In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.
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 because 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 mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.
In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose-Einstein condensates confined to highly anisotropic cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model.
A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n − 1 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
In mathematics, constraint counting is counting the number of constraints in order to compare it with the number of variables, parameters, etc. that are free to be determined, the idea being that in most cases the number of independent choices that can be made is the excess of the latter over the former.
In mathematics and physics, the Kadomtsev–Petviashvili equation – or KP equation, named after Boris Borisovich Kadomtsev and Vladimir Iosifovich Petviashvili – is a partial differential equation to describe nonlinear wave motion. The KP equation is usually written as:
In mathematics, a first-order partial differential equation is a partial differential equation that involves only first derivatives of the unknown function of n variables. The equation takes the form
In mathematics, a (real) Monge–Ampère equation is a nonlinear second-order partial differential equation of special kind. A second-order equation for the unknown function u of two variables x,y is of Monge–Ampère type if it is linear in the determinant of the Hessian matrix of u and in the second-order partial derivatives of u. The independent variables (x,y) vary over a given domain D of R2. The term also applies to analogous equations with n independent variables. The most complete results so far have been obtained when the equation is elliptic.
In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. It is one of the most important developments in mathematical physics in the past 40 years. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to solve many linear partial differential equations. The name "inverse scattering method" comes from the key idea of recovering the time evolution of a potential from the time evolution of its scattering data: inverse scattering refers to the problem of recovering a potential from its scattering matrix, as opposed to the direct scattering problem of finding the scattering matrix from the potential.
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.
In the theory of stochastic processes, the filtering problem is a mathematical model for a number of state estimation problems in signal processing and related fields. The general idea is to establish a "best estimate" for the true value of some system from an incomplete, potentially noisy set of observations on that system. The problem of optimal non-linear filtering was solved by Ruslan L. Stratonovich, see also Harold J. Kushner's work and Moshe Zakai's, who introduced a simplified dynamics for the unnormalized conditional law of the filter known as Zakai equation. The solution, however, is infinite-dimensional in the general case. Certain approximations and special cases are well understood: for example, the linear filters are optimal for Gaussian random variables, and are known as the Wiener filter and the Kalman-Bucy filter. More generally, as the solution is infinite dimensional, it requires finite dimensional approximations to be implemented in a computer with finite memory. A finite dimensional approximated nonlinear filter may be more based on heuristics, such as the Extended Kalman Filter or the Assumed Density Filters, or more methodologically oriented such as for example the Projection Filters, some sub-families of which are shown to coincide with the Assumed Density Filters.
Dispersionless limits of integrable partial differential equations (PDE) arise in various problems of mathematics and physics and have been intensively studied in recent literature. They typically arise when considering slowly modulated long waves of an integrable dispersive PDE system.
In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.
In mathematics, the Novikov–Veselov equation is a natural (2+1)-dimensional analogue of the Korteweg–de Vries (KdV) equation. Unlike another (2+1)-dimensional analogue of KdV, the Kadomtsev–Petviashvili equation, it is integrable via the inverse scattering transform for the 2-dimensional stationary Schrödinger equation. Similarly, the Korteweg–de Vries equation is integrable via the inverse scattering transform for the 1-dimensional Schrödinger equation. The equation is named after S.P. Novikov and A.P. Veselov who published it in Novikov & Veselov (1984).
The Tzitzeica equation is a nonlinear partial differential equation devised by Gheorghe Țițeica in 1907 in the study of differential geometry, describing surfaces of constant affine curvature. The Tzitzeica equation has also been used in nonlinear physics, being an integrable 1+1 dimensional Lorentz invariant system.
The Estevez–Mansfield–Clarkson equation is a nonlinear partial differential equation introduced by Pilar Estevez, Elizabeth Mansfield, and Peter Clarkson.
The Gibbons–Tsarev equation is an integrable second order nonlinear partial differential equation. In its simplest form, in two dimensions, it may be written as follows:
