List of named differential equations

Short description: none

Differential equations

Navier–Stokes differential equations used to simulate airflow around an obstruction.

Scope

* Natural sciences
* Engineering

* Astronomy
* Physics
* Chemistry

Biology
* Geology

Applied mathematics

* Continuum mechanics
* Chaos theory
* Dynamical systems

Social sciences

* Economics
* Population dynamics

Classification

Types

* Ordinary
* Partial
* Differential-algebraic
* Integro-differential
* Fractional
* Linear
* Non-linear

By variable type

* Dependent and independent variables

* Autonomous
* Complex
* Coupled / Decoupled
* Exact
* Homogeneous / Nonhomogeneous

Features

* Order
* Operator

* Notation

Relation to processes

* Difference (discrete analogue)

* Stochastic
* Stochastic partial
* Delay

Solution

General topics

* Picard–Lindelöf theorem
* Wronskian

* Phase portrait
* Phase space

* Lyapunov / Asymptotic / Exponential stability
* Rate of convergence
* Series / Integral solutions

* Numerical integration
* Dirac delta function

Solution methods

* Inspection
* Method of characteristics

Euler
* Exponential response formula
* Finite difference (Crank–Nicolson)
* Finite element
* Infinite element
* Finite volume
* Galerkin
* Petrov–Galerkin
* Integrating factor
* Integral transforms
* Perturbation theory
* Runge–Kutta

* Separation of variables
* Undetermined coefficients
* Variation of parameters

People

* Isaac Newton
* Gottfried Leibniz
* Leonhard Euler
* Émile Picard
* Józef Maria Hoene-Wroński
* Ernst Lindelöf
* Rudolf Lipschitz
* Augustin-Louis Cauchy
* John Crank
* Phyllis Nicolson
* Carl David Tolmé Runge
* Martin Kutta

* v
* t
* e

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.

## Contents

* 1 Mathematics
* 1.1 Algebraic geometry
* 1.2 Complex analysis
* 1.3 Differential geometry
* 1.4 Dynamical systems and Chaos theory
* 1.5 Mathematical physics
* 1.6 Ordinary Differential Equations (ODEs)
* 1.7 Riemannian geometry
* 2 Physics
* 2.1 Astrophysics
* 2.2 Classical mechanics
* 2.3 Electromagnetism
* 2.4 Fluid dynamics and hydrology
* 2.5 General relativity
* 2.6 Materials science
* 2.7 Nuclear physics
* 2.8 Plasma physics
* 2.9 Quantum mechanics and quantum field theory
* 2.10 Thermodynamics and statistical mechanics
* 2.11 Waves (mechanical or electromagnetic)
* 3 Engineering
* 3.1 Electrical and Electronic Engineering
* 3.2 Game theory
* 3.3 Mechanical engineering
* 3.4 Nuclear engineering
* 3.5 Optimal control
* 3.6 Orbital mechanics
* 3.7 Signal processing
* 3.8 Transportation engineering
* 4 Chemistry
* 5 Biology and medicine
* 5.1 Population dynamics
* 6 Economics and finance
* 7 Linguistics
* 8 Military strategy
* 9 References

## 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

* Lorenz equations
* 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

* Equation of motion
* Euler's rotation equations in rigid body dynamics
* Euler–Lagrange equation
* Beltrami identity
* Hamilton's equations
* Hamilton-Jacobi equation
* Lorenz equations in chaos theory
* n-body problem in celestial mechanics
* Wave action in continuum mechanics

### Electromagnetism

* Bloch equations
* Continuity equation for conservation laws
* Maxwell's equations
* Poynting's theorem

### Fluid dynamics and hydrology

* Acoustic theory
* Benjamin–Bona–Mahony equation
* Biharmonic equation
* Blasius boundary layer
* Boussinesq approximation (buoyancy)
* Boussinesq approximation (water waves)
* Buckley–Leverett equation
* Camassa–Holm equation
* Chaplygin's equation
* Continuity equation for conservation laws
* Convection–diffusion equation
* Double diffusive convection
* Davey–Stewartson equation
* Euler–Tricomi equation
* Falkner–Skan boundary layer
* Gardner equation in hydrodynamics
* General equation of heat transfer
* Geophysical fluid dynamics
* Potential vorticity
* Quasi-geostrophic equations
* Shallow water equations
* Taylor–Goldstein equation
* Groundwater flow equation
* Richards equation
* Hicks equation
* Kadomtsev–Petviashvili equation in nonlinear wave motion
* KdV equation
* Magnetohydrodynamics
* Grad–Shafranov equation
* Navier–Stokes equations
* Euler equations
* Burgers' equation
* Nonlinear Schrödinger equation in water waves
* Omega equation
* Orr–Sommerfeld equation
* Porous medium equation
* Potential flow
* Rayleigh–Bénard convection
* Rayleigh–Plesset equation
* Reynolds-averaged Navier–Stokes (RANS) equations
* Reynolds equation
* Reynolds transport theorem
* Riemann problem
* Taylor–von Neumann–Sedov blast wave
* Turbulence modeling
* Turbulence kinetic energy (TKE)
* K-epsilon turbulence model
* k–omega turbulence model
* Spalart–Allmaras turbulence model
* Vorticity equation
* Whitham equation
* Zebiak-Cane model[1] for El Niño–Southern Oscillation
* Zeldovich–Taylor flow

### 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
* Parker transport 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 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
* Planar reentry 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

* Allee effect in population ecology
* Bidomain model in cardiology
* Chemotaxis[7] in wound healing
* Compartmental models in epidemiology
* SIR model
* SIS model
* Hagen–Poiseuille equation in blood flow
* Hodgkin–Huxley model in neural action potentials
* Kardar–Parisi–Zhang equation for bacteria surface growth models
* Kermack-McKendrick theory in infectious disease epidemiology
* Kuramoto model in biological and chemical oscillations
* Mackey-Glass equations
* McKendrick–von Foerster equation in age structure modeling
* Nernst–Planck equation in ion flux across biological membranes
* Price equation in evolutionary biology
* Reaction-diffusion equation in theoretical biology
* Fisher–KPP equation in nonlinear traveling waves
* FitzHugh–Nagumo model in neural activation
* Swift-Hohenberg equation in pattern formation
* Replicator dynamics in theoretical biology
* Verhulst equation in biological population growth
* von Bertalanffy model in biological individual growth
* Wilson–Cowan model in computational neuroscience

### Population dynamics

* Arditi–Ginzburg equations to describe predator–prey dynamics
* Kolmogorov–Petrovsky–Piskunov equation (also known as Fisher's equation) to model population growth
* Logistic differential equation, for population growth
* Generalized logistic differential equation for growth modeling
* Lotka–Volterra equations to describe the dynamics of biological systems in which two species interact

## Economics and finance

* Bass diffusion model
* Black–Scholes equation
* Economic growth
* Solow–Swan model
* k′(t)=s[k(t)]α−δk(t)
* Ramsey–Cass–Koopmans model
* Dynamic stochastic general equilibrium[8]
* Feynman–Kac formula
* Black–Scholes equation
* Affine term structure modeling[9]
* Fokker–Planck equation
* Dupire equation (local volatility)
* Hamilton–Jacobi–Bellman equation
* Merton's portfolio problem
* Optimal stopping
* Malthusian growth model
* Mean field game theory[10]
* Optimal rotation age
* Sovereign debt accumulation
* D˙=rD+G(t)−T(t)
* Stochastic differential equation
* Geometric Brownian motion
* Ornstein–Uhlenbeck process
* Cox–Ingersoll–Ross model
* Vidale–Wolfe advertising model

## Linguistics

* Replicator dynamics in evolutionary linguistics

## Military strategy

* Lanchester's laws in combat modeling

## References

1. ↑ 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.
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.
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.

* v
* t
* e

Differential equations

Classification|

| Operations|

* Differential operator
* Notation for differentiation
* Ordinary
* Partial
* Differential-algebraic
* Integro-differential
* Fractional
* Linear
* Non-linear
* Holonomic

|

Attributes of variables|

* Dependent and independent variables
* Homogeneous
* Nonhomogeneous
* Coupled
* Decoupled
* Order
* Degree
* Autonomous
* Exact differential equation
* Complex differential equation

Relation to processes|

* Difference (discrete analogue)
* Stochastic
* Stochastic partial
* Delay

Solutions|

| Solution topics|

* Picard–Lindelöf theorem (existence and uniqueness)
* Wronskian
* Phase portrait
* Phase space
* Lyapunov stability
* Asymptotic stability
* Exponential stability
* Rate of convergence
* Series solutions
* Integral solutions
* Numerical integration
* Dirac delta function

|

Solution methods|

* Inspection
* Separation of variables
* Method of undetermined coefficients
* Variation of parameters
* Integrating factor
* Integral transforms
* Euler method
* Finite difference method
* Crank–Nicolson method
* Runge–Kutta methods
* Finite element method
* Finite volume method
* Galerkin method
* Perturbation theory

Applications|

* List of named differential equations

Mathematicians|

* Isaac Newton
* Gottfried Wilhelm Leibniz
* Leonhard Euler
* Jacob Bernoulli
* Émile Picard
* Józef Maria Hoene-Wroński
* Ernst Lindelöf
* Rudolf Lipschitz
* Joseph-Louis Lagrange
* Augustin-Louis Cauchy
* John Crank
* Phyllis Nicolson
* Carl David Tolmé Runge
* Martin Kutta

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/List of named differential equations. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=List_of_named_differential_equations&oldid=4230297"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template