sol the question

Problem: 2 (5 points] Let us consider the following boundary value problem on the unit interval xe (0, 1) (X(x)u'(1))'=f(x) with u(0)= 1 and s(1) =0. (0.0.1) here u(x) is the unknown function satisfying the boundary conditions and describes the displacement of a bar at the position x. K(x) represents the material property of the bar. One of the physical backgrounds of the problem is the hanging bar problem frequently used in biomechanical systems. For our computation let us assume K(x) =1 and /(x) = -2 a constant external force. The exact analytic solution of such a configuration is u(x) = (x - 1)]. 1. Use the collocation method based on Chebyshev polynomials Tm(x) to find an approximation for the boundary value problem. For the approximation use the first 4 Chobyshev polynomials. 2. Compute the moments of the solution M. = [ w(x) w(x)* dx for k= 1, 2, 3, 4 by a Gauss integration based on Chebyshev polynomials of order 4. 3. Since the external force f(x) can vary induced by the physical conditions we will keep the material properties K(x) = 1 and vary the force to f(x) =-(1 +2x)/(1 - x)". The exact solution for this case is u(x) =1/\\1 - x . Approximate this solution by collocation of the first 6 Chebyshev polynomials. 4. Compute for both cases the global error $= ([w(x) (u(x)-u, (x] dx) - using your Gauss Chebyshev integration of order 4. 5/8