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Math 445: Applied PDEs: models, problems, methods

Math 445: Applied PDEs: models, problems, methods. D. Gurarie. Models: processes. Transport 1-st order linear (quasi-linear) PDE in space-time. Heat-diffusion 1-st order in t, 2-nd order in x, called parabolic.

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Math 445: Applied PDEs: models, problems, methods

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  1. Math 445: Applied PDEs: models, problems, methods D. Gurarie

  2. Models: processes Transport 1-st order linear (quasi-linear) PDE in space-time Heat-diffusion 1-st order in t, 2-nd order in x, called parabolic Similar equations apply to Stochastic Processes (Brownian motion): u(x,t) - Probability to find particle at point x time t

  3. Wave equation 2-st order in x, t (hyperbolic) Vibrating strings, membranes,…: u – vertical displacement (from rest) Elasticity: medium displacement components (P,S –waves) Acoustics: u – velocity/pressure/density perturbation in gas/fluid Optics, E-M propagation: u – component(s) of E-M field, or potentials Laplace’s (elliptic) equation Stationary heat distribution Potential theory (gravitational, Electro-static, electro-dynamic, fluid,…)

  4. Nonlinear models - Fisher-Kolmogorov (genetic drift) - Burgers (sticky matter) - KdV (integrable Hamiltonian system)

  5. PDE systems: Fluid dynamics Electro-magnetism: Elasticity Acoustics

  6. Basic Problems: • Initial and Boundary value problems (well posedness) • Solution methods: • exact; approximate; • analytic/numeric; • general or special solutions (equilibria, periodic et al) • Analysis: stability, parameter dependence, bifurcations • Applications • Prediction and control • Mechanical (propagation of heat, waves/signals) • Chemical, bio-medical, • Other…

  7. Solution methods • Analytic • Method of characteristics (1-st and higher order PDE) • Separation of variables, reduction to ODE • Expansion and transform methods (Fourier, Laplace et al); special functions • Green’s functions and fundamental solutions (integral equations) • Approximate and asymptotic methods • Variational methods • Numeric methods (Mathematica/Matlab) • Other techniques (change of variables, symmetry reduction, Integrable models,…)

  8. Examples (with Mathematica) Analytic 2D incompressible fluid Shear instability Computational Time evolution of traffic jam for initial Gaussian profile Vorticity Stream f.

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