Overview
This lecture introduces the main components of mathematical modeling, focusing on deterministic and stochastic models, conservation laws, constituent relations, and the importance of units and non-dimensionalization in building and analyzing models.
Classes of Mathematical Models
- There are two main classes of models: deterministic and stochastic.
- Deterministic models have no randomness; outcomes are fully determined by initial conditions.
- Examples of deterministic systems use calculus, linear algebra, and ordinary differential equations.
- Stochastic models include randomness; outcomes vary due to random variables.
- Stochastic models are analyzed using probability and statistics.
Deterministic vs. Stochastic Models
- Deterministic models yield a specific, unique answer for given conditions.
- Stochastic models provide expected or probable outcomes rather than fixed results.
- Markov chains are an example of stochastic models, where questions are framed in terms of probabilities.
Essential Components in Modeling
- Conservation laws ensure certain quantities (e.g., mass, number of animals) remain constant in the model.
- Constituent relations use known equations from various fields (e.g., Newton’s second law, Hooke’s law, ideal gas law).
- Borrowing established relations from other disciplines is common in modeling.
Validating and Using Models
- After building a model, conduct quantitative and qualitative studies to check robustness and realism.
- Parameter values (constants) should be sourced from literature or determined experimentally.
- Unit analysis is vital for ensuring the self-consistency of models.
Units and Non-Dimensionalization
- Keeping consistent units throughout a model checks for errors in formulation.
- Units must match on both sides of equations; this helps detect mistakes.
- Non-dimensionalization simplifies models by rescaling variables, often removing constants to make equations easier to analyze.
Example: Non-Dimensionalizing a Differential Equation
- Introduce new scaled variables to eliminate constants in the differential equation.
- Choose scaling factors so that all coefficients in the equation are set to one.
- The method is formalized in Buckingham Pi’s theorem.
Key Terms & Definitions
- Deterministic Model — A model without randomness; future states are uniquely determined by initial conditions.
- Stochastic Model — A model that incorporates randomness; outcomes are described probabilistically.
- Conservation Law — Principle asserting certain quantities remain unchanged over time.
- Constituent Relation — A known physical or empirical equation used to relate variables in a model.
- Non-dimensionalization — The process of removing units and constants from equations by rescaling variables.
- Unit Analysis — Checking equations for consistent units as a self-check for correctness.
Action Items / Next Steps
- Review conservation laws and constituent equations relevant to your field.
- Practice unit analysis and non-dimensionalization on sample differential equations.
- Prepare for the next lecture on optimization.