PHYS 598 hw6 - 1 Flexible rod again A flexible rod is...

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Physics 498/MMA Handout 6 Fall 2007 Mathematical Methods in Physics I http://w3.physics.uiuc.edu/ m-stone5 Prof. M. Stone 2117 ESB University of Illinois 1) Flexible rod again : A flexible rod is supported near its ends by means of knife edges that constrain its position, but not its slope or curvature. It is acted on by by a force F ( x ). F(x) x=0 x=1 y x Simply supported rod. The deflection of the rod is found by solving the the boundary value problem d 4 y dx 4 = F ( x ) , y (0) = y (1) = 0 , y 00 (0) = y 00 (1) = 0 . We wish to find the Green function G ( x, y ) that facilitates the solution of this problem. a) If the differential operator and domain (boundary conditions) above is denoted by L , what is the operator and domain for L ? Is the problem self-adjoint? b) Are there any zero-modes? Does F have to satisfy any conditions for the solution to exist? c) Write down the conditions, if any, obeyed by G ( x, y ) and its derivatives x G ( x, y ), 2 xx G ( x, y ), 3 xxx G ( x, y ) at x = 0, x = y , and x = 1.
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