) = 0 , where a 0 6 = 0, a 1 6 = 0 and Q ( x ) > 0 in the interval ( 0 , 1 ) . Discuss the role of b 0 and b 1 . Compare to the exact solution for Q ( x ) = 1. 2 For large σ , ﬁnd approximations to the eigenvalues of the equation y 00 + σ 2 ( x 2 - 4 )( x 2 - 1 ) y = Ey corresponding to solutions that are bounded for large | x | . Bonus: Compute the approximate eigenvalues numerically. Bonus 2 : Compute the exact eigenvalues numerically. 3 Obtain a Liouville–Green type expansion for the fourth-order equation y ( 4 ) + a ( x ) y 00 + b ( x ) y 0 + λ 2 c ( x ) y = 0 , λ ± 1 . Find approximate eigenvalues for arbitrary a ( x ) , b ( x ) and c ( x ) > 0 on the interval ( 0 , 1 ) with bound-
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Unformatted text preview: ary conditions y ( ) = y ( ) = y ( 1 ) = y ( 1 ) . Why is there no loss of generality in not having a y 000 term in the equation? 4 The Helmholtz equation is ∇ 2 f + k 2 n 2 ( x ) f = , where k is a non-dimensional wavenumber and n ( x ) is a positive function. For large k , substitute the ansatz f = exp [ i k φ ( x )] ∞ ∑ n = A n ( x ) ( i k ) n into the equation and show that the equations satisﬁed by φ and A are | ∇φ | 2 = n 2 , ∇ · ( A 2 ∇φ ) = . The former is called the eikonal equation while the latter is called the transport equation. 1...
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