Kursus · 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 unit description in lists 19 lesson 121 soalan 180 minit terakhir

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.

Datang selepas: Multivariable-Calculus, Differential-Equations, Linear-Algebra, Analysis

Kursus

  1. Classification and first-order equations

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

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

    3 lesson
    Well-posedness in the sense of HadamardA priori estimates, local and global existenceScaling and criticality: why dimension matters
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Ujian akhir.

Soalan dari setiap unit, yang paling sukar terakhir. Jawapan yang ditandakan oleh sistem algebra; bukti dan kerja yang difoto dibaca oleh model; kemudian seseorang meninjau keseluruhan kertas sebelum hasil anda dikeluarkan. lulus dan anda mendapat sijil dengan pautan sendiri. Tanda lulus: 70%.

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Harga

Laluan

$39

Setiap unit kuiz, peperiksaan akhir dengan satu retake, dan 20 periksa kerja anda sendiri, untuk satu kursus.

Beli lawatan

Satu peperiksaan akhir

$15

Satu sesi akhir satu kursus, ditanda dan dikaji semula untuk seseorang yang sudah tahu bahan dan mahu diuji.

Beli satu terakhir

Setiap kursus

$19/bulan

Semua soalan dan akhir dalam setiap kursus (dua akhir setiap kursus setiap bulan) dan 100 periksa kerja setiap bulan. Batalkan dari akaun anda.

Langganan

Semak kerja

$9

Sepuluh periksa: muat naik foto apa-apa yang berfungsi, atau taipkannya, dan lihat di mana ia salah.

Beli sepuluh cek

Periksa bahagian kerja awak sendiri.