Reference no: EM131069316
1. The potential energy of a simply supported beam of length l sustaining a load at the center is given by
V = (1/2)EI0∫1(d2y/dx2)dx - Py(x = l/2)
Use the Ritz method to determine the coefficients for the approximation y- = n=1Σ3an sin(nπx/l).
2. For the following boundary value problem, use the collocation method to determine a solution of the form of a second order polynomial y = a+bx+cx2.
d2y/dx2 = x,0 < x < 1; y(0) = 1; dy/dx(1) + y(1) = 0
Hint: One approach is to choose a first function that satisfies the non-homogeneous boundary conditions and the rest that just satisfy homogeneous bc's, Another approach would be to just use quadratic y = a + bx + cx, directly, and use the BC's to determine some of the coefficients.
3. Determine an approximate solution of the following BVP below using the Least Squares method.
-d/dx[k(1 + x)du/dx] = x, 0 < x <1, u(0)= u(1)= 0
Use the following series: u- = a1x(1-x)+a2((1/2)- x)(1- x)
4. Repeat problem 3 using the Galerkin method.
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