List of named differential equations

Fields

List of named differential equations

Types

By variable type
Ordinary Partial Differential-algebraic Integro-differential Fractional Linear Non-linear
Dependent and independent variables Autonomous Coupled / Decoupled Exact Homogeneous / Nonhomogeneous
Features
Order Operator Notation

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

Ordinary Differential Equations (ODEs)

Riemannian geometry

Gauss–Codazzi equations

Physics

Astrophysics

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

General relativity

Materials science

Nuclear physics

Radioactive decay equations

Plasma physics

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^[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

Nuclear engineering

Neutron diffusion equation^[3]

Optimal control

Orbital mechanics

Signal processing

Transportation engineering

Law of conservation in the kinematic wave model of traffic flow theory

Chemistry

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
Vidale–Wolfe advertising model

Linguistics

Replicator dynamics in evolutionary linguistics

Military strategy

Lanchester's laws in combat modeling

References

^ Zebiak, Stephen E.; Cane, Mark A. (1987). "A Model El Niño–Southern Oscillation". Monthly Weather Review. 115 (10): 2262–2278. doi:10.1175/1520-0493(1987)115<2262:AMENO>2.0.CO;2. ISSN 1520-0493.
^ Griffiths, David J. (2004), Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, pp. 1–2, ISBN 0-13-111892-7
^ Ragheb, M. (2017). "Neutron Diffusion Theory" (PDF).
^ Choi, Youngsoo (2011). "PDE-constrained Optimization and Beyond" (PDF).
^ Heinkenschloss, Matthias (2008). "PDE Constrained Optimization" (PDF). SIAM Conference on Optimization.
^ Rudin, Leonid I.; Osher, Stanley; Fatemi, Emad (1992). "Nonlinear total variation based noise removal algorithms". Physica D. 60 (1–4): 259–268. Bibcode:1992PhyD...60..259R. CiteSeerX 10.1.1.117.1675. doi:10.1016/0167-2789(92)90242-F.
^ Murray, James D. (2002). Mathematical Biology I: An Introduction (PDF). Interdisciplinary Applied Mathematics. Vol. 17 (3rd ed.). New York: Springer. pp. 395–417. doi:10.1007/b98868. ISBN 978-0-387-95223-9.
^ Fernández-Villaverde, Jesús (2010). "The econometrics of DSGE models" (PDF). SERIEs. 1 (1–2): 3–49. doi:10.1007/s13209-009-0014-7. S2CID 8631466.
^ Piazzesi, Monika (2010). "Affine Term Structure Models" (PDF).
^ Cardaliaguet, Pierre (2013). "Notes on Mean Field Games (from P.-L. Lions' lectures at Collège de France)" (PDF).

Differential equations

Classification

Operations	Differential operator Notation for differentiation Ordinary Partial Differential-algebraic Integro-differential Fractional Linear Non-linear Holonomic
Attributes of variables	Dependent and independent variables Homogeneous Nonhomogeneous Coupled Decoupled Order Degree Autonomous Exact differential equation On jet bundles
Relation to processes	Difference (discrete analogue) Stochastic Stochastic partial Delay

Solutions

Existence/uniqueness	Picard–Lindelöf theorem Peano existence theorem Carathéodory's existence theorem Cauchy–Kowalevski theorem
Solution topics	Wronskian Phase portrait Phase space Lyapunov stability Asymptotic stability Exponential stability Rate of convergence Series solutions Integral solutions Numerical integration Dirac delta function
Solution methods	Inspection Substitution Separation of variables Method of undetermined coefficients Variation of parameters Integrating factor Integral transforms Euler method Finite difference method Crank–Nicolson method Runge–Kutta methods Finite element method Finite volume method Galerkin method Perturbation theory

Examples

Mathematicians

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