MATH503 hw4 - ∂F ∂y 00 = const 3 Find the curve y x...

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Homeworks (Math 503) T1 : Div, grad, curl and all that , by Schey (4th edition) T2 : Calculus of variations T3 : Nonlinear dynamics and chaos , by Strogatz Note: Detail your work to receive full credit Homework#4: (due Wed Nov 3) 1. Find the minimizer of the functional J ( y ) = 1 2 Z 1 0 ( y 2 + y 0 2 ) dx given y (0) = 1. Note that an extra ‘natural’ condition is needed. 2. Consider the functional J ( y ) = Z b a F ( x, y, y 0 , y 00 ) dx with boundary conditions y ( a ) = A 0 , y ( b ) = B 0 , y 0 ( a ) = A 1 , y 0 ( b ) = B 1 (i) Present a detailed derivation of the Euler-Poisson equation in this case. (ii) If F does not depend on y , show that this equation is equivalent to ∂F ∂y 0 - d dx
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Unformatted text preview: ∂F ∂y 00 = const 3. Find the curve y ( x ) joining the two points (0 , 0) and (1 , 0) for which the functional J ( y ) = Z 1 y 00 2 dx is minimum if (i) y (0) = a and y (1) = b , (ii) no other conditions are prescribed. Hint: In the latter case, use two natural boundary conditions as suggested by the derivation of the Euler-Poisson equation in Problem 2. 4. Find the curve y ( x ) such the functional J ( y ) = Z 1-1 y dx has an extremum, subject to the conditions y (-1) = 0 , y (1) = 0 , Z 1-1 p 1 + y 2 dx = 4...
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This note was uploaded on 02/20/2011 for the course MATH 503 taught by Professor Schleiniger,g during the Fall '08 term at University of Delaware.

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