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.
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Differential equations
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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
Potential vorticity
Quasi-geostrophic equations
Shallow water equations
n-body problem in celestial mechanics
Navier–Stokes equations in fluid dynamics
Euler equations
Burgers' equation
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]
Dirac equation
Klein–Gordon equation
Schwinger–Dyson equation
Engineering
Chemical reaction model
Elasticity
Euler–Bernoulli beam theory
Timoshenko beam theory
Neutron diffusion[5]
Optimal control
Linear-quadratic regulator
Matrix differential equation
PDE-constrained optimization[6][7]
Shape optimization
Spherical harmonics
Associated Legendre polynomials
Telegrapher's equations
Total variation denoising (Rudin-Osher-Fatemi[8])
Traffic flow
Van der Pol oscillator
Fluid dynamics and hydrology
Acoustic theory
Blasius boundary layer
Buckley–Leverett equation
Groundwater flow equation
Richards equation
Magnetohydrodynamics
Potential flow
Rayleigh–Plesset equation
Reynolds-averaged Navier–Stokes (RANS) equations
Reynolds transport theorem
Riemann problem
Turbulence kinetic energy (TKE)
Vorticity equation
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
Solow–Swan model
\( {\textstyle k'(t)=s[k(t)]^{\alpha }-\delta k(t)} \)
Ramsey–Cass–Koopmans model
Dynamic stochastic general equilibrium[11]
Feynman–Kac formula
Black–Scholes equation
Affine term structure modeling[12]
Fokker–Planck equation
Dupire equation (local volatility)
Hamilton–Jacobi–Bellman equation
Merton's portfolio problem
Optimal stopping
Malthusian growth model
Mean field game theory[13]
Sovereign debt accumulation
D ˙ ⏟ change in debt = r D ⏟ interest on debt + G ( t ) − T ( t ) ⏟ net borrowing {\textstyle \underbrace {\dot {D}} _{\text{change in debt}}=\underbrace {rD} _{\text{interest on debt}}+\underbrace {G(t)-T(t)} _{\text{net borrowing}}} {\textstyle \underbrace {\dot {D}} _{\text{change in debt}}=\underbrace {rD} _{\text{interest on debt}}+\underbrace {G(t)-T(t)} _{\text{net borrowing}}}
Stochastic differential equation
Geometric Brownian motion
Ornstein–Uhlenbeck process
Cox–Ingersoll–Ross model
Vidale–Wolfe advertising model
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).
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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