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 jedinice 19 lekcije 121 pitanja 180 završno

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.

Dolazi nakon: Multivariable-Calculus, Differential-Equations, Linear-Algebra, Analysis

Tečaj

  1. Classification and first-order equations

    3 lekcije
    What a PDE is, and how to classify oneFirst-order linear equations and characteristicsNonlinear first-order equations and shocks
    Prijavi se da uzmeš kviz.

  2. The heat equation and Fourier series

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

  3. The wave equation

    2 lekcije
    The wave equation and d'Alembert's formulaThe vibrating string: modes and energy
    Prijavi se da uzmeš kviz.

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

    3 lekcije
    Laplace's equation and harmonic functionsMaximum principles and uniquenessGreen's functions and the fundamental solution
    Prijavi se da uzmeš kviz.

  5. Fourier transforms and energy methods

    2 lekcije
    The Fourier transform and the heat kernelEnergy methods: uniqueness, stability and decay
    Prijavi se da uzmeš kviz.

  6. Distributions and Sobolev spaces

    2 lekcije
    Distributions and weak derivativesSobolev spaces and embeddings
    Prijavi se da uzmeš kviz.

  7. Well-posedness, estimates and scaling

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

Završni ispit

Pitanja od svake jedinice, najteži zadnji. Utipkani odgovori su označeni algebra sustav; dokazi i fotografirani rad su pročitani od strane modela; onda osoba pregleda cijeli rad prije nego što je rezultat objavljen. Pass i dobijete certifikat sa vlastitim linkom. Oznaka: 70%.

Upišite se

Cijene

Prolaz kursa

$39

Svaki ispit, završni ispit s jednim ponovnim, i 20 provjera vlastitog rada, za jedan kurs.

Jedan završni ispit.

$15

Jedno sjedenje jednog tečaja je konačno, označeno i pregledano.

Svaki kurs

$19/Mo

Svi kvizovi i finale u svakom tečaju (dva finala po tečaju mjesečno) i 100 radnih provjera mjesečno. Otkaži s vašeg računa.

Provjere rada

$9

Deset čekova: ubaciti sliku bilo kojeg radnog, ili kucati, i vidjeti gdje je krenuo po zlu.

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