Masomo · 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 ↓ 19 masomo 121 maswali 180 mwisho wa dakika

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.

Nyotamkia baada: Multivariable-Calculus, Differential-Equations, Linear-Algebra, Analysis

Mwendo

  1. Classification and first-order equations

    3 masomo
    What a PDE is, and how to classify oneFirst-order linear equations and characteristicsNonlinear first-order equations and shocks
    Ishara ya kutumia mtihani

  2. The heat equation and Fourier series

    4 masomo
    The heat equation: where it comes fromSeparation of variablesFourier series: coefficients, convergence and ParsevalThe heat equation on an interval
    Ishara ya kutumia mtihani

  3. The wave equation

    2 masomo
    The wave equation and d'Alembert's formulaThe vibrating string: modes and energy
    Ishara ya kutumia mtihani

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

    3 masomo
    Laplace's equation and harmonic functionsMaximum principles and uniquenessGreen's functions and the fundamental solution
    Ishara ya kutumia mtihani

  5. Fourier transforms and energy methods

    2 masomo
    The Fourier transform and the heat kernelEnergy methods: uniqueness, stability and decay
    Ishara ya kutumia mtihani

  6. Distributions and Sobolev spaces

    2 masomo
    Distributions and weak derivativesSobolev spaces and embeddings
    Ishara ya kutumia mtihani

  7. Well-posedness, estimates and scaling

    3 masomo
    Well-posedness in the sense of HadamardA priori estimates, local and global existenceScaling and criticality: why dimension matters
    Ishara ya kutumia mtihani

Mtihani wa mwisho

Questions from every unit, the hardest ones last. Typed answers are marked by the algebra system; proofs and photographed working are read by a model; then a person reviews the whole paper before your result is released. Pass and you get a certificate with its own link. Njia za kupita: 70%.

Ishara Katika

Bei

Mwendo unapita

$39

Kila tafiti, mtihani wa mwisho wenye kufanyiwa tena jambo moja, na uchunguzi 20 wa kazi yako mwenyewe, kwa mtaala mmoja.

Mtihani mmoja wa mwisho

$15

Kiti kimoja cha mwisho cha mtaala mmoja, kilitia alama na kupitiwa.

Kila jambo

$19/mo

Kila mmoja huulizwa maswali na kumaliza katika kila mtaala (humaliza vipindi viwili kwa mwezi) na hundi 100 za kazi kwa mwezi.

Uchunguzi wa kazi

$9

Uchunguzi kumi: Pakua picha ya mtu yeyote anayefanya kazi, au chora aina yake, uone mahali ilipoharibika.

Masomo na maswali ya kwanza ya kila mtaala yanafunguliwa sasa.

Chunguza kipande cha kazi yako mwenyewe