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Mathematical Modeling

M(x)

Mathematical Modeling

Build models to understand real-world phenomena

144Modules

8Units

1

Introduction to Modeling

The art and science of mathematical modeling

19 modules

What is Mathematical Modeling?INTRO

Models Are EverywhereINTRO

The Modeling ProcessINTRO

Abstraction and SimplificationLESSON

Making AssumptionsLESSON

Making Assumptions PracticePRACTICE

Dimensional AnalysisLESSON

Dimensional Analysis ExamplesEXAMPLE

Dimensional Analysis PracticePRACTICE

Scaling and Non-dimensionalizationLESSON

Scaling ExamplesEXAMPLE

Model ValidationLESSON

Model Building WorkshopACTIVITY

Sensitivity AnalysisLESSON

Model RefinementLESSON

Weather Prediction ModelsREAL WORLD

Introduction to Modeling SummarySUMMARY

Modeling Basics ReferenceFORMULA SHEET

Introduction to Modeling QuizQUIZ

2

Discrete Models

Difference equations and discrete dynamics

21 modules

Discrete vs ContinuousINTRO

Sequences as ModelsINTRO

Difference EquationsLESSON

Difference Equations PracticePRACTICE

Linear Difference EquationsLESSON

Solving Difference EquationsLESSON

Discrete Logistic ModelLESSON

Discrete Population ModelsEXAMPLE

Financial Discrete ModelsEXAMPLE

Cobweb DiagramsLESSON

Cobweb Diagram PlotterACTIVITY

Stability of Fixed PointsLESSON

Discrete Stability PracticePRACTICE

BifurcationsLESSON

Introduction to ChaosLESSON

Markov ChainsLESSON

Stock Market ModelsREAL WORLD

Bifurcation ExplorerACTIVITY

Discrete Models SummarySUMMARY

Discrete Models ReferenceFORMULA SHEET

Discrete Models QuizQUIZ

3

Continuous Models

Differential equation models

22 modules

Continuous ModelsINTRO

Modeling Rates of ChangeINTRO

Exponential Growth and DecayLESSON

Exponential and Logistic PracticePRACTICE

Logistic GrowthLESSON

Compartmental ModelsLESSON

Epidemiological ModelsEXAMPLE

SIR Epidemic ModelLESSON

Epidemic SimulatorACTIVITY

Predator-Prey ModelsLESSON

Lotka-Volterra EquationsLESSON

Phase Plane AnalysisLESSON

Phase Portrait ToolACTIVITY

Stability of EquilibriaLESSON

Continuous Population ExamplesEXAMPLE

LinearizationLESSON

Limit CyclesLESSON

COVID-19 ModelingREAL WORLD

Systems PracticePRACTICE

Continuous Models SummarySUMMARY

Continuous Models ReferenceFORMULA SHEET

Continuous Models QuizQUIZ

4

Probabilistic Models

Stochastic modeling and simulation

20 modules

Stochastic ModelsINTRO

Randomness in ModelsINTRO

Random Variables in ModelingLESSON

Random Walk VisualizerACTIVITY

Monte Carlo MethodsLESSON

Monte Carlo ExamplesEXAMPLE

Monte Carlo SimulatorACTIVITY

Monte Carlo PracticePRACTICE

Random WalksLESSON

Brownian MotionLESSON

Stochastic ProcessesLESSON

Birth-Death ProcessesLESSON

Queuing ModelsLESSON

Queuing Model ExamplesEXAMPLE

Stochastic Epidemic ModelsLESSON

Finance and Stochastic ModelsREAL WORLD

Stochastic Models PracticePRACTICE

Probabilistic Models SummarySUMMARY

Probabilistic Models ReferenceFORMULA SHEET

Probabilistic Models QuizQUIZ

5

Optimization in Modeling

Finding optimal solutions

18 modules

Optimization in ModelsINTRO

Finding the BestINTRO

Formulating Optimization ProblemsLESSON

Logistics OptimizationREAL WORLD

Optimization SolverACTIVITY

Optimization Solving PracticePRACTICE

Calculus-Based OptimizationLESSON

Lagrange MultipliersLESSON

Linear ProgrammingLESSON

Sensitivity in OptimizationLESSON

Constrained OptimizationLESSON

Introduction to Optimal ControlLESSON

Optimal Design ExamplesEXAMPLE

Resource Allocation ExamplesEXAMPLE

Problem Formulation PracticePRACTICE

Optimization SummarySUMMARY

Optimization ReferenceFORMULA SHEET

Optimization in Modeling QuizQUIZ

6

Spatial Models

Partial differential equations and spatial dynamics

20 modules

Spatial ModelsINTRO

Why Space MattersINTRO

DiffusionLESSON

Diffusion SimulatorACTIVITY

Diffusion PracticePRACTICE

The Heat EquationLESSON

Heat Conduction ExamplesEXAMPLE

AdvectionLESSON

Reaction-Diffusion EquationsLESSON

Turing PatternsLESSON

Wave EquationLESSON

Climate ModelingREAL WORLD

Traveling WavesLESSON

Boundary ConditionsLESSON

Pattern Formation ExamplesEXAMPLE

Pattern GeneratorACTIVITY

Spatial Models PracticePRACTICE

Spatial Models SummarySUMMARY

Spatial Models ReferenceFORMULA SHEET

Spatial Models QuizQUIZ

7

Data-Driven Modeling

Fitting models to data

19 modules

Data-Driven ModelingINTRO

Building Models from DataINTRO

Least Squares FittingLESSON

Linear RegressionLESSON

Nonlinear Curve FittingLESSON

Curve Fitting ExamplesEXAMPLE

Curve Fitting ToolACTIVITY

Curve Fitting PracticePRACTICE

Parameter EstimationLESSON

Model SelectionLESSON

Model Comparison ExamplesEXAMPLE

Model Selection PracticePRACTICE

Overfitting and UnderfittingLESSON

Cross-ValidationLESSON

Uncertainty QuantificationLESSON

Machine Learning PreviewREAL WORLD

Data-Driven Modeling SummarySUMMARY

Data-Driven Modeling ReferenceFORMULA SHEET

Data-Driven Modeling QuizQUIZ

8

Course Resources

Comprehensive references and practice materials

5 modules

Problem Solving GuideSUMMARY

Mathematical Modeling Master ReferenceFORMULA SHEET

Complete Formula SheetFORMULA SHEET

Model CatalogFORMULA SHEET

Comprehensive Modeling TestQUIZ