List of named differential equations
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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
- 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
Classical mechanics
Electromagnetism
- 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
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's equation[2]
- Schwinger–Dyson equation
- Yang-Mills equations in gauge theory
Thermodynamics and statistical mechanics
- 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)
- 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
- Neutron diffusion equation[3]
Optimal control
- Linear-quadratic regulator
- Matrix differential equation
- PDE-constrained optimization[4][5]
- Riccati equation
- Shape optimization
Orbital mechanics
Signal processing
Transportation engineering
Chemistry
- 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
Population dynamics
- Arditi–Ginzburg equations to describe predator–prey dynamics
- Fisher's equation to model population growth
- Kolmogorov–Petrovsky–Piskunov equation to model population growth
- Lotka–Volterra equations to describe the dynamics of biological systems in which two species interact
- Predator–prey 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
- ↑ 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. https://journals.ametsoc.org/view/journals/mwre/115/10/1520-0493_1987_115_2262_ameno_2_0_co_2.xml.
- ↑ 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". https://mragheb.com/NPRE%20402%20ME%20405%20Nuclear%20Power%20Engineering/Neutron%20Diffusion%20Theory.pdf.
- ↑ Choi, Youngsoo (2011). "PDE-constrained Optimization and Beyond". https://web.stanford.edu/class/cme334/docs/2011-11-08-choi_pdeopt.pdf.
- ↑ Heinkenschloss, Matthias (2008). "PDE Constrained Optimization". SIAM Conference on Optimization. https://archive.siam.org/meetings/op08/Heinkenschloss.pdf.
- ↑ Rudin, Leonid I.; Osher, Stanley; Fatemi, Emad (1992). "Nonlinear total variation based noise removal algorithms". Physica D 60 (1–4): 259–268. doi:10.1016/0167-2789(92)90242-F. Bibcode: 1992PhyD...60..259R.
- ↑ Murray, James D. (2002). Mathematical Biology I: An Introduction. Interdisciplinary Applied Mathematics. 17 (3rd ed.). New York: Springer. pp. 395–417. doi:10.1007/b98868. ISBN 978-0-387-95223-9. https://www.ucl.ac.uk/~rmjbale/3307/Reading_Chemotaxis1.pdf.
- ↑ Fernández-Villaverde, Jesús (2010). "The econometrics of DSGE models". SERIEs 1 (1–2): 3–49. doi:10.1007/s13209-009-0014-7. https://www.sas.upenn.edu/~jesusfv/econometricsDSGE.pdf.
- ↑ Piazzesi, Monika (2010). "Affine Term Structure Models". https://web.stanford.edu/~piazzesi/s.pdf.
- ↑ Cardaliaguet, Pierre (2013). "Notes on Mean Field Games (from P.-L. Lions' lectures at Collège de France)". https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf.
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