课程 · 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 单位 19 经验教训 121 问题 180 定时数

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.

之后的来: Multivariable-Calculus, Differential-Equations, Linear-Algebra, Analysis

课程

  1. Classification and first-order equations

    3 经验教训
    What a PDE is, and how to classify oneFirst-order linear equations and characteristicsNonlinear first-order equations and shocks
    签名参加测验

  2. The heat equation and Fourier series

    4 经验教训
    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 经验教训
    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 经验教训
    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 经验教训
    The Fourier transform and the heat kernelEnergy methods: uniqueness, stability and decay
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  6. Distributions and Sobolev spaces

    2 经验教训
    Distributions and weak derivativesSobolev spaces and embeddings
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  7. Well-posedness, estimates and scaling

    3 经验教训
    Well-posedness in the sense of HadamardA priori estimates, local and global existenceScaling and criticality: why dimension matters
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期末考试

每个单位的问题,最难回答的问题最后。输入的答案以代数系统为标志;证据和照片工作用模型读取;然后一个人在结果发布之前审查整份文件。通过并获得带有其链接的证书。 通行证: 70%.

签署日期

价格

课程通过

$39

每个单元测验,期末考试 一次再考, 和20检查 自己工作, 一次课程。

一次期末考试

$15

单次的一门课程的终极、标记和审查。对于已经知道材料并想要测试材料的人来说,这是一次。

每门课程

$19/以 单位

每门课程的所有测验和决赛(每月两次决赛)以及每月100次工作检查。取消您的账户。

工作检查

$9

十次检查:上传任何工作的照片,或输入它,看看它哪里出错。

课程开放后很快开始购买。每个课程的课目和第一个测验现在都开放。

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