For the following boundary value problem use the

Question

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=1Σ³a_n 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.