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Course Criteria
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1.00 Credits
Introduces science and engineering graduate students to a variety of fundamental mathematical methods. Topics include linear algebra, complex variables, Fourier series, Fourier and Laplace transforms and their applications, ordinary differential equations, tensors, curvilinear coordinates, partial differential equations, and calculus of variations.
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1.00 Credits
Provides the basis of real analysis which is fundamental to many of the other courses in the program: metric spaces, measure theory, and the theory of integration and differentiation.
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1.00 Credits
A continuation of APMA 2110: metric spaces, Banach spaces, Hilbert spaces, the spectrum of bounded operators on Banach and Hilbert spaces, compact operators, applications to integral and differential equations.
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1.00 Credits
Solution methods and basic theory for first and second order partial differential equations. Geometrical interpretation and solution of linear and nonlinear first order equations by characteristics; formation of caustics and propagation of discontinuities. Classification of second order equations and issues of well-posed problems. Green's functions and maximum principles for elliptic systems. Characteristic methods and discontinuous solutions for hyperbolic systems.
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1.00 Credits
Integral equations. Fredholm and Volterra theory, expansions in orthogonal functions, theory of Hilbert-Schmidt. Singular integral equations, method of Wiener-Hopf. Calculus of variations and direct methods.
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1.00 Credits
Calculus of asymptotic expansions, evaluation of integrals. Solution of linear ordinary differential equations in the complex plane, WKB method, special functions.
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1.00 Credits
Topics vary according to interest of instructor and class.
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1.00 Credits
Basic theory of ordinary differential equations, flows, and maps. Two-dimensional systems. Linear systems. Hamiltonian and integrable systems. Lyapunov functions and stability. Invariant manifolds, including stable, unstable, and center manifolds. Bifurcation theory and normal forms. Nonlinear oscillations and the method of averaging. Chaotic motion, including horseshoe maps and the Melnikov method. Applications in the physical and biological sciences.
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1.00 Credits
Basic theory of ordinary differential equations, flows, and maps. Two-dimensional systems. Linear systems. Hamiltonian and integrable systems. Lyapunov functions and stability. Invariant manifolds, including stable, unstable, and center manifolds. Bifurcation theory and normal forms. Nonlinear oscillations and the method of averaging. Chaotic motion, including horseshoe maps and the Melnikov method. Applications in the physical and biological sciences.
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1.00 Credits
This course will discuss the asymptotic stability of noncharacteristic boundary layers in gas dynamics or magnetohydrodynamics equations. One of the main difficulties in the stability analysis is that there is no spectral gap between the imaginary axis and the essential spectrum of the linearized operator about the layer. Standard semi-group methods therefore do not seem to apply and, at best, algebraic temporal decay can be expected in case of stability. Pointwise estimates for the Green function are then useful and sufficient for analysis of the (linear and) nonlinear stability. The general mathematical approach to be covered in the course is the so-called pointwise semi-group or Evans function approach developed systematically by Zumbrun-Howard and Mascia-Zumbrun in their studies of (orbital) asymptotic stability of viscous shock waves. We discuss its application in the context of boundary layers; in particular, the gap/conjugation lemma, the tracking/reduction lemma, construction of the resolvent kernel, the spectral and Evans function theory, and the construction of the Green function with sharp pointwise estimates. Time permitting, we shall also discuss a few recent developments on stability of boundary layers for a more general class of hyperbolic-parabolic conservation laws in one or multi-dimensional spaces. Certain numerical evidences for stability might as well be demonstrated in the end of the course.
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