Iikhosi · 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 Iiyunithiunit description in lists 19 Iinkqubo 121 iimibuzo 180 Imini

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 emva: Multivariable-Calculus, Differential-Equations, Linear-Algebra, Analysis

Inkqubo

  1. Classification and first-order equations

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

  6. Distributions and Sobolev spaces

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

  7. Well-posedness, estimates and scaling

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

I-Exam

Imibuzo evela kwiyunithi nganye, ezinzima ziphela. Iimpendulo ezishicilelwe ziphawulwe yinkqubo ye algebra; iimvavanyo kunye nomsebenzi othathwe ngefoto ufundwa ngumzobo; ngoko umntu uvavanya iphepha lonke phambi kokuba impumelelo yakho iphume. Ugqitha kwaye ufumana isiqinisekiso kunye nekhonkco layo. Inqaku le-Pass: 70%.

Ngena

Ixabiso

Iinkqubo

$39

Iiyunithi zonke zovavanyo, uvavanyo oluphelileyo olunokuphinda luthathelwe kwakhona, kunye neentsuku ezili-20 zokuhlola umsebenzi wakho, kwinkqubo enye.

Iinkqubo zolawulo

$15

Ukuhlala olunye kwesinye isifundo esiphelelwe lixesha, esiphawulwe kwaye sihlaziywe. Kulowo sele eyaziyo i-material kwaye ufuna ukuba ivavanywe.

Iinkqubo

$19/Iiyure

Zonke iimvavanyo kunye neengxelo zesiphelo kwinkqubo nganye (iingxelo ezimbini zesiphelo kwinkqubo nganye ngenyanga) kunye neengxelo ezili-100 zomsebenzi ngenyanga.

Iinkqubo zolawulo

$9

Uvavanyo lweshumi: thumela umfanekiso wento esebenza, okanye ubhale, kwaye ubone apho ihambe khona kakubi.

Iikhosi zivuliwe ukuthengwa ngoku. Iikhosi kunye nemibuzo yokuqala yekhosi nganye ivuliwe ngoku.

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