AE/ME 330 Applied Computational Methods, Test 2, 05/15/2013 Name: Signature: Take-home nal is due noon, Friday May 17th. Please present your results in an organized and clear way with sucient explanation as you do in homework solutions. Include the program listings at the end as an Appendix. No consultation with other students is allowed. All submitted so- lutions and computer programs should be based on your individual work. Failure to obey this will be considered as an act of academic dishonesty. By signing above, you acknowledge that all submitted answers are prepared by you and you have not received any unauthorized help from other people or source. Good luck! Question 1 (20 points) A forward dierence approximation to the rst derivative of a function f(x) at xi is given by the formula df dxi = [a1f(xi) + a2f(xi + x) + a3f(xi + 2x) + a4f(xi + 3x) + a5f(xi + 4x] 12x where x is the mesh spacing. (a) Determine coecients a1, a2, a3, a4, and a5 so that the nite dierence approximation is 4th order accurate. You can use Taylor’s series expansions, the method of undetermined coecients or Lagrange polynomial approxima- tion for your derivation, however you should show each step clearly in your calculations. (b) Use the approximation derived in part (a) to determine the rst deriva- tive of the function f(x) = x3Sin(2x) ?? 2x at xi = ??3:0 for x = 0:1, 0:01, and 0:001. Also calculate the error for each x value by using the exact derivative of the function at the given point. Comment on the results you have obtained. Question 2 (30 points) Approximate the function given in part (b) of question 1 using a (a) natural cubic spline with 13 equally spaced data points on the interval ??3:0 x 3:0. (b) clamped cubic spline with 13 equally spaced data points on the interval ??3:0 x 3:0. For the boundary condition at x = ??3:0 approximate the 1
rst derivative with the nite dierence formula derived in question 1 with x = 0:5. For the boundary condition at x = 3:0, use the exact value…