List of named differential equations explained

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

Electromagnetism

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

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^[1]

Optimal control

• Linear-quadratic regulator
• Matrix differential equation
• PDE-constrained optimization^{[2] [3]}
• Riccati equation
• Shape optimization

Orbital mechanics

• Clohessy–Wiltshire equations
• Planar reentry equations

Signal processing

• Filtering theory
  • Kushner equation
  • Zakai equation
• Rudin-Osher-Fatemi equation^[4] 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

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

Linguistics

• Replicator dynamics in evolutionary linguistics

Military strategy

• Lanchester's laws in combat modeling

Notes and References

1. Web site: Neutron Diffusion Theory. Ragheb. M.. 2017.
2. Web site: PDE-constrained Optimization and Beyond. Choi. Youngsoo. 2011.
3. Web site: PDE Constrained Optimization. Heinkenschloss. Matthias. 2008. SIAM Conference on Optimization.
4. Rudin. Leonid I.. Osher. Stanley. Fatemi. Emad. 1992. Nonlinear total variation based noise removal algorithms. Physica D. 60. 1–4. 259–268. 10.1.1.117.1675. 10.1016/0167-2789(92)90242-F. 1992PhyD...60..259R.