# Engineering Science & Applied Math (ES_APPM)

ES_APPM 311-1 Methods of Applied Mathematics (1 Unit)

Ordinary differential equations; Sturm-Liouville theory, properties of special functions, solution methods including Laplace transforms. Fourier series: eigenvalue problems and expansions in orthogonal functions. Partial differential equations: classification, separation of variables, solution by series and transform methods.

Prerequisite: GEN_ENG 205-4, GEN_ENG 206-4, or MATH 250-0.

ES_APPM 311-2 Methods of Applied Mathematics (1 Unit)

Ordinary differential equations; Sturm-Liouville theory, properties of special functions, solution methods including Laplace transforms. Fourier series: eigenvalue problems and expansions in orthogonal functions. Partial differential equations: classification, separation of variables, solution by series and transform methods.

Prerequisite: GEN_ENG 205-4, GEN_ENG 206-4, or MATH 250-0.

ES_APPM 312-0 Complex Variables (1 Unit)

Imaginary numbers and complex variables, analytic functions, calculus of complex functions, contour integration with application to transform inversion, conformal mapping.

Prerequisite: GEN_ENG 205-4, GEN_ENG 206-4, or MATH 250-0.

ES_APPM 322-0 Applied Dynamical Systems (1 Unit)

Example-oriented survey of nonlinear dynamical systems, including chaos. Combines numerical exploration of differential equations describing physical problems with analytic methods and geometric concepts. Applications to mechanical, fluid dynamical, electrical, chemical, and biological systems.

Prerequisites: ES_APPM 311-1 and ES_APPM 311-2 or equivalent or consent of instructor.

ES_APPM 345-0 Applied Linear Algebra (1 Unit)

Understanding and implentation of algorithms to calculate matrix decompositions such as eigenvalue/vector, LU, QR, and SVD decompositions. Applications include data-fitting, image analysis, and ranking algorithms.

ES_APPM 346-0 Modeling and Computation in Science & Engineering (1 Unit)

Advanced techniques for initial value problems, differential algebraic systems, bifurcations, chaos, and partial differential equations. Applications drawn from different physical areas.

Prerequisites: MATH 234-0, MATH 240-0, and MATH 250-0; or GEN_ENG 205-4 and PHYSICS 135-1, PHYSICS 135-2; or equivalent; familiarity with a programming language; or consent of instructor.

ES_APPM 370-1 Introduction to Computational Neuroscience (1 Unit)

From neurons to networks. Ion channels, Hodgkin-Huxley framework, simplified models, cable equation, synapses, spike triggered average, and optimal stimulus. Feedforward and recurrent firing rate networks. Statistical approach, Bayesian modeling. Brief introduction to numerical methods.

ES_APPM 375-1 Quantitative Biology I: Experiments, Data, Models, and Analysis (1 Unit)

High-resolution, high-throughput, and dynamic imaging and sequencing data is the substrate of modern biology. The course consists of case-studies where we learn how to computational work with, analyze, and make sense of experimental dataset using fundamental principles of mathematics, statistics, and physics. No formal course.

prerequisites. Programming in python.

ES_APPM 375-2 Quantitative Biology II: Experiments, Data, Models, and Analysis (1 Unit)

High-resolution, high-throughput, and dynamic imaging and sequencing data is the substrate of modern biology. In this course we learn how to perform experiments, and computational work with, analyze, and make sense of experimental dataset using fundamental principles of mathematics, statistics, and physics. No formal course.

prerequisites. Programming in python.

ES_APPM 395-0 Special Topics (1 Unit)

ES_APPM 398-0 Introduction to Applied Math Research (0 Unit)

This is a seminar course where ESAM faculty present their current and planned research topics in applied mathematics.

ES_APPM 401-0 Options Pricing: Theory and Applications (1 Unit)

Consideration of ordinary and elementary partial differential equations models of problems in science and engineering, arising in various areas of application.

Prerequisites: Permission of instructor and department.

ES_APPM 411-1 Differential Equations of Mathematical Physics (1 Unit)

Methods for solving linear, ordinary, and partial differential equations of mathematical physics. Green's functions, distribution theory,integral equations, transforms, potential theory, diffusion equation, wave equation, maximum principles, and variational methods.

ES_APPM 411-2 Differential Equations of Mathematical Physics (1 Unit)

Methods for solving linear, ordinary, and partial differential equations of mathematical physics. Green's functions, distribution theory,integral equations, transforms, potential theory, diffusion equation, wave equation, maximum principles, and variational methods.

ES_APPM 411-3 Differential Equations of Mathematical Physics (1 Unit)

Methods for solving linear, ordinary, and partial differential equations of mathematical physics. Green's functions, distribution theory,integral equations, transforms, potential theory, diffusion equation, wave equation, maximum principles, and variational methods.

ES_APPM 412-0 Methods of Nonlinear Analysis (1 Unit)

Methods for analyzing nonlinear problems in science and engineering. Constructive approach to bifurcation theory and stability theory, dynamical response of nonlinear systems, nonlinear oscillations and phase plane analysis, nonlinear wave propagation, and perturbation methods. Applications.

ES_APPM 420-1 Asymptotic and Perturbation Methods in Applied Mathematics (1 Unit)

Asymptotic expansions of integrals. Regular and singular perturbation methods for ordinary and partial differential equations. Boundary layer theory. Matched asymptotic expansions. Homogenization. Two-time and uniform expansions. Wave propagation and WKBJ method. Turning point theory. Nonlinear oscillations. Bifurcation and stability theory.

ES_APPM 420-2 Asymptotic and Perturbation Methods in Applied Mathematics (1 Unit)

Asymptotic expansions of integrals. Regular and singular perturbation methods for ordinary and partial differential equations. Boundary layer theory. Matched asymptotic expansions. Homogenization. Two-time and uniform expansions. Wave propagation and WKBJ method. Turning point theory. Nonlinear oscillations. Bifurcation and stability theory.

ES_APPM 420-3 Asymptotic and Perturbation Methods in Applied Mathematics (1 Unit)

Asymptotic expansions of integrals. Regular and singular perturbation methods for ordinary and partial differential equations. Boundary layer theory. Matched asymptotic expansions. Homogenization. Two-time and uniform expansions. Wave propagation and WKBJ method. Turning point theory. Nonlinear oscillations. Bifurcation and stability theory.

ES_APPM 421-1 Models in Applied Mathematics (1 Unit)

Applications to illustrate typical problems and methods of applied mathematics. Mathematical formulation of models for phenomena in science and engineering, problem solution, and interpretation of results. Examples from solid and fluid mechanics, combustion, diffusion phenomena, chemical and nuclear reactors, and biological processes.

ES_APPM 426-0 Theory of Flows With Small Inertia (1 Unit)

Asymptotic methods for flows with small inertia: flows past bodies and matching procedures. Slowly varying flows: lubrication theory and Hele-Shaw flow; swimming of microorganisms and suspension of particles.

ES_APPM 429-0 Hydrodynamic Stability Theory (1 Unit)

Mathematical theory of hydrodynamic states; energy methods, linear theories, and nonlinear bifurcation theories. Convective, centrifugal, and shear flow instabilities. Instability of unsteady flows and systems having interfaces. Physical mechanisms and results of experiments.

ES_APPM 430-0 Wave Propagation (1 Unit)

Problems of linear wave propagation; applications to acoustics, optics, electromagnetics, elasticity, and fluids. Radiation, transmission, scattering, diffraction, dispersion, layered media, wave-guides, coupled fluid solid waves, and inverse problems. Development and application of perturbation, asymptotic, numerical, and integral transform methods.

ES_APPM 440-0 Integral Equations & Applications (1 Unit)

Integral equations in various scientific theories and their relation to differential equations. Methods of solving linear problems with Hilbert-Schmidt, Cauchy, and Wiener-Hopf type kernels; applications. Nonlinear problems in bifurcation phenomena.

ES_APPM 442-0 Stochastic Differential Equations (1 Unit)

Brownian motion and Langevin's equation. Ito and Stratonovich stochastic integrals. Stochastic calculus and Ito's formula. SDEs and PDEs of Kolmogorov, Fokker-Planck, and Dynkin. Boundary conditions, exit times, exit distributions, stability. Asymptotic analysis of SDE, the Smoluchowski-Kramers approximation, and diffusion approximation to Markov chains. Applications.

ES_APPM 444-0 High Performance Scientific Computing (1 Unit)

Solving partial differential equations using high performance computing platforms. Basic C programming. Distributed computing using MPI. GPU programming using CUDA. Adaptation of algorithms for solving PDE's to different architectures.

ES_APPM 445-0 Iterative Methods for Elliptic Equations (1 Unit)

Analysis and application of numerical methods for solving elliptic equations. Stationary iterative, multigrid, conjugate gradient, GMRES methods and preconditioners.

ES_APPM 446-1 Numerical Solution of Partial Differential Equations (1 Unit)

Numerical solution of partial differential equations by finite difference methods and spectral methods. Construction of algorithms, consistency, convergence, and stability of numerical methods. Matrix iterative analysis.

ES_APPM 446-2 Numerical Solution of Partial Differential Equations (1 Unit)

Numerical solution of partial differential equations by finite difference methods and spectral methods. Construction of algorithms, consistency, convergence, and stability of numerical methods. Matrix iterative analysis.

ES_APPM 447-0 Boundary Integral Method (1 Unit)

Numerical solution of Fredholm and Volterra integral equations and of integro-differential equations. Convergence and stability of algorithms.

ES_APPM 448-0 Numerical Methods for Random Processes (1 Unit)

Analysis and implementation of numerical methods for random processes: random number generators, Monte Carlo methods, Markov chains, stochastic differential equations, and applications.

ES_APPM 449-0 Numerical Methods for Moving Interfaces (1 Unit)

methods for simulating sharp interfaces. Marker particle, level set, fast marching, volume of fluid, and phase fields methods.

ES_APPM 495-0 Selected Topics in Applied Mathematics (0.5-1 Unit)

Topics selected from research of current interest in applied mathematics.

ES_APPM 499-0 Projects (1 Unit)

Special projects to be carried out under faculty direction. Permission of instructor and department required.

ES_APPM 519-0 Responsible Conduct of Research Training (0 Unit)

ES_APPM 590-0 Research (1-4 Units)

Independent investigation of selected problems pertaining to thesis or dissertation. May be repeated for credit.