• ES Español (Latinoamérica)
• EN English

4.3. Mathematical Modeling and Simulation (MMS)

This area bridges theoretical mathematics and real-world engineering applications. It focuses on formulating, analyzing, and simulating mathematical representations of physical systems using differential equations, dynamical systems theory, and control theory.

Knowledge Area (KA)	Core Tier1	Core Tier2
4.3.1 First-Order Ordinary Differential Equations (ODEs)	1	1
4.3.2 Higher-Order Differential Equations and Systems (ODEs)	1	1
4.3.3 Partial Differential Equations (PDEs)	1	1
4.3.4 Mathematical Fluid Dynamics	1	1
4.3.5 Asymptotic Analysis and Perturbation Methods	1	1
4.3.6 Dynamical Systems and Bifurcation Theory	Elective
4.3.7 Control Theory and Optimal Control	Elective

4.3.1. MMS/First-Order Ordinary Differential Equations (ODEs)  (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top

Methods for solving and analyzing first-order ordinary differential equations, fundamental to modeling simple engineering systems.
Topics:
Core

• First-order equations: separable, linear, and exact forms

Learning Outcomes:
Core:

1. Solve linear first-order ODEs using integrating factors [Usage]

4.3.2. MMS/Higher-Order Differential Equations and Systems (ODEs)  (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top

Methods for solving and analyzing linear higher-order differential equations, systems of ODEs, and the Laplace transform, fundamental to mechanical, electrical, and thermal engineering systems.
Topics:
Core

• Linear higher-order equations with constant coefficients
• Systems of first-order linear ODEs and state-space formulation
• Laplace transform method for solving engineering ODEs with initial conditions
• Modeling of mass-spring-damper, RLC circuits, and heat conduction with ODEs

Learning Outcomes:
Core:

1. Apply the Laplace transform method to solve linear engineering ODEs with initial conditions [Usage]
2. Model a damped mechanical oscillator or RLC circuit as a second-order linear ODE [Usage]
3. Analyze the stability and transient response of engineering systems described by systems of ODEs [Assessment]
4. Formulate state-space representations for multi-variable engineering dynamic systems [Usage]

4.3.3. MMS/Partial Differential Equations (PDEs)  (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top

Equations involving multiple independent variables essential for describing heat transfer, structural vibration, wave propagation, and potential fields in engineering.
Topics:
Core

• Separation of variables for heat and wave equations in engineering geometries
• Laplace transform methods applied to initial-boundary value problems
• Classification of PDEs: elliptic, parabolic, and hyperbolic types and their physical significance
• Formulation and physical interpretation of boundary conditions in engineering problems

Non Core

• Green's functions and boundary value problems for advanced engineering analysis

Learning Outcomes:
Core:

1. Apply separation of variables to solve the 1D heat conduction and wave equations [Usage]
2. Classify a given PDE and select the appropriate analytical or numerical solution approach [Assessment]
3. Formulate boundary conditions for a heat transfer or stress analysis problem [Assessment]
4. Derive the governing PDE for steady-state heat conduction using Fourier's law [Usage]

4.3.4. MMS/Mathematical Fluid Dynamics  (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top

Continuum mechanics, Euler and Navier-Stokes equations, vorticity, potential flow, and turbulence modeling.
Topics:
Non Core

• Continuum hypothesis, material derivative, and the Reynolds transport theorem
• Euler equations for inviscid flow: conservation form, vorticity, and potential flow
• Navier-Stokes equations: derivation, exact solutions, and dimensional analysis
• Boundary layer theory: Prandtl equations and the Blasius solution
• Turbulence: Reynolds averaging, the \(k\)-\(\varepsilon\) model, and direct numerical simulation

Learning Outcomes:
NonCore:

1. Derive the Navier-Stokes equations from conservation principles and identify each term's physical meaning [Familiarity]
2. Analyze inviscid flows using the Euler equations and apply Bernoulli's principle [Usage]
3. Apply boundary layer theory to estimate drag on a flat plate and compare with the Blasius solution [Assessment]

4.3.5. MMS/Asymptotic Analysis and Perturbation Methods  (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top

Asymptotic expansions, regular and singular perturbation theory, boundary layers, and multiple scales, for engineering models with a small parameter.
Topics:
Non Core

• Asymptotic expansions and the Landau order symbols (\(O\), \(o\), \(\sim\))
• Regular perturbation theory for algebraic equations and ODEs
• Singular perturbation theory and matched asymptotic expansions (boundary layers)
• Method of multiple scales and the WKB approximation

Learning Outcomes:
NonCore:

1. Construct a regular perturbation expansion for an ODE with a small parameter [Familiarity]
2. Apply matched asymptotic expansions to solve singular perturbation problems with boundary layers [Usage]

4.3.6. MMS/Dynamical Systems and Bifurcation Theory ↑ Back to top

Long-term behavior of engineering systems modeled as autonomous differential equations, focusing on stability, limit cycles, and parameter-dependent behavior.
Topics:
Core

• Phase portraits and linearization of nonlinear engineering systems
• Stability of equilibrium points via eigenvalue analysis and Lyapunov methods
• Limit cycles and self-sustained oscillations in mechanical and electrical systems

Non Core

• Bifurcation analysis and parameter sensitivity in engineering systems

Learning Outcomes:
Core:

1. Sketch phase portraits for two-dimensional nonlinear engineering systems [Usage]
2. Classify the stability of equilibrium points for a linearized engineering system [Assessment]
3. Apply Lyapunov's direct method to assess stability of a nonlinear control system [Usage]

NonCore:

1. Identify bifurcation points and predict system behavior changes under varying engineering parameters [Assessment]

4.3.7. MMS/Control Theory and Optimal Control ↑ Back to top

Mathematical framework for designing feedback systems to regulate the dynamic behavior of engineering processes and machines.
Topics:
Core

• Feedback control, PID controllers, and closed-loop stability analysis
• Transfer functions, frequency response, and Bode diagrams
• State-space representation, controllability, and observability
• Root locus technique and frequency-domain design methods

Non Core

• Pontryagin's Maximum Principle for optimal engineering control

Learning Outcomes:
Core:

1. Design a PID controller for a given engineering plant and assess closed-loop stability [Usage]
2. Evaluate the controllability and observability of a state-space model [Assessment]
3. Construct Bode diagrams and use frequency-domain criteria for stability assessment [Usage]
4. Analyze the root locus of a feedback system to determine gain settings for desired transient response [Assessment]