Kursi · 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 %s: Mitir 19 Waxbarasho 121 su'aalo 180 daqiiqo

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.

Ka dib: Multivariable-Calculus, Differential-Equations, Linear-Algebra, Analysis

Kursiga

  1. Classification and first-order equations

    3 Waxbarasho
    What a PDE is, and how to classify oneFirst-order linear equations and characteristicsNonlinear first-order equations and shocks
    Ku soo gal si aad u qaadato imtixaankan

  2. The heat equation and Fourier series

    4 Waxbarasho
    The heat equation: where it comes fromSeparation of variablesFourier series: coefficients, convergence and ParsevalThe heat equation on an interval
    Ku soo gal si aad u qaadato imtixaankan

  3. The wave equation

    2 Waxbarasho
    The wave equation and d'Alembert's formulaThe vibrating string: modes and energy
    Ku soo gal si aad u qaadato imtixaankan

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

    3 Waxbarasho
    Laplace's equation and harmonic functionsMaximum principles and uniquenessGreen's functions and the fundamental solution
    Ku soo gal si aad u qaadato imtixaankan

  5. Fourier transforms and energy methods

    2 Waxbarasho
    The Fourier transform and the heat kernelEnergy methods: uniqueness, stability and decay
    Ku soo gal si aad u qaadato imtixaankan

  6. Distributions and Sobolev spaces

    2 Waxbarasho
    Distributions and weak derivativesSobolev spaces and embeddings
    Ku soo gal si aad u qaadato imtixaankan

  7. Well-posedness, estimates and scaling

    3 Waxbarasho
    Well-posedness in the sense of HadamardA priori estimates, local and global existenceScaling and criticality: why dimension matters
    Ku soo gal si aad u qaadato imtixaankan

Imtixaankii ugu dambeeyay

Su'aalaha ka mid ah mid kasta oo ka mid ah, kuwa ugu adag ee ugu dambeeyay. jawaabaha la daabacay waxaa lagu calaamadeeyay nidaamka algebra; caddayn iyo sawirka shaqada waxaa akhriya qaab; markaas qof ayaa dib u eegista warbixinta oo dhan ka hor inta aad natiijooyinka la sii daayay. Calaamada ka gudubka: 70%.

Ku soo gal

Qiimaha

Kursi

$39

Mid kasta oo iskuulka, imtixaanka ugu dambeeyay oo leh mid ka mid ah inta kale, iyo 20 tijaabooyinka shaqadaada, si ay u mid ka mid ah.

Iibso fasalka

Imtixaan dhamaadka ah

$15

A fadhi kaliya ee dhamaadka mid ka mid ah fasalka, calaamadeeyay oo dib loo eego. Si qof ka horeeya u yaqaan waxyaabaha iyo doonista in ay la tijaabiyay.

Iibso mid dhamaadka ah

Kursi kasta

$19/mi

Dhammaan su'aalaha iyo finals in fasal kasta (laba finals fasal kasta bishii) iyo 100 shahaadooyinka shaqada bishii. Cancel ka xisaabtaada.

Ku biir

Shaqada baaritaan

$9

Toban tijaabooyin: soo deji sawir kasta oo shaqada ah, ama ku qor, oo eeg meesha ay ku dhacay.

Iibso toban warqad

Ka hubi qayb ka mid ah shaqadaada