- Problem 3 Important For Both Problem 3 And Problem 4 Below You Can Work The Problem Entirely By Hand Ie Paper Pe 1 (100.53 KiB) Viewed 121 times
need Problem 4 completed, NOT Problem 3!!
I will give a big thumbs up to a very detailed and neat
solution!
Problem 3 IMPORTANT: For both Problem 3 and Problem 4 (below), you can work the problem entirely by hand, ie., paper, pencil and calculator. At the same time, if you invest the time to self-learn the symbolic toolbox in MATLAB, so that you learn how to differentiate, integrate and solve equations symbolically, then these two problems will be much easier to solve. I recommend the latter option and that is how I solved these two problems. Given the strong form: d [(-1- x) dx] = 0, for 0 < x < 3 (x u(0) = 1, u(3) = 7 obtain the weak form which should look like: du = dx = Bu, w) = (w) = Using this weak form, let us seek to find a three-parameter approximate solution to this problem that looks like: un = po(x) + C101(x) + C202(x) + C303(x) i.e., our problem now reduces to finding the best values for C1, C2 and C3. Let me provide: po(x) = 1 + 2x 01(x) = x(3 - x) 02(x) = x2(3 - x) and 03(x) = x(3 - x) Solve for C1, C2 and c3 using the Ritz method (as discussed in class). Hint: partial answer for this problem: C1 = 254/355 Problem 4 For the same problem (as in Problem 3), and using the same three-parameter approximation for un, write the integral form I(u) = {B(u, u) – l(u) In the above, when you replace u by un, then I will become a function of C1, C2, C3. Now, solve for C1, C2, C3 by minimizing this I, i.e., solve minimize I(C1,C2, C3) 41.62,63 Recall that this is a task that you have seen previously in multivariable calculus, i.e., how to minimize a function of more than one variable (three variables here). 2