Tečaji · 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 enote 19 učne izkušnje 121 vprašanja 180 minutna končna

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.

Pride po: Multivariable-Calculus, Differential-Equations, Linear-Algebra, Analysis

Tečaj

  1. Classification and first-order equations

    3 učne izkušnje
    What a PDE is, and how to classify oneFirst-order linear equations and characteristicsNonlinear first-order equations and shocks
    Prijavi se, da boš vzel kviz.

  2. The heat equation and Fourier series

    4 učne izkušnje
    The heat equation: where it comes fromSeparation of variablesFourier series: coefficients, convergence and ParsevalThe heat equation on an interval
    Prijavi se, da boš vzel kviz.

  3. The wave equation

    2 učne izkušnje
    The wave equation and d'Alembert's formulaThe vibrating string: modes and energy
    Prijavi se, da boš vzel kviz.

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

    3 učne izkušnje
    Laplace's equation and harmonic functionsMaximum principles and uniquenessGreen's functions and the fundamental solution
    Prijavi se, da boš vzel kviz.

  5. Fourier transforms and energy methods

    2 učne izkušnje
    The Fourier transform and the heat kernelEnergy methods: uniqueness, stability and decay
    Prijavi se, da boš vzel kviz.

  6. Distributions and Sobolev spaces

    2 učne izkušnje
    Distributions and weak derivativesSobolev spaces and embeddings
    Prijavi se, da boš vzel kviz.

  7. Well-posedness, estimates and scaling

    3 učne izkušnje
    Well-posedness in the sense of HadamardA priori estimates, local and global existenceScaling and criticality: why dimension matters
    Prijavi se, da boš vzel kviz.

Končni izpit

Vprašanja vsake enote, najtežji zadnji. Vtisnjeni odgovori so označeni z algebra sistem; dokazi in fotografirani delo se preberejo s modelom; potem oseba pregleda celoten časopis, preden je vaš rezultat objavljen. Prejmite in dobite certifikat z lastno povezavo. Oddaljenost: 70%.

Vpišite se

Cene

Prehod smeri

$39

Vsak kviz, končni izpit z eno ponovno in 20 čekov svojega dela, za en tečaj.

Eden končni izpit.

$15

Eno sejo enega tečaja je končno, označeno in pregledano, za nekoga, ki že pozna material in želi, da se testira.

Vsaka smer

$19/Mo

Vsi kvizi in finale v vsakem tečaju (dva finala na tečaj na mesec) in 100 delovnih pregledov na mesec. Prekliči s vašega računa.

Delovni pregledi

$9

Deset čekov: naložite sliko katerega koli dela, ali jo natipkajte, in poglejte, kam je šla narobe.

Tečaji so odprti za nakup kmalu. Lekcije in prvi kviz vsakega tečaja so odprti zdaj.

Preverite del svojega dela