Applications of Partial Differential Equations

G. Hernandez

Department of MathematicsUniversity of Wisconsin

Math 322

AcousticsSignal processing can be applied to psychoacoustics. Otherapplications of acoustics are quality improvement, noisereduction, image enhancement and echo cancellation, andspeech recognition.

Aerodynamics: Euler equationsPDEs can be used to model air passing complicatedgeometries like aircraftÕs wing. This is how airbus designedwings that save up to 30% in fuel.

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Figure: (Vortex created by the passage of an aircraft wing: NASALangley Research Center), and vorticity of simulation described bythe Boussinesq equations

Fluid Dynamics: Shallow water equationsPDEs like the shallow water equations can be used to modeltsunamis. Currently the models are expensive and take severalhours to predict coast that will be affected. Parallel computingcan be added to accelerate the simulations. Picture by RandyLeVeque (University of Washington)

Meteorology

Models used for forecast and to predict hurricanes trajectoriesare described by PDEs (Anelastic equations, Boussinesqequations, etc). Variables in the equation are velocity, watervapor mixing ratio, rain water, ice, etc.

Quantum mechanicsTunneling effect, etc.

Other applications

Other applications are: Aneurysm treatments, predicting traffic(traffic equation), multi-phase flows (water, oil and gas phases)models can be used to find oil, inverse problems (inversescattering) to improve cancer detection devices, hearing a fetalhearthbeat, etc etc ect.Google SIAM Math Matters Apply It.