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Course Criteria
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5.00 Credits
Survey of practical solution techniques for ordinary differential equations. Linear systems of equations including nondiagonable case. Nonlinear systems; stability phase plane analysis. Asymptotic expansions. Regular and singular perturbations. Recommended: 402 or equivalent. Offered: W.
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5.00 Credits
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. Offered: Sp.
Prerequisite:
AMATH 403, AMATH 568 or MATH 428 or permission of instructor
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5.00 Credits
Asymptotics for integrals, perturbation and multiple-scale analysis. Singular perturbations: matched asymptotic expansions, boundary layers, shock layers, uniformly valid solutions. Offered: A.
Prerequisite:
AMATH 567, AMATH 568, AMATH 569, or permission of instructor
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5.00 Credits
Analysis and application of spectral methods for the numerical solution of differential equations. Fourier methods and the FFT; collocation methods; polynomial interpolation and Chebyshev series; approximation theory and spectral accuracy; boundary conditions. Offered: W.
Prerequisite:
AMATH 584, AMATH 585, AMATH 586, or permission of instructor
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5.00 Credits
Introduction to the theory of probability and stochasitc processes based on differential equations with applications to science and engineering. Poisson processes and continuous-time Markov processes, Brownian motions and diffusion. Offered: Sp.
Prerequisite:
AMATH/ STAT 506, AMATH 402, or equivalent knowledge of probability and ordinary differential equations
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5.00 Credits
Methods for integrable and near-integrable nonlinear partial differential equations such as the Korteweg-de Vries equation and the Nonlinear Schrodinger equation; symmetry reductions and solitons; soliton interactions; infinite-dimensional Hamiltonian systems; Lax pairs and inverse scattering; Painleve analysis. Offered: A.
Prerequisite:
AMATH 569, or permission of instructor
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5.00 Credits
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. Offered: W.
Prerequisite:
AMATH 586 or permission of instructor
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5.00 Credits
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:
AMATH 568 or equivalent
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3.00 Credits
Regular perturbations. Singular perturbations: matched asymptotic expansions, boundary layers, shock layers, uniformly valid solutions. The methods of multiple scales and averaging, weakly nonlinear wave propagation problems and resonance phenomena, homogenization, nonlinear wave propagation in fluid, solid, and particle mechanics. Offered: even years.
Prerequisite:
AMATH 567, AMATH 568, AMATH 569, or equivalent
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3.00 Credits
Regular perturbations. Singular perturbations: matched asymptotic expansions, boundary layers, shock layers, uniformly valid solutions. The methods of multiple scales and averaging, weakly nonlinear wave propagation problems and resonance phenomena, homogenization, nonlinear wave propagation in fluid, solid, and particle mechanics. Offered: even years.
Prerequisite:
AMATH 567, AMATH 568, AMATH 569, or equivalent
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