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# Problem 1 (25 pts) Consider an innite elastic beam ( &amp;lt; x &amp;lt; ) subjected to a concentrated force at x = 0. The displacement of the beam,

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Problem 1 (25 pts) Consider an inﬁnite elastic beam ( -∞ < x < ) subjected to a concentrated force at x = 0. The displacement of the beam, y ( x ), is governed by d 4 y dx 4 + 4 y = δ ( x ) . Use Fourier transform methods to calculate y ( x ) for the portion of the beam: 0 < x < . Hint: What assumption can you make about y and its derivatives as x → ±∞ ? Problem 2 (25 pts) Consider the nonhomogeneous boundary value problem d 2 u dx 2 + π 2 L 2 u = α + x, 0 < x < L, α is a real constant u (0) = 0 u ( L ) = 0 a) Does a unique solution exist (why or why not)? b) For which values of α does a solution exist? c) Use an eigenfunction expansion to ﬁnd the form of the solution when it does exist. Problem 3 (25 pts) Consider the singular Sturm-Liouville eigenvalue problem d dx ± x du dx ² + λx - 1 u = 0 , 0 < x < 1 , u (1) = 0 . a) Specify an appropriate boundary condition at x = 0 to ensure that the problem is self-adjoint. b) Use the Rayleigh quotient to show that there are no negative eigenvalues. c) Find two linearly independent solutions to the diﬀerential equation. d) Find the spectrum and eigenfunctions.
Problem 4 (25 pts) Consider the Sturm-Liouville eigenvalue problem d 2 u dx 2 + λu = 0 , 0 < x < 2 π with mixed boundary conditions u (0) = u (2 π ) , du dx (0) = du dx (2 π ) . a) Is the problem self-adjoint (why or why not)? b) Does each eigenvalue correspond to one linearly independent eigenfunction (why or why not)? c) Noting that the eigenvalues are real and non-negative, use an eigenfunction expansion to ﬁnd the Green’s function for the nonhomogeneous boundary value problem d 2 u dx 2 + e 2 u = f ( x ) , 0 < x < 2 π u (0) = u (2 π ) , du dx (0) = du dx (2 π ) . Problem 5 (25 pts) Suppose we want to approximate a piecewise continuous function f ( x ) on the interval - π x π by the “trigonometric polynomial”: F n ( x ) = α 0 2 + n X k =1 ( α k cos kx + β k sin kx ) . a) Determine the form of the coeﬃcients ( α 0 1 2 ,...,α n and β 1 2 ,...,β n ) that minimizes the total square deviation Z π - π [ f ( x ) - F n ( x )] 2 dx of the approximation. Hint: Exploit orthogonality and recall that at a strict local minimum, a smooth function g ( γ ) satisﬁes g 0 ( γ ) = 0 and g 00 ( γ ) > 0. b) After calculating optimal coeﬃcients for a given value of n , suppose we decide to increase n to obtain a better approximation with more terms. What modiﬁcation is needed for the previously calculated coeﬃcients?

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