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.

Gulani kuphunzira kwa kuphunzira

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.

Gulani imodzi yachiwiri

Mfundo zonse

$19/mphindi

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

Kulembetsa

Kugwira ntchito zoyeserera

$9

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

Gulani ma cheque asanu

Onani gawo la ntchito yanu yokha