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 nondimensional 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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 Winter '09
 MAE294B
 Numerical Analysis, Approximation, Partial Differential Equations, Applied Mechanics, approximate eigenvalues

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