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Partial Differential Equations
Explore equations and data; model equations; weak solutions and discretization. Concepts and connections are available independently of video coverage.
Prerequisites and connections
Partial derivatives
A partial derivative varies one coordinate while holding the others fixed.
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Equations and data
Partial differential equations
A PDE relates a multivariable unknown function to its partial derivatives.
Video explanations to be addedBoundary value problems
Boundary data restrict a differential equation's solution on the edge of its domain.
Video explanations to be addedPDE classification
Second-order principal terms distinguish elliptic, parabolic and hyperbolic behavior under appropriate assumptions.
Video explanations to be addedWell-posedness
A well-posed problem has a solution, uniqueness and continuous dependence on data.
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Model equations
Heat equation
The heat equation models diffusion through time and spatial second derivatives.
Video explanations to be addedWave equation
The wave equation links temporal and spatial second derivatives to describe propagation.
Video explanations to be addedLaplace and Poisson equations
The Laplacian equals zero for harmonic fields or a source term in Poisson's equation.
Video explanations to be addedSeparation of variables
Product-form trial solutions reduce suitable PDEs to ordinary differential equations.
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Weak solutions and discretization
Method of characteristics
Characteristic curves reduce suitable first-order PDEs to equations along paths.
Video explanations to be addedWeak solutions
Integral identities define solutions whose classical derivatives may not exist.
Video explanations to be addedFinite element method
Piecewise basis functions approximate weak formulations over a subdivided domain.
Video explanations to be addedEnergy estimates
Bounds on integrated solution quantities help establish stability and uniqueness.
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