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.

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.

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.

Shaqada baaritaan

$9

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

Kursiyada furan iibka soon. Waxbarashada iyo su'aalaha ugu horeysay ee fasal kasta oo hadda waa la furay.

Ka hubi qayb ka mid ah shaqadaada