Math Modeling: Getting Started & Getting Solutions
by Karen M. Bliss, Kathleen R. Fowler, Benjamin J. Galluzzo · Society for Industrial and Applied Mathematics (SIAM)
Free SIAM guidebook walking through the full modeling cycle: defining an open-ended problem statement, making and justifying assumptions, choosing variables, building and solving models, then assessing results. Worked examples throughout; no prerequisites beyond secondary algebra.
More resources on Mathematical Modeling
Street-Fighting Mathematics (MIT 18.098)
The art of guessing answers without proofs or exact calculation: extreme cases, dimensional analysis, successive approximation, discretization, generalization and pictorial reasoning, applied to integrals, series, geometry and differential equations. Includes Mahajan's open textbook and problem sets with solutions.
Introduction to Modeling and Simulation (MIT 3.021J)
Introduces continuum methods, atomistic and molecular simulation, and quantum mechanical modeling applied to engineering and materials problems, using web-based simulation tools rather than heavy programming. Includes 11 lecture videos, lecture slides and problem sets for hands-on practice with each method.
Mathematical Models in Biology
SIAM Classics volume deriving discrete and continuous models of population dynamics, molecular interactions, nerve conduction, and spatial pattern formation, with phase-plane and stability analysis worked out in full. Assumes calculus and ordinary differential equations; exercises carry much of the derivation.
Nonlinear Dynamics and Chaos (Cornell MAE 5790, full lecture series)
Twenty-five filmed Cornell lectures covering fixed points, bifurcations, phase planes, limit cycles, chaos, and strange attractors, with modeling examples drawn from lasers, insect outbreaks, and Josephson junctions. Follows Strogatz's textbook chapter by chapter; calculus and linear algebra assumed.
The Art of Insight in Science and Engineering: Mastering Complexity (MIT RES.6-011)
Presents approximation tools for turning complex science and engineering problems into tractable estimates: divide and conquer, abstraction, symmetry and conservation, proportional reasoning, dimensional analysis, easy cases, lumping and probabilistic reasoning. The page provides the complete open textbook, building judgment for quick quantitative modeling.
Modeling and Simulation in Python
Free online textbook with runnable Jupyter notebooks that builds physical and biological models in Python: population growth, epidemics, vaccination, pharmacokinetics, thermal systems, and projectile motion, emphasizing units, validation against data, and iterative refinement of assumptions.