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7.26. Calculus IV (Mandatory)

7.26.1. Justification ↑ Back to top

This course covers ordinary differential equations, an introduction to partial differential equations, and asymptotic and perturbation methods, extending the multivariable calculus foundations built in previous courses. It develops the analytical tools needed to model and solve differential equations arising in physics and engineering, and to approximate the solutions of problems that cannot be solved in closed form.

7.26.2. Generales Goals ↑ Back to top

1. Solve first-order ordinary differential equations and model physical systems with them.
2. Solve higher-order linear ordinary differential equations and systems of ODEs.
3. Classify and solve elementary partial differential equations using separation of variables and Fourier methods.
4. Construct asymptotic expansions and apply regular and singular perturbation methods to approximate solutions.

7.26.3. Contribution to Outcomes ↑ Back to top

ABET-1) An ability to identify, formulate, and solve complex engineering problems by applying principles of engineering, science, and mathematics. (Familiarity)

7.26.4. Content ↑ Back to top

7.26.4.1. First-Order Ordinary Differential Equations (ODEs) (30 hours) [Skills ABET-1] ↑ Back to top

Bibliography: (Boyce et al., 2017; Stewart, 2016)

Topics

1. First-order equations: separable, linear, and exact forms

Learning Outcomes

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

7.26.4.2. Higher-Order Differential Equations and Systems (ODEs) (20 hours) [Skills ABET-1] ↑ Back to top

Bibliography: (Boyce et al., 2017)

Topics

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

Learning Outcomes

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]

7.26.4.3. Partial Differential Equations (PDEs) (16 hours) [Skills ABET-1] ↑ Back to top

Bibliography: (Strauss, 2008; Stewart, 2016)

Topics

1. Separation of variables for heat and wave equations in engineering geometries
2. Laplace transform methods applied to initial-boundary value problems
3. Classification of PDEs: elliptic, parabolic, and hyperbolic types and their physical significance
4. Formulation and physical interpretation of boundary conditions in engineering problems
5. Green's functions and boundary value problems for advanced engineering analysis

Learning Outcomes

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]

7.26.4.4. Asymptotic Analysis and Perturbation Methods (4 hours) [Skills ABET-1] ↑ Back to top

Bibliography: (Bender and Orszag, 1999)

Topics

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

Learning Outcomes

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]

7.26.5. Bibliography ↑ Back to top

Boyce, W. E., DiPrima, R. C., and Meade, D. B. (2017). Ecuaciones Diferenciales y Problemas con Valores en la Frontera. Limusa Wiley, 11th edition.

Stewart, J. (2016). Cálculo de una variable: Trascendentes tempranas. Cengage Learning, 8th edition.

Strauss, W. A. (2008). Partial Differential Equations: An Introduction. Wiley, 2nd edition.

Bender, C. M. and Orszag, S. A. (1999). Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory. Springer.