Mathematical Modeling
- Professor
- Gustavo Buscaglia.
- e-mail: gustavo.buscaglia at
icmc.usp.br
- Summary:
- The idea behind this course is to sinthetize several themes that
are studied in the applied mathematics curriculum.
- The first topic
discussed concerns the modeling of mechanical systems. The spring-mass
system and the pendulum are used to discuss ordinary differential
equations, equilibria, phase-space diagrams and stability.
Numerics can be incorporated by suggesting simulations.
With enthousiastic students the problem of controling a pendulum
can be addressed.
- The second topic concerns models of population growth.
This is an excellent topic to revisite probabilistic models,
and then obtain a dynamical system by (master equation)
tending to the continuous-time
limit. Direct simulation with binomial probability, then
Monte Carlo (Langevin)
simulations for large $N$ are encouraged. The
alternative based on Poisson's distribution (Solari and coworkers)
is also introduced and a good project is to compare all three.
Going back to the deterministic system, the equilibria
are obtained and their stability discussed (the usual predator-prey
discussion).
- The third and last topic concerns mathematical modeling
with finantial application. Models of option pricing are introduced
and discussed. Here the goal is to arrive at understanding
Black-Scholes theory, risk-free portfolios, and hedging.
This part is based on the contributions of Prof. Dorival Lećo
Pinto Jr. and some companies acting on the market are invited
to illustrate the topic (in 2011 a presentation was delivered
by Banco Itaś).
- Text: Book by Richard Haberman (SIAM).
Last update: 2011