A comprehensive treatment of the advanced methods of applied mathematics. Designed to strengthen the mathematical abilities of graduate students and train them to think on their own. Review of elementary methods in complex analysis, ordinary differential equations, and partial differential equations. Expansions around regular and irregular singular points; asymptotic evaluation of integrals, regular perturbations; WKB method; multiple scale method; boundary-layer techniques.
Functions of a complex variable; calculus of residues. Ordinary differential equations; Bessel and Legendre functions; Sturm-Liouville theory; partial differential equations.
Introduction to computational techniques arising in aerospace engineering. Applications drawn from aerospace structures, aerodynamics, dynamics and control, and aerospace systems. Techniques include: numerical integration of systems of ordinary differential equations; finite-difference, finite-volume, and finite-element discretization of partial differential equations; numerical linear algebra; eigenvalue problems; and optimization with constraints.
Study of ordinary differential equations, including modeling of physical problems and interpretation of their solutions. Standard solution methods for single first-order equations, including graphical and numerical methods. Higher-order forced linear equations with constant coefficients. Complex numbers and exponentials. Matrix methods for first-order linear systems with constant coefficients. Non-linear autonomous systems; phase plane analysis. Fourier series; Laplace transforms.
Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time.
Introduction to nonlinear deterministic dynamical systems. Nonlinear ordinary differential equations. Planar autonomous systems. Fundamental theory: Picard iteration, contraction mapping theorem, and Bellman-Gronwall lemma. Stability of equilibria by Lyapunov's first and second methods. Feedback linearization. Application to nonlinear circuits and control systems. Alternate years. Description from course website: This course provides an introduction to nonlinear deterministic dynamical systems. Topics covered include: nonlinear ordinary differential equations; planar autonomous systems; fundamental theory: Picard iteration, contraction mapping theorem, and Bellman-Gronwall lemma; stability of equilibria by Lyapunov's first and second methods; feedback linearization; and application to nonlinear circuits and control systems.
IEEE-standard, iterative and direct linear system solution methods, eigendecomposition and model-order reduction, fast Fourier transforms, multigrid, wavelets and other multiresolution methods, matrix sparsification. Nonlinear root finding (Newton's method). Numerical interpolation and extrapolation. Quadrature.
The class will cover mathematical techniques necessary for understanding of materials science and engineering topics such as energetics, materials structure and symmetry, materials response to applied fields, mechanics and physics of solids and soft materials. The class uses examples from 3.012 to introduce mathematical concepts and materials-related problem solving skills. Topics include linear algebra and orthonormal basis, eigenvalues and eigenvectors, quadratic forms, tensor operations, symmetry operations, calculus of several variables, introduction to complex analysis, ordinary and partial differential equations, theory of distributions, fourier analysis and random walks.
Modeling a Changing World written by mathematics professor Tim Chartier and his student Nick Dovidio presents curricular material in an OSP Launcher package to motivate the need for numerically solving ordinary differential equations. The package discusses such applications as a mass-spring system and its connection to computer simulation for movies. An interactive model that simulates a two-body gravitational model of the moon and earth allows for exploring the topic of numerical error. Other models explore topics that include slope fields, numerical integration and numerical solvers for ordinary differential equations.
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