List of named differential equations

Differential equations play a prominent role in many scientific areas: mathematics, physics, engineering, chemistry, biology, medicine, economics, etc. This list presents differential equations that have received specific names, area by area.

Mathematics

 * Ablowitz-Kaup-Newell-Segur (AKNS) system
 * Clairaut's equation
 * Hypergeometric differential equation
 * Jimbo–Miwa–Ueno isomonodromy equations
 * Painlevé equations
 * Picard–Fuchs equation to describe the periods of elliptic curves
 * Schlesinger's equations
 * Sine-Gordon equation
 * Sturm–Liouville theory of orthogonal polynomials and separable partial differential equations
 * Universal differential equation

Algebraic geometry

 * Calabi flow in the study of Calabi-Yau manifolds

Complex analysis

 * Cauchy–Riemann equations

Differential geometry

 * Equations for a minimal surface
 * Liouville's equation
 * Ricci flow, used to prove the Poincaré conjecture
 * Tzitzeica equation

Dynamical systems and Chaos theory

 * Rabinovich–Fabrikant equations

Mathematical physics

 * General Legendre equation
 * Heat equation
 * Laplace's equation in potential theory
 * Poisson's equation in potential theory

Ordinary Differential Equations (ODEs)

 * Bernoulli differential equation
 * Cauchy–Euler equation
 * Riccati equation
 * Hill differential equation

Riemannian geometry

 * Gauss–Codazzi equations

Astrophysics

 * Chandrasekhar's white dwarf equation
 * Lane-Emden equation
 * Emden–Chandrasekhar equation
 * Hénon–Heiles system

Classical mechanics
• Equation of motion

• Euler's rotation equations in rigid body dynamics

• Euler–Lagrange equation

• * Beltrami identity

• Hamilton's equations

• Hamilton-Jacobi equation

• Lorenz equations in chaos theory

• n-body problem in celestial mechanics

• Wave action in continuum mechanics

Electromagnetism

 * Continuity equation for conservation laws
 * Maxwell's equations
 * Poynting's theorem

Fluid dynamics and hydrology
• Acoustic theory

• Benjamin–Bona–Mahony equation

• Biharmonic equation

• Blasius boundary layer

• Boussinesq approximation (buoyancy)

• Boussinesq approximation (water waves)

• Buckley–Leverett equation

• Camassa–Holm equation

• Chaplygin's equation

• Continuity equation for conservation laws

• Convection–diffusion equation

• * Double diffusive convection

• Davey–Stewartson equation

• Euler–Tricomi equation

• Falkner–Skan boundary layer

• Gardner equation in hydrodynamics

• General equation of heat transfer

• Geophysical fluid dynamics

• * Potential vorticity

• * Quasi-geostrophic equations

• * Shallow water equations

• * Taylor–Goldstein equation

• Groundwater flow equation

• * Richards equation

• Hicks equation

• Kadomtsev–Petviashvili equation in nonlinear wave motion

• KdV equation

• Magnetohydrodynamics

• *Grad–Shafranov equation

• Navier–Stokes equations

• *Euler equations

• *Burgers' equation

• Nonlinear Schrödinger equation in water waves

• Orr–Sommerfeld equation

• Porous medium equation

• Potential flow

• Rayleigh–Bénard convection

• Rayleigh–Plesset equation

• Reynolds-averaged Navier–Stokes (RANS) equations

• Reynolds transport theorem

• Riemann problem

• Taylor–von Neumann–Sedov blast wave

• Turbulence modeling

• * Turbulence kinetic energy (TKE)

• * K-epsilon turbulence model

• * k–omega turbulence model

• * Spalart–Allmaras turbulence model

• Vorticity equation

• Whitham equation

• Zebiak-Cane model for El Niño–Southern Oscillation

• Zeldovich–Taylor flow

General relativity

 * Einstein field equations
 * Friedmann equations
 * Geodesic equation
 * Mathisson–Papapetrou–Dixon equations
 * Schrödinger–Newton equation

Materials science

 * Ginzburg–Landau equations in superconductivity
 * London equations in superconductivity
 * Poisson–Boltzmann equation in molecular dynamics

Nuclear physics

 * Radioactive decay equations

Plasma physics

 * Gardner equation
 * Hasegawa–Mima equation
 * KdV equation
 * Kuramoto–Sivashinsky equation
 * Vlasov equation

Quantum mechanics and quantum field theory

 * Dirac equation, the relativistic wave equation for electrons and positrons
 * Gardner equation
 * Klein–Gordon equation
 * Knizhnik–Zamolodchikov equations in quantum field theory
 * Nonlinear Schrödinger equation in quantum mechanics
 * Schrödinger's equation
 * Schwinger–Dyson equation
 * Yang-Mills equations in gauge theory

Thermodynamics and statistical mechanics

 * Boltzmann equation
 * Continuity equation for conservation laws
 * Diffusion equation
 * Heat equation
 * Kardar-Parisi-Zhang equation
 * Kuramoto–Sivashinsky equation
 * Liñán's equation as a model of diffusion flame
 * Maxwell relations
 * Zeldovich–Frank-Kamenetskii equation to model flame propagation

Waves (mechanical or electromagnetic)

 * D'Alembert's wave equation
 * Eikonal equation in wave propagation
 * Euler–Poisson–Darboux equation in wave theory
 * Helmholtz equation

Electrical and Electronic Engineering

 * Chua's circuit
 * Liénard equation to model oscillating circuits
 * Nonlinear Schrödinger equation in fiber optics
 * Telegrapher's equations
 * Van der Pol oscillator

Game theory

 * Differential game equations

Mechanical engineering

 * Euler–Bernoulli beam theory
 * Timoshenko beam theory

Nuclear engineering

 * Neutron diffusion equation

Optimal control

 * Linear-quadratic regulator
 * Matrix differential equation
 * PDE-constrained optimization
 * Riccati equation
 * Shape optimization

Orbital mechanics

 * Clohessy–Wiltshire equations

Signal processing

 * Filtering theory
 * Kushner equation
 * Zakai equation
 * Rudin-Osher-Fatemi equation in total variation denoising

Transportation engineering

 * Law of conservation in the kinematic wave model of traffic flow theory

Chemistry

 * Allen–Cahn equation in phase separation
 * Cahn–Hilliard equation in phase separation
 * Chemical reaction model
 * Brusselator
 * Oregonator
 * Master equation
 * Rate equation
 * Streeter–Phelps equation in water quality modeling

Biology and medicine
• Allee effect in population ecology

• Chemotaxis in wound healing

• Compartmental models in epidemiology

• * SIR model

• * SIS model

• Hagen–Poiseuille equation in blood flow

• Hodgkin–Huxley model in neural action potentials

• Kardar–Parisi–Zhang equation for bacteria surface growth models

• Kermack-McKendrick theory in infectious disease epidemiology

• Kuramoto model in biological and chemical oscillations

• Mackey-Glass equations

• McKendrick–von Foerster equation in age structure modeling

• Nernst–Planck equation in ion flux across biological membranes

• Price equation in evolutionary biology

• Reaction-diffusion equation in theoretical biology

• * Fisher–KPP equation in nonlinear traveling waves

• * FitzHugh–Nagumo model in neural activation

• Replicator dynamics in theoretical biology

• Verhulst equation in biological population growth

• von Bertalanffy model in biological individual growth

• Wilson–Cowan model in computational neuroscience

Population dynamics

 * Arditi–Ginzburg equations to describe predator–prey dynamics
 * Kolmogorov–Petrovsky–Piskunov equation (also known as Fisher's equation) to model population growth
 * Lotka–Volterra equations to describe the dynamics of biological systems in which two species interact

Economics and finance
• Bass diffusion model

• Black–Scholes equation

• Economic growth

• * Solow–Swan model

• ** $k'(t) = s [k(t)]^\alpha - \delta k(t)$

• * Ramsey–Cass–Koopmans model

• * Dynamic stochastic general equilibrium

• Feynman–Kac formula

• * Black–Scholes equation

• * Affine term structure modeling

• Fokker–Planck equation

• * Dupire equation (local volatility)

• Hamilton–Jacobi–Bellman equation

• * Merton's portfolio problem

• * Optimal stopping

• Malthusian growth model

• Mean field game theory

• Optimal rotation age

• Sovereign debt accumulation

• * $\dot{D} = rD + G(t)-T(t)$

• Stochastic differential equation

• * Geometric Brownian motion

• * Ornstein–Uhlenbeck process

• * Cox–Ingersoll–Ross model

• Vidale–Wolfe advertising model

Linguistics

 * Replicator dynamics in evolutionary linguistics

Military strategy

 * Lanchester's laws in combat modeling