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.

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.

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.

Working checks

$9

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

Courses open for purchase soon. The lessons and the first quiz of every course are open now.

Check a piece of your own working