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ES_APPM 311-1,2 Methods of Applied Mathematics (1)(1): 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. Prerequisites: Permission of instructor.
ES_APPM 311-3 Methods of Applied Mathematics: Complex Variables (1): Imaginary numbers and complex variables, analytic functions, calculus of complex functions, contour integration with application to transform inversion, conformal mapping. ES_APPM 311-3 may be taken independently of ES_APPM 311-1,2.
ES_APPM 321 Modeling Soft Matter: Networks, Membranes, Fluctuations (1): Fundamental mathematical tools (e.g., differential geometry, variational calculus) are applied to modern concepts of soft matter structure and mechnics in various fields (e.g., biological membranes, polymers).
ES_APPM 322 Applied Dynamical Systems (1): An example-oriented survey of nonlinear dynamical systems, including chaos. Combines numerical exploration of differential equations describing physical problems with analytic methods and geometric concepts. Emphasis is on application to mechanical, fluid dynamical, electrical, chemical, and biological systems. Prerequisites: Equivalent of ES_APPM 311-1,2, or permission of instructor.
ES_APPM 346 Modeling and Computation in Science and Engineering (1): Pricing and trading of equity and index options. Elementary and advanced trading strategies illustrated through mock trades. Modeling of stock price movement. Basic concepts of stochastic differential equations and Ito calculus. Derivation of Black-Scholes equation. Solution techniques for European and American options.
ES_APPM 401 Options Pricing: theory and Applications (1): 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 411-1,2,3 Differential Equations of Mathematical Physics (1)(1)(1): 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 412-1,2,3 Methods of Nonlinear Analysis (1)(1)(1): 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 420-1,2,3 Asymptotic and Perturbation Methods in Applied Mathematics (1)(1)(1): 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 421-1,2,3 Models in Applied Mathematics (1)(1)(1): 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 424-1,2 Combustion Theory (1)(1): Derivation of combustion models from general conservation equations, by asymptotic methods, to describe deflagration and detonation phenomena. Solutions of models. Bifurcation and stability of solutions and the transition from laminar to turbulent flame propagation. Cellular and pulsating flames. Ignition and extinction.
ES_APPM 426 Theory of Flows with Small Inertia (1): 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 427 Theory of Flows with Small Viscosity (1): Asymptotic methods for flows with small viscosity: the boundary layers equations, similarity solutions, Ekman layer, unsteady boundary layers, boundary layers at free surfaces, and compressible boundary layers. Theory of separation, stability, and transition to turbulence.
ES_APPM 429-1,2 Hydrodynamic Stability Theory (1)(1): 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 430-1,2,3 Wave Propagation (1)(1)(1): 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 431 Nonlinear Wave Propagation (1): Propagation of discontinuities; application of perturbation and asymptotic methods. Shock dynamics; applications to gas dynamics and detonation. Periodic solutions; Poincare's method. Nonlinear dispersive waves; applications to water waves and nonlinear acoustics and optics. Soliton theory, inverse scattering, Korteweg-deVries equation, and sine Gordon equation.
ES_APPM 438-1,2,3 Interdisciplinary Nonlinear Dynamics (1)(1)(1): ES_APPM 438-1: Example-oriented survey of nonlinear dynamical systems, including chaos, combining numerical, analytical and geometrical approaches to differential equations. ES_APPM 438-2,3: Interdisciplinary theoretical, computational and experimental projects involving complex systems in science and engineering directed by cross-disciplinary faculty teams.
ES_APPM 440 Integral Equations and Applications (1): 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 442-1,2,3 Stochastic Differential Equations (1)(1)(1): 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 445-0 Iterative Methods for Elliptic Equations (1): Analysis and application of numerical methods for solving elliptic equations. Stationary iterative, multigrid, conjugate gradient, GMRES methods and preconditioners.
ES_APPM 446-1,2,3 Numerical Solution of Partial Differential Equations (1)(1)(1): 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 Numerical Solution of Integral Equations and the Boundary Integral Method (1): Numerical solution of Fredholm and Volterra integral equations and of integro-differential equations. Convergence and stability of algorithms.
ES_APPM 448 Numerical Methods for Random Processes (1): 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 Numerical Methods for Moving Interfaces (1): methods for simulating sharp interfaces. Marker particle, level set, fast marching, volume of fluid, and phase fields methods.
ES_APPM 495 Selected Topics in Applied Mathematics (1): Topics selected from research of current interest in applied mathematics.
ES_APPM 499 Projects (1-3) : Special projects to be carried out under faculty direction. Permission of instructor and department required.
ES_APPM 511 Seminar: Topics in Applied Mathematics (1): Advanced methods and/or applications of mathematics to such fields as mechanics and electromagnetic theory.
ES_APPM 521 Seminar in Combustion Theory (0): N/A
ES_APPM 522 Seminar in Dynamical Systems (0): N/A
ES_APPM 523 Seminar in Fluid Mechanics (0): N/A
ES_APPM 524 Seminar in Solidification (0): N/A
ES_APPM 590 Research (1-3) : Independent investigation of selected problems pertaining to thesis or dissertation. May be repeated for credit.
Related Courses
CHEM_ENG 438-1,2,3 Interdisciplinary Nonlinear Dynamics (1)(1)(1): CHEM_ENG 438-1: Example-oriented survey of nonlinear dynamical systems, including chaos, combining numerical, analytical and geometrical approaches to differential equations. CHEM_ENG 438-2,3: Interdisciplinary theoretical, computational and experimental projects involving complex systems in science and engineering directed by cross-disciplinary faculty teams.
EECS 438-1,2,3 Interdisciplinary Nonlinear Dynamics (1)(1)(1): EECS 438-1: Example-oriented survey of nonlinear dynamical systems, including chaos, combining numerical, analytical and geometrical approaches to differential equations. EECS 438-2,3: Interdisciplinary theoretical, computational and experimental projects involving complex systems in science and engineering directed by cross-disciplinary faculty teams.
MATH 438-1,2,3 Interdisciplinary Nonlinear Dynamics (1): First quarter: Example-oriented survey of nonlinear dynamical systems, including chaos, combining numerical, analytical and geometrical approaches to differential equations. Second and third quarters: Interdisciplinary theoretical, computational and experimental projects involving complex systems in science and engineering directed by cross-disciplinary faculty teams.
MECH_ENG 438-1,2,3 Interdisciplinary Nonlinear Dynamics (1)(1)(1): MECH_ENG 438-1: Example-oriented survey of nonlinear dynamical systems, including chaos, combining numerical, analytical and geometrical approaches to differential equations. MECH_ENG 438-2,3: Interdisciplinary theoretical, computational and experimental projects involving complex systems in science and engineering directed by cross-disciplinary faculty teams.
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