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[]
- Cauchy–Riemann equations – complex analysis
- Ricci flow – used to prove the Poincaré conjecture
- Sturm–Liouville theory – orthogonal polynomials in linearly separable PDEs
Physics[]
- Continuity equation for conservation laws in electromagnetism, fluid dynamics, and thermodynamics
- Diffusion equation
- Heat equation in thermodynamics
- Eikonal equation in wave propagation
- Euler–Lagrange equation in classical mechanics
- Geodesic equation
- Hamilton's equations in classical mechanics
- KdV equation in fluid dynamics and plasma physics
- Lane-Emden equation in astrophysics
- Laplace's equation in harmonic analysis
- London equations in superconductivity
- Lorenz equations in chaos theory
- Newton's law of cooling in thermodynamics
- Nonlinear Schrödinger equation in quantum mechanics, water waves, and fiber optics
- Poisson's equation
- Poisson–Boltzmann equation in molecular dynamics
- Radioactive decay in nuclear physics
- Universal differential equation
- Wave equation
- Yang-Mills equations in differential geometry and gauge theory
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.
- Convection–diffusion equation in fluid dynamics
- Geophysical fluid dynamics
- n-body problem in celestial mechanics
- Navier–Stokes equations in fluid dynamics
- Wave action 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[]
- Chemical reaction model
- Elasticity
- Euler–Bernoulli beam theory
- Timoshenko beam theory
- Neutron diffusion[5]
- Optimal control
- Spherical harmonics
- Telegrapher's equations
- Total variation denoising (Rudin-Osher-Fatemi[8])
- Traffic flow
- Van der Pol oscillator
Fluid dynamics and hydrology[]
Biology and medicine[]
- Allee effect – population ecology
- Chemotaxis – wound healing
- Compartmental models – epidemiology
- SIR model
- SIS model
- Hagen–Poiseuille equation – blood flow
- Hodgkin–Huxley model – neural action potentials
- McKendrick–von Foerster equation – age structure modeling
- Nernst–Planck equation – ion flux across biological membranes
- Price equation - evolutionary biology
- Reaction-diffusion equation – theoretical biology
- Fisher–KPP equation – nonlinear traveling waves
- FitzHugh–Nagumo model – neural activation
- Replicator dynamics – found in theoretical biology and evolutionary linguistics
- Verhulst equation – biological population growth
- von Bertalanffy model – biological individual growth
- Wilson–Cowan model – computational neuroscience
- Young–Laplace equation – cardiovascular physiology
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[]
- Bass diffusion model
- Economic growth
- Feynman–Kac formula
- Fokker–Planck equation
- Dupire equation (local volatility)
- Hamilton–Jacobi–Bellman equation
- Malthusian growth model
- Mean field game theory[13]
- Sovereign debt accumulation
- Stochastic differential equation
- Vidale–Wolfe advertising model
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References[]
- ^ 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.
- ^ Einstein, Albert (November 25, 1915). "Die Feldgleichungen der Gravitation". Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin: 844–847. Retrieved 2006-09-12.
- ^ 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.
- ^ 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.
- ^ IUPAC Gold Book definition of rate law. See also: According to IUPAC Compendium of Chemical Terminology.
- ^ Kenneth A. Connors Chemical Kinetics, the study of reaction rates in solution, 1991, VCH Publishers.
- ^ 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