Equations relating a function to its derivatives. Ordinary (ODE, single independent variable) vs partial (PDE, multiple variables). The mathematical language of classical mechanics (Newton's F=ma is an ODE), electromagnetism (Maxwell's are…
differential-equations
Ordinary differential equation (ODE)
An equation F(x, y, y', ..., y^(n)) = 0 relating a univariate function y(x) to its derivatives. Order = highest derivative.
Initial value problem (IVP)
An ODE together with values y(x_0), y'(x_0), ..., y^{(n-1)}(x_0) at a single point. Determines a unique solution under mild hypotheses.
Picard-Lindelöf theorem
If f(x, y) is Lipschitz in y on a neighborhood of (x_0, y_0), the IVP y' = f(x,y), y(x_0) = y_0 has a unique solution on some interval.
Linear ODE
Of the form a_n(x) y^(n) + ... + a_1(x) y' + a_0(x) y = g(x). Solution space is an affine subspace: general = homogeneous + particular.…
Laplace transform
Integral transform L{f}(s) = ∫_0^∞ e^{-st} f(t) dt. Converts linear ODEs with constant coefficients into algebraic equations in s; key tool…
Partial differential equation (PDE)
An equation involving partial derivatives of an unknown function u(x_1, ..., x_n). Classified by order and linearity; second-order linear…
Heat equation
u_t = α∇²u. Parabolic second-order PDE describing diffusion of heat (or probability density) in a medium. Solutions smooth instantaneously…
Wave equation (PDE)
u_tt = c²∇²u. Hyperbolic second-order PDE governing sound, light, strings. Solutions propagate at finite speed c with no smoothing.
Laplace equation
∇²u = 0. Elliptic PDE for equilibrium/steady-state fields: electrostatic potential in vacuum, incompressible flow, harmonic functions.…
Method of characteristics
Solves first-order PDEs by reducing to a family of ODEs along characteristic curves along which the unknown is constant or satisfies a…
Separation of variables
Ansatz u(x,t) = X(x)T(t) converts certain linear PDEs on product domains into systems of ODEs with eigenvalue coupling. Foundation of…
Fourier series
A periodic function with period 2π can be expanded as Σ(a_n cos nx + b_n sin nx), with coefficients given by orthogonality integrals.…
Fourier transform
ℱ{f}(ξ) = ∫ f(x) e^{-2πiξx} dx. Continuous analog of Fourier series; moves between position/time and frequency domains. Invertible on L¹ ∩…
Sturm–Liouville theory
Second-order ODE eigenvalue problem (py')' + qy + λwy = 0 on [a,b] with homogeneous boundary conditions; eigenvalues are real, countable,…
Bessel functions J_ν, Y_ν
Solutions to Bessel's equation x²y″ + xy′ + (x² − ν²)y = 0. Arise from separation of variables in cylindrical coordinates —…
Legendre polynomials P_n
Orthogonal polynomials on [−1,1] solving (1−x²)y″ − 2xy′ + n(n+1)y = 0. Generated by 1/√(1−2xt+t²); spherical harmonics of ℝ³.
Poisson equation
The elliptic PDE −Δu = f on Ω (plus boundary data). Classical solvability via Green's functions; variational formulation via minimising…
Green's function
A fundamental solution G(x,y) to a linear differential operator L satisfying L_x G(x,y) = δ(x−y) with prescribed boundary conditions. …
Weak solution
A function u solving a PDE only in the distributional/Sobolev sense: ∫ u L*φ = ∫ f φ for all test functions φ. Enables existence in L² or…
Sobolev embedding theorem
W^{k,p}(ℝ^n) embeds into L^q for 1/q = 1/p − k/n (when positive), or into C^{k−[n/p], α} when k>n/p. Backbone of elliptic regularity and…
Hamilton–Jacobi equation
First-order nonlinear PDE ∂_t S + H(x, ∇S, t) = 0 for the action S. Characteristics coincide with Hamilton's equations; Crandall–Lions…
Burgers' equation
The nonlinear PDE u_t + uu_x = νu_{xx}. ν > 0: dissipative; ν → 0⁺: inviscid limit develops shocks. A model problem for gas dynamics and…
Reaction-diffusion equation
u_t = D Δu + R(u): linear diffusion plus nonlinear local reaction. Generates Turing patterns, travelling waves (KPP/Fisher), combustion…
Helmholtz equation
Time-independent form of the wave equation: (∇² + k²)u = 0, where k is the wavenumber. Arises from separating time dependence u(x,t) =…
Sturm-Liouville theory
Regular SL problem -(p y')' + q y = λ w y with BCs → complete orthogonal eigenbasis. Template for PDE separation; Bessel, Legendre, Hermite.
Green's function
Impulse response satisfying L G(x,ξ) = δ(x-ξ) + BCs; solution to inhomogeneous problem is u(x) = ∫ G(x,ξ) f(ξ) dξ. Ubiquitous in physics.
Peano existence theorem
Continuous f ⟹ existence (not uniqueness) of local solution. Uses Arzelà-Ascoli compactness of approximating broken-line solutions.
Stable / unstable manifold
Hyperbolic fixed point of flow: W^s = local invariant manifold of orbits converging forward. Stable manifold theorem (Hadamard-Perron).
Hopf bifurcation
System ẋ = f(x,μ): complex-conjugate eigenvalues cross imaginary axis → birth of periodic orbit. Normal form classification;…
Navier-Stokes (open)
Existence + smoothness of global solutions to 3D incompressible NS from smooth data. Millennium problem; 2D resolved (Leray 1934).
Method of characteristics
First-order quasilinear PDE a u_x + b u_y = c: reduce to ODE on characteristic curves. Shock formation via characteristic crossing.
Hyperbolic conservation laws
u_t + f(u)_x = 0. Weak solutions (Rankine-Hugoniot); entropy conditions (Lax, Kruzhkov) select physical shocks. Burgers' u_t + uu_x = 0.
Viscosity solutions
Crandall-Lions 1983 notion for fully nonlinear 1st/2nd order PDE. Handles non-differentiable solutions; HJB equations, optimal control.
Schauder estimates
Elliptic a^{ij} D_i D_j u + b^i D_i u + cu = f with Hölder coefficients: ||u||_{C^{2,α}} ≤ C(||f||_{C^α} + ||u||_C⁰). Regularity bootstrap.
De Giorgi-Nash-Moser
Weak solutions to divergence-form elliptic equations with bounded measurable coefficients are Hölder continuous. Solves Hilbert's 19th.
Strichartz estimates
||e^{it Δ} f||_{L^q_t L^r_x} ≤ C ||f||_{L²} for admissible (q,r). Dispersive PDE: Schrödinger, wave, KdV, NLS wellposedness.
Strongly-continuous semigroups (C_0)
T(t): X→X with T(0)=I, T(t+s)=T(t)T(s), strong continuity. Hille-Yosida: A generator iff closed, densely defined, resolvent bounds. Models…
Grönwall's inequality
If u, β are non-negative continuous on [0, T] and u(t) ≤ α + ∫₀ᵗ β(s)u(s) ds, then u(t) ≤ α · exp(∫₀ᵗ β(s) ds). Grönwall 1919 (integral…
Variation of parameters
Lagrange's technique to solve inhomogeneous linear ODEs given a fundamental set (y_1, y_2) of the homogeneous equation. Wronskian W = y_1…
Duhamel's principle
Solution of an inhomogeneous linear evolution equation as a convolution with the homogeneous Green's function. Scalar, matrix, and…
Hartman-Grobman theorem
Near a hyperbolic fixed point (no eigenvalues of Df(x*) on the imaginary axis), a smooth flow is locally topologically conjugate (C⁰) to…
Poincaré-Bendixson theorem
For a C¹ planar flow, any non-empty compact ω-limit set that contains no fixed points is a periodic orbit. Poincaré 1881, Bendixson 1901. …
Lyapunov stability (direct method)
Lyapunov's direct method: if V is positive definite and V̇ = ∇V·f is negative semidefinite near a fixed point x*, then x* is stable; strict…
Heat kernel
Fundamental solution of the heat equation u_t = Δu on ℝⁿ: K(t,x) = (4πt)^{−n/2} exp(−|x|²/(4t)). Gaussian semigroup e^{tΔ}. Solves u_t =…
Maximum principle (elliptic)
Weak form: a subharmonic function u (Δu ≥ 0) on a bounded domain attains its supremum on the boundary. Strong form (E. Hopf 1927): if u…