410HW3 - 1 Homework#3 1 Suppose F x,u,p = u 2 p 2 and F u =...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 Homework #3 1. Suppose F ( x,u,p ) = u 2 p 2 , and F ( u ) = R b a F ( x,u ( x ) ,u ( x )) dx (a) Show all extremals are parabolas. Solution The Euler equation for this functional is 2 up 2- d dx (2 u 2 p ) = 0 2 up 2- (4 up 2 + 2 u 2 p ) = 0 u ( u ) 2 + u 2 u 00 = 0 u ( ( u ) 2 + uu 00 ) = 0 = ⇒ u = 0 or ( u ) 2 + uu 00 = 0 Notice that ( u 2 ) = 2 uu and ( u 2 ) 00 = 2( u ) 2 + 2 uu 00 . Therefore, either u = 0 or ( u 2 ) 00 = 0 = ⇒ u ( x ) 2 = C 1 x + C 2 (b) Find u , the extremal satisfying the boundary conditions u ( a ) = 1 and u ( b ) = 0, where a < b . Sketch a graph of the extremal. Solution With u ( a ) = 1 ,u ( b ) = 0 1 2 = C 1 a + C 2 , 2 = C 1 b + C 2 = ⇒ C 1 = 1 a- b ,C 2 =- b a- b u ( x ) 2 = x- b a- b (c) Compute F ( u ) and F ( u ) where u ( x ) is the linear function satisfying u ( a ) = 1 and u ( b )- 0. Solution u ( x ) 2 = x- b a- b 2 u u = 1 a- b = ⇒ ( u ) 2 = 1 (2 u ( a- b )) 2 1 Thus F ( u ) = Z b a u 2 ( u ) 2 dx (1) = Z b a u 2 4 u 2 ( a- b ) 2 dx (2) = Z b a 1 4( a- b ) 2 dx (3) = 1 4( a- b ) 2 x | b a (4) =- 1 4( a- b ) (5) For the linear function, u = x- b a- b F ( u ) = Z b a u 2 ( u ) 2 dx (6) = Z b a ( x- b ) 2 ( a- b ) 2 1 ( a- b ) 2 dx...
View Full Document

This note was uploaded on 09/09/2009 for the course MATH 410 taught by Professor Staff during the Spring '08 term at Maryland.

Page1 / 5

410HW3 - 1 Homework#3 1 Suppose F x,u,p = u 2 p 2 and F u =...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online