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

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 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^[1] 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^[2]
• 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^[3]

Optimal control

• Linear-quadratic regulator
• Matrix differential equation
• PDE-constrained optimization^[4][5]
• Riccati equation
• Shape optimization

Orbital mechanics

• Clohessy–Wiltshire equations
• Planar reentry equations

Signal processing

• Filtering theory
  • Kushner equation
  • Zakai equation
• Rudin-Osher-Fatemi equation^[6] 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^[7] 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
    • ${\textstyle k'(t)=s[k(t)]^{\alpha }-\delta k(t)}$
  • Ramsey–Cass–Koopmans model
  • Dynamic stochastic general equilibrium^[8]
• Feynman–Kac formula
  • Black–Scholes equation
  • Affine term structure modeling^[9]
• Fokker–Planck equation
  • Dupire equation (local volatility)
• Hamilton–Jacobi–Bellman equation
  • Merton's portfolio problem
  • Optimal stopping
• Malthusian growth model
• Mean field game theory^[10]
• 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