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 Similar equations apply to Stochastic Processes (Brownian motion): u(x,t) - Probability to find particle at point x time t

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,…)

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

PDE systems: Fluid dynamicsElectro-magnetism: Elasticity Acoustics

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…

Solution methods 1.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) 2.Approximate and asymptotic methods 3.Variational methods 4.Numeric methods (Mathematica/Matlab) 5.Other techniques (change of variables, symmetry reduction, Integrable models,…)

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