APPLIED MATHEMATICS

Detailed course offerings (Time Schedule) are available for

**AMATH 301 Beginning Scientific Computing (4) NSc**

Introduction to the use of computers to solve problems arising in the physical, biological, and engineering sciences. Application of mathematical judgment, programming architecture, and flow control in solving scientific problems. Introduction to MATLAB or Python routines for numerical programming, computation, and visualization. Prerequisite: either MATH 125, Q SCI 292, or MATH 135. Offered: AWSpS.

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**AMATH 342 Introduction to Neural Coding and Computation (3)**

Introduces computational neuroscience, grounded in neuronal and synaptic biophysics. Works through mathematical description of how neurons encode information, and how neural activity is produced dynamically. Uses and teaches MATLAB and/or Python as a programming language to implement models of neuronal dynamics and to perform coding analysis. Prerequisite: MATH 125 or MATH 135. Offered: W.

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**AMATH 351 Introduction to Differential Equations and Applications (3) NSc**

Introductory survey of ordinary differential equations; linear and nonlinear equations; Taylor series; and. Laplace transforms. Emphasizes on formulation, solution, and interpretation of results. Examples drawn from physical and biological sciences and engineering. Prerequisite: MATH 125 or MATH 135. Offered: AWSpS.

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**AMATH 352 Applied Linear Algebra and Numerical Analysis (3) NSc**

Analysis and application of numerical methods and algorithms to problems in the applied sciences and engineering. Applied linear algebra, including eigenvalue problems. Emphasis on use of conceptual methods in engineering, mathematics, and science. Extensive use of MATLAB and/or Python for programming and solution techniques. Prerequisite: MATH 126 or MATH 136. Offered: AWSpS.

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**AMATH 353 Partial Differential Equations and Waves (3) NSc**

Covers traveling waves of linear equations, dispersion relation, stability, superposition and Fourier analysis, d'Alembert solution, standing waves, vibrations and separation of variables, traveling waves of nonlinear equations, conservation laws, characteristics, breaking, shocks, and rarefaction. Prerequisite: either AMATH 351, MATH 136, or MATH 207. Offered: SpS.

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**AMATH 383 Introduction to Continuous Mathematical Modeling (3) NSc**

Introductory survey of applied mathematics with emphasis on modeling of physical and biological problems in terms of differential equations. Formulation, solution, and interpretation of the results. Prerequisite: either AMATH 351, MATH 136, or MATH 207. Offered: AWS.

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**AMATH 401 Vector Calculus and Complex Variables (4) NSc**

Emphasizes acquisition of solution techniques; illustrates ideas with specific example problems arising in science and engineering. Includes applications of vector differential calculus, complex variables; line-surface integrals; integral theorems; and Taylor and Laurent series, and contour integration. Prerequisite: either MATH 126 or MATH 136. Offered: A.

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**AMATH 402 Introduction to Dynamical Systems and Chaos (4) NSc**

Overview methods describing qualitative behavior of solutions on nonlinear differential equations. Phase space analysis of fixed pointed and periodic orbits. Bifurcation methods. Description of strange attractors and chaos. Introductions to maps. Applications: engineering, physics, chemistry, and biology. Prerequisite: either AMATH 351, MATH 136, or MATH 207. Offered: W.

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**AMATH 403 Methods for Partial Differential Equations (4) NSc**

Covers separation of variables, Fourier series and Fourier transforms, Sturm-Liouville theory and special functions, eigenfunction expansions, and Greens functions. Prerequisite: AMATH 401; and either AMATH 351, MATH 136, or MATH 207. Offered: Sp.

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**AMATH 422 Computational Modeling of Biological Systems (3) NSc**

Examines fundamental models that arise in biology and their analysis through modern scientific computing. Covers discrete and continuous-time dynamics, in deterministic and stochastic settings, with application from molecular biology to neuroscience to population dynamics; statistical analysis of experimental data; and MATLAB and/or Python programming from scratch. Prerequisite: either MATH 135, MATH 207, or AMATH 351. Offered: A.

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**AMATH 423 Mathematical Analysis in Biology and Medicine (3) NSc**

Focuses on developing and analyzing mechanistic, dynamic models of biological systems and processes, to better understand their behavior and function. Applications drawn from many branches of biology and medicine. Provides experiences in applying differential equations, difference equations, and dynamical systems theory to biological problems. Prerequisite: either AMATH 351, MATH 207, or MATH 135. Offered: W.

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**AMATH 481 Scientific Computing (5)**

Survey of numerical techniques for differential equations. Emphasis is on implementation of numerical schemes for application problems. For ordinary differential equations, initial value problems and second order boundary value problems are covered. Methods for partial differential equations include finite differences, finite elements and spectral methods. Requires use of a scientific programming language (e.g., MATLAB or Python). Prerequisite: AMATH 301; either AMATH 351, MATH 135, or MATH 207; and either AMATH 352, MATH 136, or MATH 208. Offered: A.

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**AMATH 482 Computational Methods for Data Analysis (5)**

Exploratory and objective data analysis methods applied to the physical, engineering, and biological sciences. Brief review of statistical methods and their computational implementation for studying time series analysis, spectral analysis, filtering methods, principal component analysis, orthogonal mode decomposition, and image processing and compression. Prerequisite: AMATH 301; and either AMATH 352, MATH 136, or MATH 208 Offered: W.

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**AMATH 483 High-Performance Scientific Computing (5)**

Introduction to hardware, software, and programming for large-scale scientific computing. Overview of multicore, cluster, and supercomputer architectures; procedure and object oriented languages; parallel computing paradigms and languages; graphics and visualization of large data sets; validation and verification; and scientific software development. Prerequisite: AMATH 481; and either AMATH 352, MATH 136 or MATH 208. Offered: Sp.

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**AMATH 490 Special Topics (1-5, max. 15)**

Topics of current interest in applied mathematics not covered by other undergraduate courses.

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**AMATH 498 Senior Project or Thesis (1-6, max. 6)**

Intended for Honors students and other advanced undergraduates completing a special project or senior thesis in applied mathematics. Offered: AWSpS.

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**AMATH 499 Undergraduate Reading and Research (1-6, max. 6)**

Credit/no-credit only. Offered: AWSpS.

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**AMATH 500 Special Studies in Applied Mathematics (*, max. 25)**

Lectures and discussions of topics of current interest in applied mathematics. May not be offered every quarter; content may vary from one offering to another. Prerequisite: permission of instructor.

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**AMATH 501 Vector Calculus and Complex Variables (5)**

Emphasizes acquisition of solution techniques; illustrates ideas with specific example problems arising in science and engineering. Includes applications of vector differential calculus, complex variables; line-surface integrals; integral theorems; and Taylor and Laurent series, and contour integration. Prerequisite: either a course in vector calculus or permission of instructor. Offered: A.

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**AMATH 502 Introduction to Dynamical Systems and Chaos (5)**

Overview methods describing qualitative behavior of solutions on nonlinear differential equations. Phase space analysis of fixed pointed and periodic orbits. Bifurcation methods. Description of strange attractors and chaos. Introductions to maps. Applications: engineering, physics, chemistry, and biology. Prerequisite: either a course in differential equations or permission of instructor. Offered: W.

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**AMATH 503 Methods for Partial Differential Equations (5)**

Covers separation of variables, Fourier series and Fourier transforms, Sturm-Liouville theory and special functions, eigenfunction expansions, and Greens functions. Prerequisite: either AMATH 501 and a course in differential equations, or permission of instructor. Offered: Sp.

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**AMATH 505 Introduction to Fluid Dynamics (4)**

Eulerian equations for mass-motion; Navier-Stokes equation for viscous fluids, stress-strain relations; Kelvin's theorem, vortex dynamics; potential flows, flows with high-low Reynolds numbers; boundary layers, surface gravity waves; sound waves, and linear instability theory. Prerequisite: either a course in partial differential equations or permission of instructor. Offered: jointly with ATM S 505/OCEAN 511; A.

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**AMATH 507 Calculus of Variations (5)**

Necessary and sufficient conditions for a weak and strong extremum. Legendre transformation, Hamiltonian systems. Constraints and Lagrange multipliers. Space-time problems with examples from elasticity, electromagnetics, and fluid mechanics. Sturm-Liouville problems. Approximate methods. Prerequisite: MATH 224; MATH 327; and either AMATH 351 or MATH 207. Offered: W, odd years.

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**AMATH 514 Networks and Combinatorial Optimization (3)**

Mathematical foundations of combinatorial and network optimization with an emphasis on structure and algorithms with proofs. Topics include combinatorial and geometric methods for optimization of network flows, matching, traveling salesmen problem, cuts, and stable sets on graphs. Special emphasis on connections to linear and integer programming, duality theory, total unimodularity, and matroids. Prerequisite: either MATH 208 or AMATH 352; and any additional 400-level MATH course. Offered: jointly with MATH 514.

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**AMATH 515 Optimization: Fundamentals and Applications (5)**

Maximization and minimization of functions of finitely many variables subject to constraints. Basic problem types and examples of applications; linear, convex, smooth, and nonsmooth programming. Optimality conditions. Saddlepoints and dual problems. Penalties, decomposition. Overview of computational approaches. Prerequisite: Proficiency in linear algebra and advanced calculus/analysis; recommended: Strongly recommended: probability and statistics. Desirable: optimization, e.g. Math 408, and scientific programming experience in Matlab, Julia or Python. Offered: jointly with IND E 515/MATH 515.

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**AMATH 516 Numerical Optimization (3)**

Methods of solving optimization problems in finitely many variables, with or without constraints. Steepest descent, quasi-Newton methods. Quadratic programming and complementarity. Exact penalty methods, multiplier methods. Sequential quadratic programming. Cutting planes and nonsmooth optimization. Offered: jointly with MATH 516.

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**AMATH 518 Theory of Optimal Control (3)**

Trajectories from ordinary differential equations with control variables. Controllability, optimality, maximum principle. Relaxation and existence of solutions. Techniques of nonsmooth analysis. Prerequisite: real analysis on the level of MATH 426; background in optimization corresponding to MATH 515. Offered: jointly with MATH 518.

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**AMATH 521 Special Topics in Mathematical Biology (5, max. 15)**

Special topics in mathematical biology. Prerequisite: permission of instructor. Offered: Sp.

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**AMATH 522 Computational Modeling of Biological Systems (5)**

Examines fundamental models that arise in biology and their analysis through modern scientific computing. Covers discrete and continuous-time dynamics, in deterministic and stochastic settings, with application from molecular biology to neuroscience to population dynamics; statistical analysis of experimental data; and MATLAB and/or Python programming from scratch. Prerequisite: either a course in differential equations or permission of instructor. Offered: A.

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**AMATH 523 Mathematical Analysis in Biology and Medicine (5)**

Focuses on developing and analyzing mechanistic, dynamic models of biological systems and processes, to better understand their behavior and function. Applications drawn from many branches of biology and medicine. Provides experiences in applying differential equations, difference equations, and dynamical systems theory to biological problems. Prerequisite: either courses in differential equations and statistics and probability, or permission of instructor. Offered: W.

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**AMATH 524 Mathematical Biology: Spatiotemporal Models (5)**

Examines partial differential equations for biological dynamics in space and time. Draws examples from molecular and cell biology, ecology, epidemiology, and neurobiology. Topics include reaction-diffusion equations for biochemical reactions, calcium wave propagation in excitable medium, and models for invading biological populations. Prerequisite: either a course in partial differential equations or permission of instructor. Offered: Sp.

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**AMATH 531 MATHEMATICAL THEORY OF CELLULAR DYNAMICS (3)**

Develops a coherent mathematical theory for processes inside living cells. Focuses on analyzing dynamics leading to functions of cellular components (gene regulation, signaling biochemistry, metabolic networks, cytoskeletal biomechanics, and epigenetic inheritance) using deterministic and stochastic models. Prerequisite: either courses in dynamical systems, partial differential equations, and probability, or permission of instructor.

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**AMATH 532 Mathematics of Genome Analysis and Molecular Modeling (5)**

Covers genome analysis, including bioinformatics and molecular modeling in terms of molecular dynamics. Prerequisite: either AMATH 506 or permission of instructor. Offered: A.

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**AMATH 533 Neural Control of Movement: A Computational Perspective (3)**

Systematic overview of sensorimotor function on multiple levels of analysis, with emphasis on the phenomenology amenable to computational modeling. Topics include musculoskeletal mechanics, neural networks, optimal control and Bayesian inference, learning and adaptation, internal models, and neural coding and decoding. Prerequisite: vector calculus, linear algebra, MATLAB, Python, or permission of instructor.

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**AMATH 534 Dynamics of Neurons and Networks (5)**

Covers mathematical analysis and simulation of neural systems - singles cells, networks, and populations - via tolls of dynamical systems, stochastic processes, and signal processing. Topics include single-neuron excitability and oscillations; network structure and synchrony; and stochastic and statistical dynamics of large cell populations. Prerequisite: either familiarity with dynamical systems and probability, or permission of instructor.

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**AMATH 535 Mathematical Ecology (5)**

Considers models, methods, and issues in population ecology. Topics include the effects of density dependence, delays, demographic stochasticity, and age structure on population growth; population interactions (predation, competition, and mutualism); and application of optimal control theory to the management of renewable resources. Prerequisite: either a course in differential equations or permission of instructor. Offered: Sp.

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**AMATH 536 Mathematical Modeling of Cancer (5)**

Introduces stochastic and deterministic methods for mathematical modeling of cancer evolution. Particular emphasis on branching process models of cancer initiation, progression and response to therapy, and their relationship to clinical, epidemiological and sequencing data. The course introduces both analytic and computational approaches for modeling cancer, and gets students acquainted with the current research in the field. Prerequisite: Previous experience with calculus, probability, ODEs and programming or permission of instructor.
Offered: Sp.

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**AMATH 561 Introduction to Probability and Random Processes (5)**

Introduces concepts in probability and stochastic dynamics needed for mathematical modeling. In addition to the basics of probability, includes martingales, Markov chains, and Chapman-Kolmogorov equations. Introduces concepts in measure theory from an applied mathematics perspective. Emphasis on presenting theories with examples and a variety of computational methods. Prerequisite: either undergraduate coursework in partial differential equations; and undergraduate coursework in probability and statistics, or permission of instructor. Offered: A.

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**AMATH 562 Advanced Stochastic Processes (5)**

Stochastic dynamical systems aimed at students in applied math. Introduces basic concepts in continuous stochastic processes including Brownian motion, stochastic differential equations, Levy processes, Kolmogorov forward and backward equations, and Hamilton-Jacobi-Bellman partial differential equations. Presents theories with applications from physics, biology, and finance. Prerequisite: AMATH 561 or permission of instructor; recommended: undergraduate course in probability and statistics. Offered: W.

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**AMATH 563 Inferring Structure of Complex Systems (5)**

Introduces fundamental concepts of network science and graph theory for complex dynamical systems. Merges concepts from model selection, information theory, statistical inference, neural networks, deep learning, and machine learning for building reduced order models of dynamical systems using sparse sampling of high-dimensional data. Prerequisite: AMATH 561 and AMATH 562, or instructor permission Offered: Sp.

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**AMATH 567 Applied Complex Analysis (5)**

Complex variable and associated topics. Branch cuts, series and product expansions. Contour integration, numerical implications. Harmonic functions. Complex potential (and singularities) in physical problems. Conformal mapping; applications and examples. Fourier and Laplace transforms and applications. Prerequisite: either AMATH 401or equivalent, or permission of instructor . Offered: A.

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**AMATH 568 Advanced Methods for Ordinary Differential Equations (5)**

Regular and singular points of differential equations. Asymptotic expansions for solutions of linear ordinary equations. Regular and singular perturbations. Asymptotic evaluation of integrals. Boundary layers and the WKB method. The method of multiple scales. Prerequisite: either a course in differential equations or permission of instructor. Offered: W.

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**AMATH 569 Advanced Methods for Partial Differential Equations (5)**

Analytical solution techniques for linear partial differential equations. Discussion of how these arise in science and engineering. Transform and Green's function methods. Classification of second-order equations, characteristics. Conservation laws, shocks. Prerequisite: either a course in partial differential equations or permission of instructor. Offered: Sp.

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**AMATH 570 Approximation Theory and Spectral Methods (5)**

Introduction to interpolation and approximation of data and functions by polynomials, piecewise polynomials, and trigonometric series. Covers aspects of implementation including FFTs and the chebfun software. Spectral methods for solving differential equations serve as main motivating application, along with other approximation problems. Prerequisite: AMATH 584; MATH 585; AMATH 586; programing experience in either Matlab or Python; or permission of instructor. Offered: A.

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**AMATH 571 Intelligent Control through Learning and Optimization (3)**

Design or near-optimal controllers for complex dynamical systems, using analytical techniques, machine learning, and optimization. Topics from deterministic and stochastic optimal control, reinforcement learning and dynamic programming, numerical optimization in the context of control, and robotics. Prerequisite: vector calculus; linear algebra; MATLAB. Offered: jointly with CSE 579.

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**AMATH 573 Coherent Structures, Pattern Formation and Solitons (5)**

Methods for nonlinear partial differential equations (PDEs) leading to coherent structures and patterns. Includes symmetries, conservations laws, stability Hamiltonian and variation methods of PDEs; interactions of structures such as waves or solitons; Lax pairs and inverse scattering; and Painleve analysis. Prerequisite: either a course in partial differential equations or permission of instructor. Offered: A, odd years.

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**AMATH 574 Conservation Laws and Finite Volume Methods (5)**

Theory of linear and nonlinear hyperbolic conservation laws modeling wave propagation in gases, fluids, and solids. Shock and rarefaction waves. Finite volume methods for numerical approximation of solutions; Godunov's method and high-resolution TVD methods. Stability, convergence, and entropy conditions. Prerequisite: either AMATH 586 or permission of instructor. Offered: W.

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**AMATH 575 Dynamical Systems (5)**

Overview of ways in which complex dynamics arise in nonlinear dynamical systems. Topics include bifurcation theory, universality, Poincare maps, routes to chaos, horseshoe maps, Hamiltonian chaos, fractal dimensions, Liapunov exponents, and the analysis of time series. Examples from biology, mechanics, and other fields. Prerequisite: either AMATH 502 or permission of instructor. Offered: Sp, odd years.

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**AMATH 581 Scientific Computing (5)**

Survey of numerical techniques for differential equations. Emphasis is on implementation of numerical schemes for application problems. For ordinary differential equations, initial value problems and second order boundary value problems are covered. Methods for partial differential equations include finite differences, finite elements and spectral methods. Requires use of a scientific programming language (e.g., MATLAB or Python). Prerequisite: either a course in numerical analysis or permission of instructor. Offered: A.

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**AMATH 582 Computational Methods for Data Analysis (5)**

Exploratory and objective data analysis methods applied to the physical, engineering, and biological sciences. Brief review of statistical methods and their computational implementation for studying time series analysis, spectral analysis, filtering methods, principal component analysis, orthogonal mode decomposition, and image processing and compression. Prerequisite: either familiarity with a scientific programming language and college-level coursework in linear algebra, or permission of instructor. Offered: W.

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**AMATH 583 High-Performance Scientific Computing (5)**

Introduction to hardware, software, and programming for large-scale scientific computing. Overview of multicore, cluster, and supercomputer architectures; procedure and object oriented languages; parallel computing paradigms and languages; graphics and visualization of large data sets; validation and verification; and scientific software development. Prerequisite: AMATH 581, or permission of instructor. Offered: Sp.

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**AMATH 584 Numerical Linear Algebra (5)**

Singular value decomposition. QR factorization and linear least squares problems. Conditioning of problems and stability of algorithms. LU (lower-upper) factorization. Eigenvalue problems. Iterative methods for solving linear systems of equations. Prerequisite: either a course in linear algebra or permission of instructor. Offered: jointly with MATH 584; A.

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**AMATH 585 Numerical Analysis (5)**

Iterative methods for solving nonlinear systems of equations. Numerical approximation and interpolation. Fast Fourier transform. Operator spectrum approximation. Numerical differentiation and integration. Numerical methods for ordinary differential equations. Prerequisite: either AMATH 352 (or equivalent), AMATH 481 (or equivalent), AMATH 581, AMATH 584/MATH 584, or permission of instructor; recommended: AMATH 584/MATH 584. Offered: jointly with MATH 585; W.

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**AMATH 586 Numerical Methods for Partial Differential Equations (5)**

Method of lines discretization. Initial and boundary value problems, including finite difference methods and spectral methods. Elliptic, parabolic, hyperbolic and dispersive equations. Stability, accuracy, and convergence theory. Prerequisite: either AMATH 352 (or equivalent), AMATH 481 (or equivalent), AMATH 569 (which may be taken concurrently), AMATH 581, AMATH 584/MATH 584, AMATH 585/MATH 585, or permission of instructor; recommended: AMATH 584/MATH 584 and AMATH 585/MATH 585. Offered: jointly with ATM S 581/MATH 586; Sp.

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**AMATH 590 Special Topics (1-5, max. 30)**

Topics of current interest in applied mathematics. Offered: AWSpS.

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**AMATH 600 Independent Research or Study (*-)**

Credit/no-credit only.

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**AMATH 601 Internship (1-10, max. 30)**

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**AMATH 700 Master's Thesis (*-)**

Credit/no-credit only.

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**AMATH 800 Doctoral Dissertation (*-)**

Credit/no-credit only.

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