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Fourier Series and PDEs

Year 2 · 5 sections · 47 concepts · 102 practice questions

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What's covered

free1 · Fourier series11 concepts

Overview 1 · The plan of this section
Proposition 1.1 · Orthogonality of the trigonometric system
Definition 1.2 · Fourier series
Example 1.3 · The square wave and the sawtooth
Theorem 1.4 · Dirichlet's convergence theorem
Proposition 1.5 · Sine and cosine series
Definition 1.6 · Complex Fourier series
Theorem 1.7 · Parseval's identity
Example 1.8 · The Gibbs phenomenon
Proposition 1.9 · Smoothness and decay of coefficients
Check 1.10 · Check: computing Fourier series

2 · Separation of variables10 concepts

Overview 2 · The plan of this section
Definition 2.1 · The heat equation
Proposition 2.2 · The superposition principle
Theorem 2.3 · Heat equation with zero end temperatures
Example 2.4 · Insulated ends
Proposition 2.5 · Inhomogeneous boundary conditions via the steady state
Theorem 2.6 · The vibrating string
Example 2.7 · Laplace's equation on a rectangle
Proposition 2.8 · Laplace's equation in a disc
Check 2.9 · Check: solving by separation of variables

3 · Fourier transform9 concepts

Overview 3 · The plan of this section
Definition 3.1 · The Fourier transform
Theorem 3.2 · Fourier inversion
Example 3.3 · The Gaussian is its own transform
Proposition 3.4 · Properties of the Fourier transform
Theorem 3.5 · The convolution theorem
Theorem 3.6 · Plancherel's theorem
Theorem 3.7 · The heat equation on the line
Check 3.8 · Check: solving PDEs by Fourier transform

4 · First-order PDEs8 concepts

Overview 4 · The plan of this section
Example 4.1 · The transport equation
Definition 4.2 · Characteristic curves
Theorem 4.3 · Solving linear first-order PDEs by characteristics
Example 4.4 · A PDE in x and y
Proposition 4.5 · Quasi-linear equations
Example 4.6 · Shock formation in Burgers' equation
Check 4.7 · Check: the method of characteristics

5 · Classification9 concepts

Overview 5 · The plan of this section
Definition 5.1 · Classification of second-order PDEs
Proposition 5.2 · Canonical forms for constant coefficients
Theorem 5.3 · d'Alembert's solution of the wave equation
Corollary 5.4 · Finite speed of propagation
Proposition 5.5 · Energy conservation for the wave equation
Theorem 5.6 · Maximum principle for the heat equation
Corollary 5.7 · Uniqueness and stability for the heat equation
Check 5.8 · Check: matching method to type

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Proposition 1.1 · Orthogonality of the trigonometric system

$\{1, \cos\frac{n\pi x}L, \sin\frac{n\pi x}L\}$ are orthogonal on $[-L, L]$ ; each has squared norm $L$ (the constant: $2L$ ).

Proof

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