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

Lorenz equations
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

Physics

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

Bloch equations
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
Omega equation
Orr–Sommerfeld equation
Porous medium equation
Potential flow
Rayleigh–Bénard convection
Rayleigh–Plesset equation
Reynolds-averaged Navier–Stokes (RANS) equations
Reynolds equation
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 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

Engineering

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
Planar reentry 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
Bidomain model in cardiology
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
- Swift-Hohenberg equation in pattern formation
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
Logistic differential equation, for 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
  - ${\textstyle k'(t)=s[k(t)]^{\alpha }-\delta k(t)}$
- 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
- ${\textstyle {\dot {D}}=rD+G(t)-T(t)}$
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