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
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
Mathematical physics
- General Legendre equation
- Heat equation
- Laplace's equation in potential theory
- Poisson's equation in potential theory
Ordinary Differential Equations (ODEs)
Riemannian geometry
Physics
Astrophysics
- Chandrasekhar's white dwarf equation
- Lane-Emden equation
- Emden–Chandrasekhar equation
- Hénon–Heiles system
Electromagnetism
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
Plasma physics
- Gardner equation
- Hasegawa–Mima equation
- KdV equation
- Kuramoto–Sivashinsky equation
- Parker transport 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
Mechanical engineering
Nuclear engineering
Optimal control
- Linear-quadratic regulator
- Matrix differential equation
- PDE-constrained optimization
- Riccati equation
- Shape optimization
Orbital mechanics
Signal processing
- Filtering theory
- * Kushner equation
- * Zakai equation
- Rudin-Osher-Fatemi equation in total variation denoising
Transportation engineering
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
Population dynamics
- Arditi–Ginzburg equations to describe predator–prey dynamics
- Kolmogorov–Petrovsky–Piskunov equation to model population growth
- Logistic differential equation, for population growth
- * Generalized logistic differential equation for growth modeling
- Lotka–Volterra equations to describe the dynamics of biological systems in which two species interact
Linguistics
Military strategy
- Lanchester's laws in combat modeling