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1. Construct the weak form of the following linear equation. Are the boundary conditions "essential" or "natural"?
x2d2u/dx2 = -f(x) 0 ≤ x ≤ L
du/dx|x=0 du/dx|x=L = 0
2. Construct the weak form of the following nonlinear equation. Identify the BC's as either "essential" or "natural."
-d/dx(udu/dx) + f = 0 ≤ x ≤ 1
(u.du/dx)|z=0 = 0 u(1) = √2
3. Construct the weak forms of the following nonlinear equations representing the Euler-Bernoulli-von Kaman nonlinear theory of beams:
-d/dx{EA [du/dx + 1/2(ds/dx)2]} = f 0 ≤ x ≤ L
-d2/dx2{EI d2s/dx2) -d/dx{a.ds/dx[du/dx + 1/2(ds/dx)2]} = q
u = s = 0 at x = 0, L
(ds/dx)|x = 0 = 0, (EId2S/dx2)| x = L = Mo
See how to find the weak forms for a system of equations.
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