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Course Criteria
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4.00 Credits
Linear vector spaces; eigenvalues and eigenvectors in discrete systems; eigenvalues and eigenvectors in continuous systems including Sturm-Liouville theory, orthogonal expansions and Fourier series, Green's functions; elementary theory of nonlinear ODEs including phase plane, stability and bifurcation; calculus of variations. Applications will be drawn from equilibrium and dynamic phenomena in science and engineering. Prerequisites/Corequisites: Prerequisites: MATH 2400 and MATH 4600. When Offered: Fall term annually. Credit Hours: 4
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4.00 Credits
This course is devoted to advanced methods rather than theory. Content includes such topics as matched asymptotic expansions, multiple scales, WKB, and homogenization. Applications are made to ODEs, PDEs, difference equations, and integral equations. The methods are illustrated using currently interesting scientific and engineering problems that involve such phenomena as boundary or shock layers, nonlinear wave propagation, bifurcation and stability, and resonance. Prerequisites/Corequisites: Prerequisites: MATH 2400 and MATH 4600 or equivalent. When Offered: Spring term even-numbered years. Credit Hours: 4
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4.00 Credits
Review of basic complex variables theory; power series, analytic functions, singularities, and integration in the complex plane. Integral transforms (Laplace, Fourier, etc.) in the complex plane, with application to solution of PDEs and integral equations. Asymptotic expansions of integrals (Laplace method, methods of steepest descent and stationary phase), with emphasis on extraction of useful information from inversion integrals of transforms. Problems to be drawn from linear models in science and engineering. Prerequisites/Corequisites: Prerequisites: MATH 4600 and familiarity with elementary ordinary and partial differential equations. When Offered: Spring term odd-numbered years. Credit Hours: 4
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4.00 Credits
This course is the second mathematical and computational finance course in a new one-year sequence for mathematics, DSES and engineering majors, and graduate students. It will cover the basics of stochastic processes, and current methods in the simulation of stochastic problems such as Monte Carlo algorithms and variance reduction tools. It will also focus on teaching the application of these stochastic simulation methods to finance. Prerequisites/Corequisites: Prerequisite: MATH 4740. When Offered: Spring term. Credit Hours: 4
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4.00 Credits
Advanced methods and/or applications of mathematics. Possible topics include: nonlinear continuum mechanics, nonlinear waves, inverse problems, nonlinear optics, combustion, acoustic wave propagation, similarity methods for differential equations, quantum field theory and statistical mechanics, stability of fluid flows, biomathematics, and finance. Prerequisites/Corequisites: Prerequisites: vary with topic. When Offered: Spring term annually . Credit Hours: 4
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4.00 Credits
Gaussian elimination, special linear systems (such as positive definite, banded, or sparse), introduction to parallel computing, iterative methods for linear systems (such as conjugate gradient and preconditioning), QR factorization and least squares problems, and eigenvalue problems. Prerequisites/Corequisites: Prerequisite: MATH 4800 or CSCI 4800 or permission of instructor. When Offered: Fall term annually. Cross Listed: Cross-listed as CSCI 6800. Students cannot obtain credit for both this course and CSCI 6800. Credit Hours: 4
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4.00 Credits
Numerical methods and analysis for ODEs with applications from mechanics, optics, and chaotic dynamics. Numerical methods for dynamical systems include Runge-Kutta, multistep and extrapolation techniques, methods for conservative and Hamiltonian systems, methods for stiff differential equations and for differential-algebraic systems. Methods for boundary value problems include shooting and orthogonalization, finite difference and collocation techniques, and special methods for problems with boundary or shock layers. Prerequisites/Corequisites: Prerequisite: MATH 4800 or CSCI 4800 or permission of instructor. When Offered: Spring term odd-numbered years. Cross Listed: Cross-listed as CSCI 6820. Students cannot obtain credit for both this course and CSCI 6820. Credit Hours: 4
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4.00 Credits
Numerical methods and analysis for linear and nonlinear PDEs with applications from heat conduction, wave propagation, solid and fluid mechanics, and other areas. Basic concepts of stability and convergence (Lax equivalence theorem, CFL condition, energy methods). Methods for parabolic problems (finite differences, method of lines, ADI, operator splitting), methods for hyperbolic problems (vector systems and characteristics, dissipation and dispersion, shock capturing and tracking schemes), methods for elliptic problems (finite difference and finite volume methods). Prerequisites/Corequisites: Prerequisite: MATH-4800 or CSCI-4800 or permission of Instructor When Offered: Fall term odd-numbered years. Cross Listed: (Cross listed as CSCI-6840. Students cannot obtain credit for both this course and CSCI-6840.) Credit Hours: 4
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4.00 Credits
Galerkin's method and extremal principles, finite element approximations (Lagrange, hierarchical and 3-D approximations, interpolation errors), mesh generation and assembly, adaptivity (h-, p-, hp-refinement). Error analysis and convergence rates. Perturbations resulting from boundary approximation, numerical integration, etc. Time dependent problems including parabolic and hyperbolic PDEs. Applications will be selected from several areas including heat conduction, wave propagation, potential theory, and solid and fluid mechanics. Prerequisites/Corequisites: Prerequisite: MATH 4800 or CSCI 4800 or permission of instructor. When Offered: Spring term even-numbered years. Cross Listed: Cross-listed as CSCI 6860. Students cannot obtain credit for both this course and CSCI 6860. Credit Hours: 4
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4.00 Credits
Advanced methods and/or applications in scientific computing. Possible topics include computational fluid dynamics, parallel computing, computational acoustics, and computer applications in medicine and biology. Prerequisites/Corequisites: Prerequisites: vary with topic. When Offered: Fall term even-numbered years. Credit Hours: 4
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