maths.free › Partial Differential Equations
Partial Differential Equations
Heat, waves, potentials and transport: Fourier methods, energy estimates, Sobolev spaces and scaling.
Leçons
does u = x^2 - y^2 satisfy u_xx + u_yy = 0
Core
First-order linear equations and characteristics
The transport equation, characteristic curves, and turning a PDE into a family of ODEs.
does u = (x-3*t)^2 satisfy u_t + 3*u_x = 0
Advanced
Nonlinear first-order equations and shocks
Burgers' equation, the breaking time, weak solutions, the Rankine-Hugoniot condition and rarefaction fans.
critical points of -2*x*exp(-x^2)
Core
The heat equation: where it comes from
Conservation of energy plus Fourier's law, diffusion as a random walk, and the diffusion length.
does u = exp(-t)*sin(x) satisfy u_t = u_xx
Core
Separation of variables
Product solutions, the eigenvalue problem X'' + λX = 0, and building the general solution by superposition.
y'' + 9y = 0
Core
Fourier series: coefficients, convergence and Parseval
Orthogonality, computing coefficients, what kind of convergence to expect, Gibbs, and Parseval's identity.
fourier series of x from -pi to pi
Core
The heat equation on an interval
Sine series solutions, long-time behaviour, instant smoothing, Neumann conditions and why time only runs forward.
integrate 2*sin(pi*x) dx from 0 to 1
Core
The wave equation and d'Alembert's formula
Factoring the wave operator, travelling waves, d'Alembert's formula and finite speed of propagation.
does u = sin(x - 2*t) satisfy u_tt = 4*u_xx
Core
The vibrating string: modes and energy
Standing waves and harmonics on a finite string, energy conservation, and uniqueness from energy.
integrate sin(x)^2 dx from 0 to pi
Core
Laplace's equation and harmonic functions
Harmonic functions, radial solutions, the mean value property and its consequences.
laplacian of x^3 - 3*x*y^2
Core
Maximum principles and uniqueness
Weak and strong maximum principles for Laplace and heat equations, comparison, uniqueness and stability.
maximum of x*(1-x)
Advanced
Green's functions and the fundamental solution
The fundamental solution of Laplace's equation, Newtonian potentials, Green's functions, images and the 1D case.
laplacian of 1/sqrt(x^2+y^2+z^2)
Advanced
The Fourier transform and the heat kernel
Derivatives become multiplication, the heat equation solved on the line, the Gaussian kernel and Plancherel.
integrate exp(-x^2/4)/sqrt(4*pi) dx from -oo to oo
Core
Energy methods: uniqueness, stability and decay
Multiply by the solution and integrate by parts: L² estimates, uniqueness, stability, Poincaré and exponential decay.
integrate exp(-2*t)*sin(x)^2 dx from 0 to pi
Advanced
Distributions and weak derivatives
Test functions, the Dirac delta, differentiating anything, weak derivatives and weak solutions.
derivative of abs(x)
Advanced
Sobolev spaces and embeddings
Functions with weak derivatives in L^p, when they are bounded or continuous, and the scaling that predicts the exponents.
integrate 4*pi*r^2*(1/16)*r^(-5/2) dr from 0 to 1
Core
Well-posedness in the sense of Hadamard
Existence, uniqueness, continuous dependence, and the classic examples that fail.
limit of exp(n^2)/n as n -> oo
Advanced
A priori estimates, local and global existence
Bounds proved before the solution is known, blow-up in finite time, continuation criteria and Gronwall.
y' = y^2
Advanced
Scaling and criticality: why dimension matters
Scaling symmetries, critical norms, subcritical and supercritical quantities, and what they say about Navier-Stokes.
solve 1 - 3/p = 0
A partial differential equation ties a function of several variables to its rates of change in each of them, which is how diffusion, vibration, gravity and fluid flow are written down. This course takes the four model equations apart by hand (characteristics, separation of variables, Fourier series and transforms, Green's functions), then builds the tools that work when no formula exists: maximum principles, energy methods, weak derivatives, Sobolev spaces, a priori estimates and scaling. Those tools are exactly what you need to read the Navier-Stokes problem.
Questions posées par les gens
What should I know before starting?
Partial derivatives, the divergence theorem and multiple integrals from multivariable calculus; linear second-order ODEs; eigenvalues from linear algebra; and enough analysis to be comfortable with uniform convergence and integrals over infinite intervals.
Why are there so few formulas for solutions?
Explicit solutions exist for linear equations with constant coefficients on simple domains. Almost everything else (curved domains, variable coefficients, nonlinear terms) has none, so the modern subject proves that a solution exists and estimates its size and smoothness without ever writing it down.
How does this course lead to the Navier-Stokes problem?
The Navier-Stokes equations are a heat equation for the velocity with a transport term and a pressure that is found from a Poisson equation. Reading the Millennium problem needs energy estimates, weak solutions, Sobolev spaces, local existence with a blow-up criterion, and the scaling argument that explains why three dimensions is hard. The last six lessons build those.
What do elliptic, parabolic and hyperbolic mean?
Elliptic equations (Laplace) describe equilibrium and smooth everything; parabolic equations (heat) describe diffusion forward in time; hyperbolic equations (waves) carry signals at finite speed and keep their sharp edges. The type decides which data make a sensible problem.
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