(3) Consider a functional of the form cb = - [ L(x,u(x), u'(x)) dx, x(a) =...

 

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(3) Consider a functional of the form cb = - [ L(x,u(x), u'(x)) dx, x(a) = A, z(b) a F[u(.)] You can assume L, u C. Lets devide the interval [a, b] using the points a = x0, x1, . . . , Xn, Xn+1 = b into n + 1 equal parts of length h = Thinking of u(x;) as ui as we approximate the functional b-a n+1. by the function F: R R: n+1 F(u,..., un)=L(xi, Ui, h i=1 Wi - Wi-) h, Ui-1 = B. Recall that the first order condition for a minimizer u = Rn of F is that (u) =0 all i = 1, .., n. dF dui (u, ..., un) dF (a) Calculate dui (b) By taking the limit as h0, show that the conditions 0 = d.F du (u) gives rise to the Euler-Lagrange equation 0 = Lz(x, u(x), u'(x)) d. U i Ui-1 Lp(x, u(x), u'(x)). Hint: you can approximate u'() as h (3) Consider a functional of the form cb = - [ L(x,u(x), u'(x)) dx, x(a) = A, z(b) a F[u(.)] You can assume L, u C. Lets devide the interval [a, b] using the points a = x0, x1, . . . , Xn, Xn+1 = b into n + 1 equal parts of length h = Thinking of u(x;) as ui as we approximate the functional b-a n+1. by the function F: R R: n+1 F(u,..., un)=L(xi, Ui, h i=1 Wi - Wi-) h, Ui-1 = B. Recall that the first order condition for a minimizer u = Rn of F is that (u) =0 all i = 1, .., n. dF dui (u, ..., un) dF (a) Calculate dui (b) By taking the limit as h0, show that the conditions 0 = d.F du (u) gives rise to the Euler-Lagrange equation 0 = Lz(x, u(x), u'(x)) d. U i Ui-1 Lp(x, u(x), u'(x)). Hint: you can approximate u'() as h