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TA7: Applied analysis and differential equations |
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The two Basic Courses here provide a thorough introduction to the theory of ordinary differential equations and dynamical systems, and to that of partial differential equations.
Outline of the contents:
Dynamical systems
◦ Flow properties of dynamical systems
◦ Omega limit sets
◦ Stability of fixed points and periodic orbits
◦ Invariant manifolds
◦ Examples of local and global bifurcations
◦ Chaotic behavior
Partial differential equations
◦ Scalar first order equation
◦ Elementary PDEs: heat equation, wave equation, Laplace equation
◦ Solutions of linear problems via orthogonal series (separation)
◦ Elliptic problems via Lax–Milgram theory, maximum principles
◦ Existence and smoothing properties of parabolic equations
◦ Hyperbolic equations in several space dimensions
◦ Semilinear equations
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