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

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

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

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^[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

Euler–Bernoulli beam theory
Timoshenko beam theory

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

Clohessy–Wiltshire equations

Signal processing

Filtering theory
- Kushner equation
- Zakai equation
Rudin-Osher-Fatemi equation^[6] 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
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:2.0.CO;2">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 402 ME 405 Nuclear Power Engineering/Neutron Diffusion Theory.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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[1] Zebiak, Stephen E.; Cane, Mark A. (1987). "A Model El Niño–Southern Oscillation". Monthly Weather Review 115 (10): 2262–2278. doi:2.0.CO;2">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.

[Griffiths2004-2] Griffiths, David J. (2004), Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, pp. 1–2, ISBN 0-13-111892-7

[3] Ragheb, M. (2017). "Neutron Diffusion Theory". https://mragheb.com/NPRE 402 ME 405 Nuclear Power Engineering/Neutron Diffusion Theory.pdf.

[4] Choi, Youngsoo (2011). "PDE-constrained Optimization and Beyond". https://web.stanford.edu/class/cme334/docs/2011-11-08-choi_pdeopt.pdf.

[5] Heinkenschloss, Matthias (2008). "PDE Constrained Optimization". SIAM Conference on Optimization. https://archive.siam.org/meetings/op08/Heinkenschloss.pdf.

[6] 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.

[:2-7] 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.

[8] 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.

[9] Piazzesi, Monika (2010). "Affine Term Structure Models". https://web.stanford.edu/~piazzesi/s.pdf.

[10] 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.