Topics:

• Modeling continuous systems using Ordinary Differential Equations (ODE).
• Application: Radiactive Decay, half-life, C14 dating. Shroud of Turin.
  www.shroud.com
  Radiocarbon Dating of the Shroud of Turin, by P.E Damon et al., Nature, Vol.337 (1989), pp. 611-615.
• Generating approximate solutions of ODEs on the computer.
• Euler's method Euler's original paper, with commentary
• How to compute e: comparison of several algorithms.
• Lecture notes by fellow student: Lecture 9-29-09 Lecture 10-1-09

Assignment:

1. Read Tutorial 1 in PHASER for computing e using the initial-value problem x' = x, x(0) = 1. Using the method in this tutorial, compute sqrt(e), 1/e. (hint: stop your computations at the appropriate time)

2. The half-life of C14 is 5730 years. Determine the value of the decay constant k (in the ODE x' = k x) using PHASER (see Week4). In 1989, fibres from the Shroud of Turin were found to contain about 92% of the level of C14 in living matter. Determine the age of the shroud using PHASER. Suppose that there was 1% error in the determination of the percentage of C14 in the sample of the shroud. What is the range of possible dates for the sample?

3. Some people do not agree with the results of the C14 dating of the Shroud. Do some research on the Web (see the link above, for example) and identify an objection or raise one of your own. Argue for or against the objection.

4. Consider the initial value problem x' = x, x(0) = 1, and compute x(1) using Euler's algorithm with various step sizes. Start with h = 0.1 and compute approximate value and determine the error. Next half the step and compute again, until you get 4 correct digits. Make a table of three columns:
   step size --- Approximate value -- error
   Can you conclude that as you decrease the step size, the error decreases proportionally?

5. Consider the continuous anolog of the logistic model using the ODE:
   x' = r x (1 - x/k)
   where x(t) is the population size at time t, r (growth rate) and k (carrying capacity) are parameters. For r = 1 and k = 1.5, several solutions of this equation with various initial conditions are displayed in the picture below. Describe what happens to the population as the initial population size varies.
   Download the following phaser Project file logisticODE.ppf by just clicking on it ( or by right-click and save it to our computer. Now load this file into Phaser). You should see the following XivsTime view:

   

   Fix the parameter r = 1 and vary the parameter k from 0.5 to 3.0. Next fix k = 1.5 and vary r from 0.5 to 4.0. Describe the results of your experiments from the biological viewpoint.

6. Consider the logistic ODE above with r = 1, k = 1.5, and x(0)=0.5 and take step size h = 0.01. By hand, compute two steps of Euler iteration by hand. Do the same again for x(0) = 1.5. Check your hand computations on PHASER.