List of named differential equations

In mathematics, differential equation is a fundamental concept that is used in many scientific areas. Many of the differential equations that are used have received specific names, which are listed in this article.

Pure mathematics

Physics

Classical mechanics

So long as the force acting on a particle is known, Newton's second law is sufficient to describe the motion of a particle. Once independent relations for each force acting on a particle are available, they can be substituted into Newton's second law to obtain an ordinary differential equation, which is called the equation of motion. Classical mechanics for particles finds its generalization in continuum mechanics.

Electrodynamics

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies. Maxwell's equations describe how electric and magnetic fields are generated and altered by each other and by charges and currents. They are named after the Scottish physicist and mathematician James Clerk Maxwell, who published an early form of those equations between 1861 and 1862.

General relativity

The Einstein field equations (EFE; also known as "Einstein's equations") are a set of ten partial differential equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy.[1] First published by Einstein in 1915[2] as a tensor equation, the EFE equate local spacetime curvature (expressed by the Einstein tensor) with the local energy and momentum within that spacetime (expressed by the stress–energy tensor).[3]

Quantum mechanics

In quantum mechanics, the analogue of Newton's law is Schrödinger's equation (a partial differential equation) for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but in general a linear partial differential equation, describing the time-evolution of the system's wave function (also called a "state function").[4]

Engineering

Fluid dynamics and hydrology

Biology and medicine

Predator–prey equations

The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order, non-linear, differential equations frequently used to describe the population dynamics of two species that interact, one as a predator and the other as prey.

Chemistry

The rate law or rate equation for a chemical reaction is a differential equation that links the reaction rate with concentrations or pressures of reactants and constant parameters (normally rate coefficients and partial reaction orders).[9] To determine the rate equation for a particular system one combines the reaction rate with a mass balance for the system.[10] In addition, a range of differential equations are present in the study of thermodynamics and quantum mechanics.

Economics and finance

References
  1. Einstein, Albert (1916). "The Foundation of the General Theory of Relativity". Annalen der Physik. 354 (7): 769. Bibcode:1916AnP...354..769E. doi:10.1002/andp.19163540702. hdl:2027/wu.89059241638. Archived from the original (PDF) on 2006-08-29.
  2. Einstein, Albert (November 25, 1915). "Die Feldgleichungen der Gravitation". Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin: 844–847. Retrieved 2006-09-12.
  3. Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 978-0-7167-0344-0. Chapter 34, p. 916.
  4. Griffiths, David J. (2004), Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, pp. 1–2, ISBN 0-13-111892-7
  5. Ragheb, M. (2017). "Neutron Diffusion Theory" (PDF).
  6. Choi, Youngsoo (2011). "PDE-constrained Optimization and Beyond" (PDF).
  7. Heinkenschloss, Matthias (2008). "PDE Constrained Optimization" (PDF). SIAM Conference on Optimization.
  8. 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.
  9. IUPAC Gold Book definition of rate law. See also: According to IUPAC Compendium of Chemical Terminology.
  10. Kenneth A. Connors Chemical Kinetics, the study of reaction rates in solution, 1991, VCH Publishers.
  11. 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.
  12. Piazzesi, Monika (2010). "Affine Term Structure Models" (PDF).
  13. Cardaliaguet, Pierre (2013). "Notes on Mean Field Games (from P.-L. Lions' lectures at Collège de France)" (PDF).
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