Korsijiet · 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 unità 19 lezzjonijiet 121 mistoqsijiet 180 minuta finali

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.

Jiġi wara: Multivariable-Calculus, Differential-Equations, Linear-Algebra, Analysis

Il-kors

  1. Classification and first-order equations

    3 lezzjonijiet
    What a PDE is, and how to classify oneFirst-order linear equations and characteristicsNonlinear first-order equations and shocks
    Idħol biex tieħu l-kwizz

  2. The heat equation and Fourier series

    4 lezzjonijiet
    The heat equation: where it comes fromSeparation of variablesFourier series: coefficients, convergence and ParsevalThe heat equation on an interval
    Idħol biex tieħu l-kwizz

  3. The wave equation

    2 lezzjonijiet
    The wave equation and d'Alembert's formulaThe vibrating string: modes and energy
    Idħol biex tieħu l-kwizz

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

    3 lezzjonijiet
    Laplace's equation and harmonic functionsMaximum principles and uniquenessGreen's functions and the fundamental solution
    Idħol biex tieħu l-kwizz

  5. Fourier transforms and energy methods

    2 lezzjonijiet
    The Fourier transform and the heat kernelEnergy methods: uniqueness, stability and decay
    Idħol biex tieħu l-kwizz

  6. Distributions and Sobolev spaces

    2 lezzjonijiet
    Distributions and weak derivativesSobolev spaces and embeddings
    Idħol biex tieħu l-kwizz

  7. Well-posedness, estimates and scaling

    3 lezzjonijiet
    Well-posedness in the sense of HadamardA priori estimates, local and global existenceScaling and criticality: why dimension matters
    Idħol biex tieħu l-kwizz

L-eżami finali

Mistoqsijiet minn kull unità, l-aktar diffiċli l-aħħar. Tweġibiet tat-tip huma mmarkati mill-sistema algebra; provi u xogħol fotografati huma maqluba minn mudell; imbagħad persuna tirrevedi l-karta kollha qabel ir-riżultat tiegħek huwa rilaxxat. Pass u inti tikseb ċertifikat mal-link tagħha stess. Marka ta’ approvazzjoni: 70%.

Idħol

Prezzijiet

Pass tal-kors

$39

Kull unità kwizz, l-eżami finali ma retake wieħed, u 20 kontrolli tax-xogħol tiegħek stess, għal kors wieħed.

Eżami finali wieħed

$15

Seduta waħda tal-finali ta' kors wieħed, immarkata u riveduta għal xi ħadd li diġà jaf il-materjal u jrid li jiġi ttestjat.

Kull kors

$19/10 mg/ kg

Il-kwizzi u l-finali kollha f’kull kors (żewġ finali għal kull kors fix-xahar) u 100 verifika tax-xogħol fix-xahar.

Kontrolli ta’ ħidma

$9

Għaxar kontrolli: ittella' ritratt ta' xi xogħol, jew ittajpjah, u ara fejn mar ħażin.

Il-korsijiet miftuħa għax-xiri dalwaqt. Il-lezzjonijiet u l-ewwel kwizz ta' kull kors huma miftuħa issa.

Iċċekkja biċċa tax-xogħol tiegħek stess