Chapter 1: Mathematical Modeling

Mathematical Models

  • Models of systems have become part of our everyday lives.
  • Range from global decisions having profound impact to local decisions about daily planning based on weather predictions.
  • Together with deeper understanding of processes involved, predictive nature of models is a key strength.

Common Model Types

  • See text for descriptions
  • Empirical (Data & Curve Fitting)
  • Statistical (Data & Distributions)
  • Stochastic (Probabilistic)
  • Deterministic (Neglect randomness)
  • Simulation (Replicates phenomena)

Aims and Objectives

  • Develop modeling skills that apply to variety of problems.
  • Focus is on differential equations models for rates of change.
  • We consider models that describe growth and decay processes (Ch 2) and heating/cooling problems (Ch 9-12).
  • Concepts and methods are applicable to other scenarios.

Modeling Process

Overview of Methods

  • The process of constructing a model is carefully considered and presented in full detail.
  • Start simple, evaluate and extend.
  • Start with compartment diagrams together with balance law to develop differential equation models.

Overview of Methods

  • Assumptions and limitations are discussed together with ramifications in interpretation of results.
  • Analytical and numerical solutions are developed.
  • Validation of the models are discussed, with emphasis on strengths, limitations, and relevance of model.