# Math 445: Applied PDEs: models, problems, methods

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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. - PowerPoint PPT Presentation

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• Math 445: Applied PDEs: models, problems, methodsD. Gurarie

• Models: processesTransport 1-st order linear (quasi-linear) PDE in space-timeHeat-diffusion 1-st order in t, 2-nd order in x, called parabolicSimilar equations apply to Stochastic Processes (Brownian motion): u(x,t) - Probability to find particle at point x time t

• Wave equation2-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/fluidOptics, E-M propagation: u component(s) of E-M field, or potentialsLaplaces (elliptic) equationStationary heat distributionPotential theory (gravitational, Electro-static, electro-dynamic, fluid,)

• Nonlinear models

• 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, bifurcationsApplicationsPrediction and controlMechanical (propagation of heat, waves/signals)Chemical, bio-medical, Other

• Solution methodsAnalyticMethod of characteristics (1-st and higher order PDE)Separation of variables, reduction to ODE Expansion and transform methods (Fourier, Laplace et al); special functionsGreens functions and fundamental solutions (integral equations)Approximate and asymptotic methodsVariational methodsNumeric methods (Mathematica/Matlab)Other techniques (change of variables, symmetry reduction, Integrable models,)

• Examples (with Mathematica)2D incompressible fluid Shear instabilityVorticityStream f.Time evolution of traffic jam for initial Gaussian profileAnalyticComputational

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