List of named differential equations
| Differential equations |
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| Scope |
| Classification |
| Solution |
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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
[edit]- 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
[edit]- Calabi flow in the study of Calabi-Yau manifolds
Complex analysis
[edit]Differential geometry
[edit]- Equations for a minimal surface
- Liouville's equation
- Ricci flow, used to prove the Poincaré conjecture
- Tzitzeica equation
Dynamical systems and Chaos theory
[edit]Mathematical physics
[edit]- General Legendre equation
- Heat equation
- Laplace's equation in potential theory
- Poisson's equation in potential theory
Ordinary Differential Equations (ODEs)
[edit]Riemannian geometry
[edit]Physics
[edit]Astrophysics
[edit]- Chandrasekhar's white dwarf equation
- Lane-Emden equation
- Emden–Chandrasekhar equation
- Hénon–Heiles system
Classical mechanics
[edit]Electromagnetism
[edit]- Continuity equation for conservation laws
- Maxwell's equations
- Poynting's theorem
Fluid dynamics and hydrology
[edit]- 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
- Davey–Stewartson equation
- Euler–Tricomi equation
- Falkner–Skan boundary layer
- Gardner equation in hydrodynamics
- General equation of heat transfer
- Geophysical fluid dynamics
- Groundwater flow equation
- Hicks equation
- Kadomtsev–Petviashvili equation in nonlinear wave motion
- KdV equation
- Magnetohydrodynamics
- Navier–Stokes equations
- 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
- Vorticity equation
- Whitham equation
- Zebiak-Cane model[1] for El Niño–Southern Oscillation
- Zeldovich–Taylor flow
General relativity
[edit]- Einstein field equations
- Friedmann equations
- Geodesic equation
- Mathisson–Papapetrou–Dixon equations
- Schrödinger–Newton equation
Materials science
[edit]- Ginzburg–Landau equations in superconductivity
- London equations in superconductivity
- Poisson–Boltzmann equation in molecular dynamics
Nuclear physics
[edit]Plasma physics
[edit]Quantum mechanics and quantum field theory
[edit]- 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[2]
- Schwinger–Dyson equation
- Yang-Mills equations in gauge theory
Thermodynamics and statistical mechanics
[edit]- Boltzmann equation
- Continuity equation for conservation laws
- Diffusion 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)
[edit]- D'Alembert's wave equation
- Eikonal equation in wave propagation
- Euler–Poisson–Darboux equation in wave theory
- Helmholtz equation
Engineering
[edit]Electrical and Electronic Engineering
[edit]- Chua's circuit
- Liénard equation to model oscillating circuits
- Nonlinear Schrödinger equation in fiber optics
- Telegrapher's equations
- Van der Pol oscillator
Game theory
[edit]Mechanical engineering
[edit]Nuclear engineering
[edit]Optimal control
[edit]- Linear-quadratic regulator
- Matrix differential equation
- PDE-constrained optimization[4][5]
- Riccati equation
- Shape optimization
Orbital mechanics
[edit]Signal processing
[edit]Transportation engineering
[edit]Chemistry
[edit]- Allen–Cahn equation in phase separation
- Cahn–Hilliard equation in phase separation
- Chemical reaction model
- Master equation
- Rate equation
- Streeter–Phelps equation in water quality modeling
Biology and medicine
[edit]- 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
[edit]- 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
[edit]- Bass diffusion model
- Black–Scholes equation
- Economic growth
- Feynman–Kac formula
- Fokker–Planck equation
- Dupire equation (local volatility)
- Hamilton–Jacobi–Bellman equation
- Malthusian growth model
- Mean field game theory[10]
- Optimal rotation age
- Sovereign debt accumulation
- Stochastic differential equation
- Vidale–Wolfe advertising model
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Linguistics
[edit]Military strategy
[edit]- Lanchester's laws in combat modeling
References
[edit]- ^ 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).