### Math. 450-01, Fall, 2002 Assignments

This list is updated as the assignments are made.
R. Baker Kearfott

For Friday, September 6:
1. Solve y' = y, y(0) = 1, first using Euler's method, then using the second-order Taylor method. In each case, form a table with h=1/2n, n = 0 to n = 23. The table should have columns listing the error E(n) = y(1) - {computed approximation at 1}as well as the ratios E(n+1) / E(n).
2. Search the web, and in particular Netlib, to find software packages for solving initial value problems for systems of ordinary differential equations and differential-algebraic systems.  Describe when for what types of system each particular package is applicable and how each package is used.
For Monday, September 16:
Do the following problems from pp. 468--469 of Boyce and diPrima (seventh edition):
2, 4, 8, 14, 17.  I suggest plotting vector fields in the phase plane, in addition to plotting a few trajectories.  (See http://interval.louisiana.edu/courses/450/fall-2002-math-450-notes-and-notebooks.html for example notebooks for plotting both trajectories and phase planes.)
For Monday, September 22:
Do from pp. 487--489 of Boyce and diPrima (seventh edition): 2, 6, 22, 23
For Friday, October 18:
Experiment (with Mathematica and NDSolve) to recreate the figures in section 9.8 of Boyce and diPrima (seventh edition).  Explain any differences you see.  Do additional experiments to discover the nature of the solutions for different r and different initial conditions.  Explain what you see.
For Friday, November 1:
Using the derivation in Appendix A of Boyce and diPrima as a pattern, derive, from first principles, the heat equation over a two-dimensional square object with side lengths L and depth d.  (Assume that the temperature is uniform throughout the depth, and the square is insulated on the top and bottom surface, so the heat conduction is in two dimensions x and y.)  You may use references, but the derivation should be in your own words, and you should relate your derivation to the derivation in Appendix A of Boyce and diPrima.
For Friday, November 15:
Modify the optimization problem discussed in class related to the Schaeffer model (problem 20, page 86 of Boyce and diPrima), to include a term in the objective function (discussed in class)  that takes account of sport use of the resource. Try various models, with realistic parameter values (as discussed in class on October 25), to see what effect different values of the parameter have on the optimal amount of the resource to harvest. Use K = 100,000 and r = 0.5 as in the example in class.  Discuss the ease of computing the optimal value using different models, and discuss whether different models are equally easy to handle if numerical (as opposed to purely algebraic or analytic) methods can be used.
For Wednesday, November 27:
Use least squares to find the carrying capacity K and unconstrained growth rate r in the logistic equation y' = ry(1-y/K), y(0) = 20,000, with the following data.  How accurate do you think the data is?  How realistic do you think the model is?  Based on your results, what can you say about the carrying capacity of the environment and the reproduction rate of the resource?
 t y 0.0 20,000 0.5 22,865 1.0 26,080 1.5 29,680 2.0 33,675 2.5 38,685 3.0 42,925 3.5 48,200 4.0 53,900 4.5 60,000 5.0 66,485 5.5 73,300 6.0 80,395 6.5 87,700 7.0 95,140 7.5 102,630 8.0 110,105 8.5 117,461 9.0 124,620 9.5 131,525 10. 138,115