# List of named differential equations

Differential equations :
Scope :
Fields
Natural sciencesEngineering
Astronomy
Physics
Chemistry
Biology
Geology
Applied mathematics
Continuum mechanics
Chaos theory
Dynamical systems
Social sciences
Economics
Population dynamics
List of named differential equations :
Classification :
Types
Ordinary
Partial
Differential-algebraic
Integro-differential
Fractional
Linear
Non-linear
By variable type
Dependent and independent variables
Autonomous
Coupled / Decoupled
Exact
Homogeneous / Nonhomogeneous
Features
Order
Operator
Notation :
Relation to processes
Difference (discrete analogue)
Stochastic
Stochastic partial
Delay :
Solution :
Existence and uniqueness
Picard–Lindelöf theorem
Peano existence theorem
Carathéodory's existence theorem
Cauchy–Kowalevski theorem :
General topics
Initial conditions
Boundary values
Dirichlet
Neumann
Robin
Cauchy problem
Wronskian
Phase portrait
Lyapunov / Asymptotic / Exponential stability
Rate of convergence
Series / Integral solutions
Numerical integration
Dirac delta function :
Solution methods
Inspection
Method of characteristics
Euler
Exponential response formula
Finite difference (Crank–Nicolson)
Finite element
Infinite element
Finite volume
Galerkin
Petrov–Galerkin
Green's function
Integrating factor
Integral transforms
Perturbation theory
Runge–Kutta
Separation of variables
Undetermined coefficients
Variation of parameters :
People :
List
Isaac Newton
Gottfried Leibniz
Jacob Bernoulli
Leonhard Euler
Joseph-Louis Lagrange
Józef Maria Hoene-Wroński
Joseph Fourier
Augustin-Louis Cauchy
George Green
Carl David Tolmé Runge
Martin Kutta
Rudolf Lipschitz
Ernst Lindelöf
Émile Picard
Phyllis Nicolson
John Crank :
vte :
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 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

## 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
* 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)]$^α$−δk(t){\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
** D$^˙$=rD+G(t)−T(t){\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