See also Nonlinear partial differential equation, List of partial differential equation topics and List of nonlinear ordinary differential equations.
Name | Dim | Equation | Applications |
---|---|---|---|
G equation | 1+3 | turbulent combustion | |
Generic scalar transport | 1+3 | transport | |
Ginzburg–Landau | 1+3 | Superconductivity | |
Gross–Pitaevskii | 1 + n | Bose–Einstein condensate | |
Gyrokinetics equation | 1 + 5 | Microturbulence in plasma | |
Guzmán | 1 + n | Hamilton–Jacobi–Bellman equation for risk aversion | |
Hartree equation | Any | ||
Hasegawa–Mima | 1+3 | Turbulence in plasma | |
Heisenberg ferromagnet | 1+1 | Magnetism | |
Hicks | 1+1 | Fluid dynamics | |
Hunter–Saxton | 1+1 | Liquid crystals | |
Ishimori equation | 1+2 | Integrable systems | |
Kadomtsev –Petviashvili | 1+2 | Shallow water waves | |
Kardar–Parisi–Zhang equation | 1+3 | Stochastics | |
von Karman | 2 | ||
Kaup | 1+1 | ||
Kaup–Kupershmidt | 1+1 | Integrable systems | |
Klein–Gordon–Maxwell | any | ||
Klein–Gordon (nonlinear) | any | Relativistic quantum mechanics | |
Khokhlov–Zabolotskaya | 1+2 | ||
Korteweg–de Vries (KdV) | 1+1 | Shallow waves, Integrable systems | |
KdV (super) | 1+1 | ||
There are more minor variations listed in the article on KdV equations. | |||
Kuramoto–Sivashinsky equation | 1 + n | Combustion |
Name | Dim | Equation | Applications |
---|---|---|---|
Landau–Lifshitz model | 1+n | Magnetic field in solids | |
Lin–Tsien equation | 1+2 | ||
Liouville equation | any | ||
Liouville–Bratu–Gelfand equation | any | combustion, astrophysics | |
Logarithmic Schrödinger equation | any | Superfluids, Quantum gravity | |
Minimal surface | 3 | minimal surfaces | |
Monge–Ampère | any | lower order terms | |
Navier–Stokes (and its derivation) | 1+3 | + mass conservation: | Fluid flow, gas flow |
Nonlinear Schrödinger (cubic) | 1+1 | optics, water waves | |
Nonlinear Schrödinger (derivative) | 1+1 | optics, water waves | |
Omega equation | 1+3 | atmospheric physics | |
Plateau | 2 | minimal surfaces | |
Pohlmeyer–Lund–Regge | 2 | ||
Porous medium | 1+n | diffusion | |
Prandtl | 1+2 | , | boundary layer |
In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function.
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real characteristic directions.
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
In mathematics, a hyperbolic partial differential equation of order is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface. Many of the equations of mechanics are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. The model hyperbolic equation is the wave equation. In one spatial dimension, this is
Louis Nirenberg was a Canadian-American mathematician, considered one of the most outstanding mathematicians of the 20th century.
Luis Ángel Caffarelli is an Argentine-American mathematician. He studies partial differential equations and their applications.
Lawrence Craig Evans is an American mathematician and Professor of Mathematics at the University of California, Berkeley.
In mathematics, in the theory of differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if the linearization of the equation at this solution has the form , where r is the perturbation to the steady state, A is a linear operator whose spectrum contains eigenvalues with positive real part. If all the eigenvalues have negative real part, then the solution is called linearlystable. Other names for linear stability include exponential stability or stability in terms of first approximation. If there exist an eigenvalue with zero real part then the question about stability cannot be solved on the basis of the first approximation and we approach the so-called "centre and focus problem".
In mathematics, secondary calculus is a proposed expansion of classical differential calculus on manifolds, to the "space" of solutions of a (nonlinear) partial differential equation. It is a sophisticated theory at the level of jet spaces and employing algebraic methods.
A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena, including heat conduction, particle diffusion, and pricing of derivative investment instruments.
Neil Sidney Trudinger is an Australian mathematician, known particularly for his work in the field of nonlinear elliptic partial differential equations.
In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear terms. They describe many different physical systems, ranging from gravitation to fluid dynamics, and have been used in mathematics to solve problems such as the Poincaré conjecture and the Calabi conjecture. They are difficult to study: almost no general techniques exist that work for all such equations, and usually each individual equation has to be studied as a separate problem.
J. (Jean) François Treves is an American mathematician, specializing in partial differential equations.
The Gardner equation is an integrable nonlinear partial differential equation introduced by the mathematician Clifford Gardner in 1968 to generalize KdV equation and modified KdV equation. The Gardner equation has applications in hydrodynamics, plasma physics and quantum field theory
The modified KdV–Burgers equation is a nonlinear partial differential equation
The Chafee–Infante equation is a nonlinear partial differential equation introduced by Nathaniel Chafee and Ettore Infante.
The Drinfeld–Sokolov–Wilson (DSW) equations are an integrable system of two coupled nonlinear partial differential equations proposed by Vladimir Drinfeld and Vladimir Sokolov, and independently by George Wilson:
Irena Lasiecka is a Polish-American mathematician, a Distinguished University Professor of mathematics and chair of the mathematics department at the University of Memphis. She is also co-editor-in-chief of two academic journals, Applied Mathematics & Optimization and Evolution Equations & Control Theory.