Maphunziro · 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 Maolaunit description in lists 19 Maphunziro 121 mafunso 180 minute final

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.

Koma pambuyo: Multivariable-Calculus, Differential-Equations, Linear-Algebra, Analysis

Mfundo

  1. Classification and first-order equations

    3 Maphunziro
    What a PDE is, and how to classify oneFirst-order linear equations and characteristicsNonlinear first-order equations and shocks
    Kulowa kuti atenge chidziwitso

  2. The heat equation and Fourier series

    4 Maphunziro
    The heat equation: where it comes fromSeparation of variablesFourier series: coefficients, convergence and ParsevalThe heat equation on an interval
    Kulowa kuti atenge chidziwitso

  3. The wave equation

    2 Maphunziro
    The wave equation and d'Alembert's formulaThe vibrating string: modes and energy
    Kulowa kuti atenge chidziwitso

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

    3 Maphunziro
    Laplace's equation and harmonic functionsMaximum principles and uniquenessGreen's functions and the fundamental solution
    Kulowa kuti atenge chidziwitso

  5. Fourier transforms and energy methods

    2 Maphunziro
    The Fourier transform and the heat kernelEnergy methods: uniqueness, stability and decay
    Kulowa kuti atenge chidziwitso

  6. Distributions and Sobolev spaces

    2 Maphunziro
    Distributions and weak derivativesSobolev spaces and embeddings
    Kulowa kuti atenge chidziwitso

  7. Well-posedness, estimates and scaling

    3 Maphunziro
    Well-posedness in the sense of HadamardA priori estimates, local and global existenceScaling and criticality: why dimension matters
    Kulowa kuti atenge chidziwitso

The final exam

Mafunso a m’gulu lililonse, ovuta kwambiri ndi omwe amamaliza. Mafunso olemba amadziwika ndi dongosolo la algebra; zitsimikiziro ndi ntchito zojambula zithunzi zimawerengedwa ndi model; kenako munthu amafufuza nkhani yonseyo m’mbuyomu kuti zotsatira zanu zituluke. Kusintha: 70%.

Kulowa

Mitengo

Course pass

$39

M’chilankhulo chilichonse pali mafunso a unit, chiyeso cha finali ndi mfundo imodzi yopitanso, ndi 20 zoyeserera za ntchito yanu, kwa m’chilankhulo chimodzi.

One final exam

$15

Kukhala ndi nthawi yokha ya kumaliza kwa phunziro limodzi, kumarked ndi kufufuzidwa. Kwa munthu amene kale amadziwa zinthuzo ndipo akufuna kuti ayesedwe.

Mfundo zonse

$19/mphindi

Zomwe zimachitika pazokambirana zonse ndi zomaliza pazokambirana zonse (zomaliza ziwiri pazokambirana mwezi uliwonse) ndi 100 zoyeserera zogwira ntchito mwezi uliwonse.

Kugwira ntchito zoyeserera

$9

Kuyesa kwa makumi asanu: kutsitsa chithunzi cha ntchito iliyonse, kapena kuyika, ndikuwona komwe kunali koipa.

Maphunziro atsegulidwa kuti agwiritse ntchito. Maphunziro ndi chidziwitso choyamba cha maphunziro onse atsegulidwa pano.

Onani gawo la ntchito yanu yokha