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 Ishimori equation, an integrable nonlinear spin field equation in 2+1 dimensions 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
Electromagnetism Bloch equations Continuity equation for conservation laws Maxwell's equations Poynting's theorem
Fluid dynamics and hydrology
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 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 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
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 Generalized logistic differential equation for growth modeling Lotka–Volterra equations to describe the dynamics of biological systems in which two species interact
Economics and finance
Linguistics Replicator dynamics in evolutionary linguistics
Military strategy Lanchester's laws in combat modeling
References