Izifundo · 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 Ama-units 19 Izifundo 121 imibuzo 180 imizuzu edlule

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.

Ivela ngemuva: Multivariable-Calculus, Differential-Equations, Linear-Algebra, Analysis

Ibanga

  1. Classification and first-order equations

    3 Izifundo
    What a PDE is, and how to classify oneFirst-order linear equations and characteristicsNonlinear first-order equations and shocks
    Ngena ukuze uthathe isifundo

  2. The heat equation and Fourier series

    4 Izifundo
    The heat equation: where it comes fromSeparation of variablesFourier series: coefficients, convergence and ParsevalThe heat equation on an interval
    Ngena ukuze uthathe isifundo

  3. The wave equation

    2 Izifundo
    The wave equation and d'Alembert's formulaThe vibrating string: modes and energy
    Ngena ukuze uthathe isifundo

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

    3 Izifundo
    Laplace's equation and harmonic functionsMaximum principles and uniquenessGreen's functions and the fundamental solution
    Ngena ukuze uthathe isifundo

  5. Fourier transforms and energy methods

    2 Izifundo
    The Fourier transform and the heat kernelEnergy methods: uniqueness, stability and decay
    Ngena ukuze uthathe isifundo

  6. Distributions and Sobolev spaces

    2 Izifundo
    Distributions and weak derivativesSobolev spaces and embeddings
    Ngena ukuze uthathe isifundo

  7. Well-posedness, estimates and scaling

    3 Izifundo
    Well-posedness in the sense of HadamardA priori estimates, local and global existenceScaling and criticality: why dimension matters
    Ngena ukuze uthathe isifundo

Ukuhlolwa kokuphela

Imibuzo evela kunoma iyiphi iyunithi, enzima kakhulu ihlala. Izixazululo ezishicilelwe ziphawulwa nge-algorithm system; iziqinisekiso kanye nokusebenza okuphothiwe kufundeka yimodeli; bese umuntu ubheka iphepha lonke ngaphambi kokuba imiphumela yakho ikhishwa. Udlula futhi uthola isitifiketi esixhunywe kuyo. Iphutha lokuphawula: 70%.

Ngena

Amantengo

I-course pass

$39

Iyunithi ngayinye yokuzibuza, isifundo sokugcina esinomqulu owodwa, kanye nezibuyekezo ezingu-20 zomsebenzi wakho, kuhlelo olulodwa.

Ukuhlolwa kokuphela

$15

Ukuhlala okufanayo kwesigaba esisodwa sokugcina, okuphawulwe futhi okubuyekezwe. Kumuntu owaziyo ngaphambili izinto futhi ofuna ukuzihlola.

Isigaba

$19/ihora

Zonke izifundo neziqu eziphezulu ezikuzo zonke izifundo (iziqu eziphezulu ezimbili ngasinye isifundo ngasinye ngenyanga) kanye ne-100 yokubheka umsebenzi ngenyanga. Cisha kusuka ku-akhawunti yakho.

Ukuhlolwa kokusebenzela

$9

Ukuhlolwa okungu-10: khipha isithombe somsebenzi, noma ubhale, bese ubona lapho kudlule khona.

Izifundo zivuliwe ukuthengwa ngokushesha. Izifundo kanye nemibuzo yokuqala yesifundo ngasinye ivuliwe manje.

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