A rough idea of the topics to be covered follows. Some topics may be covered in more detail than this suggests, or the reverse. Some topics may be skipped completely and others may be included if needed. This is just to give you an idea of the flavor of the course.
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Computers and numerical issues, with a brief introduction to MATLAB®.
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Good and bad numerical schemes. Fourier Series and von Neumann stability analysis. Topics: Consistency and Stability of Numerical Schemes; von Neumann Stability Analysis; Associated Equation to a Numerical Scheme; Short Wave Stability Analysis; Discrete Fourier Transform (DFT); Fourier Series; Fourier Transform; Spectral Methods.
Part I. Some Basic Topics in Nonlinear Waves
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Shock waves and hydraulic jumps. Description and various physical set ups where they occur: traffic flow, shallow water. What is a wave?
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Traffic flow (TF). Continuum hypothesis. Conservation and derivation of the mathematical model. Integral and differential forms. Other examples of systems where conservation is used to derive the model equations (in nonlinear elasticity, fluids, etc.)
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Linearization of equations of TF and solutions. Meaning and interpretations. Solution of the fully nonlinear TF problem. Method of characteristics, graphical interpretation of the solutions, wave breaking. Weak discontinuities, shock waves and rarefaction fans. Envelope of characteristics. Irreversibility in the model.
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Quasilinear First Order PDE's.
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Shock structure, diffusivity. Burger's equation. Explicit solution by the Cole-Hopf transformation. The heat equation. Derivation of the equation and solution. Application to the Burger's equation: Inviscid limit and Laplace's method. Dimensional analysis. Nonlinear diffusion and fronts.
Part II. Dynamical Systems
Part III. Some of the following topics will be covered if time permits
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Random walks, brownian motion, diffusion.
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Shallow water waves. Derivation of the equations. Linearization and solution. Radiation conditions. More on characteristics and shocks, now for all Shallow Water equations.
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Water waves. Derivation of the equations and linearization. Notions of dispersion and group speed. Weak nonlinearity and solitary waves. Perturbation expansions.
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Linear and nonlinear oscillations, relaxation. Phase plane methods and multiple scales. Application to celestial mechanics and mechanical vibrations.
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Dynamical systems examples from mathematical biology and population dynamics.
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