Course on Mathematical Modeling

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 Leão Pinto Jr. and some companies acting on the market are invited to illustrate the topic (in 2011 a presentation was delivered by Banco Itaú).; Text: Book by Richard Haberman (SIAM).