Construct the weak form of the linear equation are the

Question

Find the solution:

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  = M_o

See how to find the weak forms for a system of equations.