# hw3(1) - H x is not di±erentiable • Show that the...

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Homework assignment 3 * Due date: Wednesday April 2, 2008. 1. (parts of # 5.8.8) Consider the BVP y ′′ + λy = 0 , y (0 , 1) , y (0) - y (0) = y (1) + y (1) = 0 . Use the Raleigh quotient to show that λ 0. Explain why λ > 0. Show that tan( λ ) = 2 λ λ - 1 . Determine the eigenvalues graphically, and estimate the large eigenvalues. 2. (adaptation of # 9.3.5) Find the Green’s function for y ′′ = f, x (0 ,L ) , y (0) = y ( L ) = 0 , in 2 ways: using the method of variation of parameters. using the method of eigenfunction expansion. 3. Show that the n th order distributional derivative of the Dirac delta function δ ( x ) is given by δ ( n ) ( x )[ f ] = ( - 1) n f ( n ) (0) , where f C c , the space of smooth functions on R with compact support. 4. Let g be a smooth function on R and H ( x ) be the Heaviside function. Show that the distributional derivative of g ( x ) H ( x ) is given by g ( x ) H ( x ) + g ( x ) δ ( x ). (Notice that formally this is the result one expects from Leibniz’s law for a product; but keep in mind that this law is not applicable here since
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Unformatted text preview: H ( x ) is not di±erentiable) • Show that the distributional integral i x −∞ g ( y ) δ ( y-x ) dy is given by g ( x ) H ( x-x ). 5. (parts of of # 9.3.6) Solve for the Green’s function directly: d 2 G dx 2 = δ ( x-x ) , x ∈ (0 ,L ) , G (0 ,x ) = dG dx ( L,x ) = 0 , and compare your result to that of problem 2 above. 6. (parts of # 9.3.11) Considering the Helmholtz equation and assuming that L is no multiple of π , solve for the Green’s function directly: d 2 G dx 2 + G = δ ( x-x ) , x ∈ (0 ,L ) , G (0 ,x ) = G ( L,x ) = 0 . Show that G is symmetric: G ( x,x ) = G ( x ,x ). (Note: use the variation of parameters method to solve for G directly, and use the rules of distributional integration; for instance, you might ²nd that the results of problem 4 come in handy.) * MAP 4341; Instructor: Patrick De Leenheer. 1...
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