Unit 12 of 13

Fourier Methods and Partial Differential Equations

  • 8 lessons
  • Videos 67 to 74

Fourier series expand a function in sines and cosines and lead to partial differential equations. This unit covers Fourier series, transport, heat, wave and Laplace equations and Fourier transforms.

Fourier coefficients are orthogonal projections, and at a jump of a piecewise smooth function the series converges to the midpoint of the jump, with an overshoot nearby. Second-order equations are elliptic, parabolic or hyperbolic, and a well-posed problem has a unique solution depending continuously on data. Characteristics solve transport equations. Separation of variables gives decaying heat and oscillating wave modes, d'Alembert's formula shows finite propagation speed, and the maximum principle gives uniqueness for Laplace's equation. Fourier transforms solve diffusion on the whole line.

Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Fourier Series

  1. Fourier Series and Orthogonal Expansions

    How Fourier coefficients arise as orthogonal projections onto sines and cosines of period 2L, why the partial sums are the best mean-square approximations, and how Parseval's identity accounts for a function's squared size.

    Video 67

  2. Fourier Convergence, Gibbs Phenomenon, and Half-Range Series

    Where the Fourier series of a piecewise smooth function converges and to what, why it gives the average of the one-sided limits at a jump, what the Gibbs overshoot is, and how odd and even extensions lead to half-range series.

    Video 68

PDE Classification and Transport

  1. PDE Types, Conditions, and Well-Posedness

    How the principal part makes a second-order equation in two variables elliptic, parabolic or hyperbolic, which data fit each type, and why a well-posed problem needs compatible data and continuous dependence on the data.

    Video 69

  2. First-Order Transport and Characteristics

    How the method of characteristics reduces transport equations to ODEs along curves, where inflow boundary data belong, and how variable speed, a source term and Burgers' equation with weak solutions change the picture.

    Video 70

Heat, Wave and Laplace Equations

  1. The Heat Equation and Separation of Variables

    How the heat equation and thermal diffusivity model diffusion, how separation of variables builds solutions from Dirichlet and Neumann modes, why insulated ends conserve the mean, and how boundary lifting handles end temperatures.

    Video 71

  2. The Wave Equation, Normal Modes, and d'Alembert's Formula

    How the wave equation uses initial displacement and velocity, how standing modes solve it with fixed or free ends, and how traveling waves and d'Alembert's formula give the domain of dependence and finite propagation speed.

    Video 72

  3. Laplace and Poisson Equations

    How Laplace's and Poisson's equations describe steady states, how separation solves them in a rectangle and a disk, why the maximum principle makes Dirichlet solutions unique, and why pure Neumann data need compatibility.

    Video 73

Fourier Transforms on the Whole Line

  1. Fourier Transforms and PDEs on Unbounded Domains

    How a stated Fourier transform convention and its inverse solve the heat equation on the whole line, how derivatives and Gaussians behave under it, what the heat kernel shows, and which decay conditions each step needs.

    Video 74