This is the summary of a presentation given at the Joint Mathematics Meetings, January 10-13, 1996, Orlando, Florida.

At JMU we have an elementary numerical analysis class that students can take after a year of calculus and as their introduction to programming. The course covers iteration, root finding, numerical linear algebra, interpolating polynomials, numerical differentiation and integration and finishes with numerical solutions to ODE's. The students learn to program algorithms in Fortran 90. Error estimation is covered for each of these topics. Since many of the students have not had a formal course in ODE's, I give the formal definition of an ODE and its solution. I show the students some ODE's, their solutions, the phase portrait and the graph of solutions.

I introduce the students to Euler's Methods (Forward, Backward and Centered), Taylor's Methods, Runge-Kutta Methods, and Picard's Methods. I like the Taylor and Picard Methods best because the students get to use Taylor Series and numerical integration and differentiation and can determine error bounds for the numerical solutions. My favorite solver is Picard's. The reasons are

1. it is an application of the Fundamental Theorem of Integral Calculus,

2. by converting an ODE to a system with a polynomial generator Picard's method using exact integration and a strategic modification gives the Taylor Polynomial approximations to the solution to the ODE (this can be done for any ODE that appears in elementary ODE texts),

3. the integrals in Picard's method can be approximated numerically and this algorithm allows variable time steps and iteration count on the iterates at each time an approximation is determined,

4. the algorithms in (2) and (3) are easy to program in Mathematica, Matlab and Fortran 90,

5. the students are broken up in groups to program one of the solvers presented in class and then each group discusses the strengths and weaknesses of the methods they have worked on. Many of the students also enjoy learning the Picard Method and are fascinated with the fact that one can generate the Taylor Polynomials using the modified iteration.

In this talk I present how I introduce ODE's into the numerical analysis course, how Picard's Method is a natural introduction into the course and demonstrate (with examples) how to use the modified Picard method. This method is an excellent capstone for the course.

James Sochacki
Department of Mathematics
James Madison University
Harrisonburg, VA 22807
(540)568-6614
jim@math.jmu.edu