Partial Differential Equations

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• What a PDE is: order, linearity, homogeneity ~20 min
• The superposition principle for linear PDEs ~16 min
• Physical derivation: the heat equation ~22 min
• Physical derivation: the wave equation ~20 min
• Physical derivation: Laplace's equation ~18 min
• Initial and boundary conditions: Dirichlet, Neumann, Robin ~20 min
• Classifying second-order PDEs: the discriminant test ~18 min
• Well-posedness: Hadamard's three conditions ~20 min
• A map of the course: which technique solves which kind of problem ~16 min

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• Linear first-order PDEs and the associated ODE ~22 min
• Solving by characteristics: constant-coefficient case ~24 min
• Variable-coefficient characteristics ~24 min
• The initial-value (Cauchy) problem for first-order PDEs ~22 min
• When characteristics fail: tangency and non-uniqueness ~22 min
• Quasilinear first-order PDEs ~24 min
• Conservation laws and the traffic-flow model ~24 min
• Shock formation: characteristics crossing ~24 min
• The Rankine-Hugoniot jump condition, informally ~22 min
• Using characteristics to reduce a 2nd-order PDE to canonical form ~26 min

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• Orthogonality of the sine and cosine functions ~22 min
• Fourier sine series ~24 min
• Fourier cosine series ~22 min
• The full Fourier series ~24 min
• Even, odd, and periodic extensions ~20 min
• Complex Fourier series ~22 min
• Convergence of a Fourier series at a point ~20 min
• The Gibbs phenomenon ~18 min
• Term-by-term differentiation and integration ~22 min
• Parseval's equality and mean-square convergence ~22 min

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• The regular Sturm-Liouville problem ~24 min
• Orthogonality of eigenfunctions ~26 min
• Reality and positivity of eigenvalues ~26 min
• Eigenfunction expansions (generalized Fourier series) ~26 min
• Convergence of an eigenfunction expansion ~22 min
• The Rayleigh quotient ~22 min
• Estimating the lowest eigenvalue ~24 min
• Singular Sturm-Liouville problems, previewed ~20 min
• Why this is the engine behind separation of variables ~18 min

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• The separation ansatz: splitting a PDE into two ODEs ~24 min
• From boundary conditions to an eigenvalue problem ~22 min
• The heat equation on a finite rod, Dirichlet conditions ~26 min
• The heat equation, Neumann conditions ~24 min
• The heat equation, Robin conditions ~24 min
• Nonhomogeneous boundary data: shifting to homogeneous ~22 min
• The heat equation with a source term ~24 min
• Heat conduction in two and three dimensions ~24 min
• Steady-state solutions and equilibrium ~22 min
• Choosing the right series for a given problem ~22 min

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• Separation of variables for the vibrating string ~24 min
• Initial position and initial velocity together ~24 min
• Energy of a vibrating string ~22 min
• Conservation of energy and uniqueness ~24 min
• Reflections at a boundary ~22 min
• The vibrating rectangular membrane ~24 min
• Nodal lines and vibration modes ~20 min
• Forced vibrations and resonance on a bounded domain ~24 min
• Comparing the heat and wave equations, revisited ~22 min

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• Laplace's equation, revisited ~20 min
• Separation of variables on a rectangle ~24 min
• Separation of variables on a disk ~26 min
• The Poisson kernel, derived ~22 min
• Laplace's equation on a wedge and an annulus ~24 min
• The Neumann problem and its solvability condition ~20 min
• Superposing to handle all four sides of a rectangle at once ~20 min
• Laplace's equation in a box (3-D separation) ~24 min
• Recognizing when separation of variables applies ~16 min

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• D'Alembert's formula for the wave equation ~22 min
• Domain of dependence, domain of influence, and causality ~18 min
• The heat kernel: a self-similarity derivation ~24 min
• The heat equation on the whole line ~20 min
• Duhamel's principle for the heat equation ~20 min
• Duhamel's principle for the wave equation ~20 min
• The Fourier transform on the whole line ~22 min
• Solving the heat equation by Fourier transform ~20 min
• The Laplace transform in time, revisited for PDEs ~20 min
• Choosing between series, transform, and characteristic methods ~16 min
• Causality in three dimensions: the solid light cone ~24 min
• The method of descent, Kirchhoff's formula, and Huygens' principle ~24 min

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• Separating the Laplacian eigenvalue problem in polar coordinates ~20 min
• Bessel's equation and Bessel functions ~22 min
• Properties of Bessel functions: zeros, recurrence, orthogonality ~22 min
• The vibrating circular drumhead ~24 min
• Separating the Laplacian eigenvalue problem in spherical coordinates ~22 min
• Legendre's equation and Legendre polynomials ~22 min
• Orthogonality of Legendre polynomials, two ways ~20 min
• Spherical harmonics ~22 min
• Heat and wave problems on a disk or ball ~22 min
• Choosing coordinates to match a domain's symmetry ~16 min

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• The Green's function for the Dirichlet problem, revisited ~18 min
• The method of images: half-space and ball ~22 min
• The method of images: additional domains (slab, wedge) ~20 min
• The heat kernel as a time-dependent Green's function ~16 min
• The Green's function for the wave equation ~18 min
• Duhamel's principle, unified via Green's functions ~18 min
• The Dirac delta as a source term ~16 min
• Representing a solution as a convolution ~20 min
• When a Green's function does and doesn't exist in closed form ~16 min

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• Writing a first-order system in matrix form ~18 min
• Hyperbolicity: real, distinct eigenvalues ~18 min
• Diagonalizing to canonical form ~20 min
• Characteristics for a system ~18 min
• Riemann invariants ~16 min
• Simple waves ~16 min
• Linear transmission-line and acoustics examples ~20 min
• From linear to quasilinear systems ~18 min
• Isentropic gas flow: Riemann invariants and shock formation ~30 min

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• Weak solutions, informally ~18 min
• The Rankine-Hugoniot condition, revisited in full ~18 min
• Entropy conditions and admissible shocks ~20 min
• Rarefaction waves ~18 min
• The traffic-flow model, worked through ~20 min
• Burgers' equation ~20 min
• The calculus of variations connection: Dirichlet's principle, revisited ~18 min
• A survey of what lies beyond this course ~14 min

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• Why numerical methods, and what can go wrong ~16 min
• Finite differences for the heat equation: the explicit scheme ~18 min
• Stability of the explicit scheme ~20 min
• The Crank-Nicolson scheme ~20 min
• Finite differences for the wave equation ~18 min
• The CFL condition ~20 min
• Finite differences for Laplace's equation: the five-point stencil ~18 min
• Iterative solvers: Jacobi, Gauss-Seidel, SOR ~20 min
• A first look at the finite element method ~16 min