Courses · 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 units 19 lessons 121 questions 180 minute final

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.

Comes after: Multivariable-Calculus, Differential-Equations, Linear-Algebra, Analysis

The course

  1. Classification and first-order equations

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

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

    3 lessons
    Well-posedness in the sense of HadamardA priori estimates, local and global existenceScaling and criticality: why dimension matters
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The final exam

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. Pass mark: 70%.

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Prices

Course pass

$39

Every unit quiz, the final exam with one retake, and 20 checks of your own working, for one course.

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One final exam

$15

A single sitting of one course's final, marked and reviewed. For someone who already knows the material and wants it tested.

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Every course

$19/mo

All quizzes and finals in every course (two finals per course a month) and 100 working checks a month. Cancel from your account.

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Working checks

$9

Ten checks: upload a photo of any working, or type it, and see where it went wrong.

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Check a piece of your own working