Natuurlik · 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 eenhede 19 lesse 121 vrae 180 minuut finale

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.

Kome na: Multivariable-Calculus, Differential-Equations, Linear-Algebra, Analysis

Die kursus

  1. Classification and first-order equations

    3 lesse
    What a PDE is, and how to classify oneFirst-order linear equations and characteristicsNonlinear first-order equations and shocks
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  2. The heat equation and Fourier series

    4 lesse
    The heat equation: where it comes fromSeparation of variablesFourier series: coefficients, convergence and ParsevalThe heat equation on an interval
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  3. The wave equation

    2 lesse
    The wave equation and d'Alembert's formulaThe vibrating string: modes and energy
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  4. Laplace's equation, maximum principles and Green's functions

    3 lesse
    Laplace's equation and harmonic functionsMaximum principles and uniquenessGreen's functions and the fundamental solution
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  5. Fourier transforms and energy methods

    2 lesse
    The Fourier transform and the heat kernelEnergy methods: uniqueness, stability and decay
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  6. Distributions and Sobolev spaces

    2 lesse
    Distributions and weak derivativesSobolev spaces and embeddings
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  7. Well-posedness, estimates and scaling

    3 lesse
    Well-posedness in the sense of HadamardA priori estimates, local and global existenceScaling and criticality: why dimension matters
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Die finale eksamen

Vrae van elke eenheid, die moeilikstes laaste. Getipe antwoorde word deur die algebra-stelsel aangedui; bewyse en gefotografeerde werk word deur 'n model gelees; dan hersien 'n persoon die hele papier voordat jou resultaat vrygestel word. Gee en jy kry' n sertifikaat met sy eie skakel. Herhaling merk: 70%.

Meld aan

Pryse

Natuurlik verby

$39

Elke eenheid vra, die finale eksamen met een herhaal en 20 tjeks van jou eie werk, vir een kursus.

Een finale eksamen

$15

Vir iemand wat reeds die materiaal ken en wil hê dat dit getoets moet word, is dit'n teken en hersiening.

Elke kursus

$19/mo

Alle vasvrae en finale in elke kursus (twee punte per kursus per maand) en 100 werkskontroles per maand. Kanselleer van jou rekening.

Werkskontroles

$9

Tien tjeks: Laai'n foto op van enige werk, of tik dit, en kyk waar dit verkeerd geloop het.

Die lesse en die eerste vasvra van elke kursus is nou oop.

Gaan 'n stuk van jou eie werk na