Seha-pianarana · Graduate

Partial Differential Equations

Transport, heat, wave and Laplace equations, Fourier methods, energy and maximum principles, weak derivatives, Sobolev spaces, well-posedness and scaling.

7 %sunits 19 Fianarana 121 fanontaniana 180 minitra farany

A graduate course in partial differential equations that starts with classification and characteristics and ends with the tools used to study the Navier-Stokes equations. You solve the four model equations by hand (characteristics, separation of variables, Fourier series and transforms, d'Alembert, Green's functions), then learn the methods that work without formulas: maximum principles, energy estimates, distributions and weak derivatives, Sobolev spaces, a priori estimates and scaling.

It assumes multivariable calculus, ordinary differential equations, linear algebra and a first course in analysis, and it prepares you for the fluid dynamics course.

Mitranga aorian'ny: Multivariable-Calculus, Differential-Equations, Linear-Algebra, Analysis

Ny fianarana

  1. Classification and first-order equations

    3 Fianarana
    What a PDE is, and how to classify oneFirst-order linear equations and characteristicsNonlinear first-order equations and shocks
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  2. The heat equation and Fourier series

    4 Fianarana
    The heat equation: where it comes fromSeparation of variablesFourier series: coefficients, convergence and ParsevalThe heat equation on an interval
    Hiditra mba handray ny fanontaniana

  3. The wave equation

    2 Fianarana
    The wave equation and d'Alembert's formulaThe vibrating string: modes and energy
    Hiditra mba handray ny fanontaniana

  4. Laplace's equation, maximum principles and Green's functions

    3 Fianarana
    Laplace's equation and harmonic functionsMaximum principles and uniquenessGreen's functions and the fundamental solution
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  5. Fourier transforms and energy methods

    2 Fianarana
    The Fourier transform and the heat kernelEnergy methods: uniqueness, stability and decay
    Hiditra mba handray ny fanontaniana

  6. Distributions and Sobolev spaces

    2 Fianarana
    Distributions and weak derivativesSobolev spaces and embeddings
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  7. Well-posedness, estimates and scaling

    3 Fianarana
    Well-posedness in the sense of HadamardA priori estimates, local and global existenceScaling and criticality: why dimension matters
    Hiditra mba handray ny fanontaniana

Ny fanadinana farany

Fanontaniana avy amin'ny kilasy tsirairay, ny sarotra indrindra no farany. Voamarika amin'ny rafitra algebra ny valinteny voasoratra; vakian'ny modely ny porofo sy ny asa nalaina sary; avy eo misy olona iray mijery ny lahatsoratry ny taratasy manontolo alohan'ny hamoahana ny vokatry ny asa. Lasa ianao ary mahazo fankatoavana miaraka amin'ny rohy manokana. Famantarana ny fankatoavana: 70%.

Hiditra

Haben'ny vidiny

Fidirana amin'ny seha-pianarana

$39

Ny fanontaniana isaky ny vondrona, ny fanadinana farany miaraka amin'ny fanadinana iray indray, ary ny 20 fitsapana ny asa ataonao, ho an'ny fianarana iray.

Mividy ny fidirana amin'ny sekoly

Fitsapana farany iray

$15

Fihaonana tokana amin'ny faran'ny fianarana iray, nanamarika sy nijery indray. Ho an'ny olona efa mahalala ny fitaovana ary te-hanandrana azy.

Mividy iray farany

Ny kilasy rehetra

$19/mo

Ny fanontaniana rehetra sy ny fihodinana farany amin'ny kilasy tsirairay (fihodinana roa isan-kilasy isam-bolana) ary ny fitsapana 100 isan-bolana. Fafao amin'ny kaontinao.

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Fitsapana asa

$9

Fanaraha-maso folo: mandefa sary ny zavatra miasa, na manoratra azy, ary jereo izay tsy nety.

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