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Path integral formulation

The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical traject

Separable partial differential equation

A separable partial differential equation is one that can be broken into a set of separate equations of lower dimensionality (fewer independent variables) by a method of separation of variables. This

Differential equation

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 der

Delay differential equation

In mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the functio

Fuchsian theory

The Fuchsian theory of linear differential equations, which is named after Lazarus Immanuel Fuchs, provides a characterization of various types of singularities and the relations among them. At any or

Nullcline

In mathematical analysis, nullclines, sometimes called zero-growth isoclines, are encountered in a system of ordinary differential equations where here represents a derivative of with respect to anoth

Stochastic differential equation

A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used

Hicks equation

In fluid dynamics, Hicks equation or sometimes also referred as Bragg–Hawthorne equation or Squire–Long equation is a partial differential equation that describes the distribution of stream function f

Malgrange–Ehrenpreis theorem

In mathematics, the Malgrange–Ehrenpreis theorem states that every non-zero linear differential operator with constant coefficients has a Green's function. It was first proved independently by Leon Eh

Liouville–Bratu–Gelfand equation

In mathematics, Liouville–Bratu–Gelfand equation or Liouville's equation is a non-linear Poisson equation, named after the mathematicians Joseph Liouville, G. Bratu and Israel Gelfand. The equation re

Hurwitz matrix

In mathematics, a Hurwitz matrix, or Routh–Hurwitz matrix, in engineering stability matrix, is a structured real square matrix constructed with coefficients of a real polynomial.

Algebraic differential equation

In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several such notions, according to the concept of differ

Jet bundle

In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sect

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

Kuramoto–Sivashinsky equation

In mathematics, the Kuramoto–Sivashinsky equation (also called the KS equation or flame equation) is a fourth-order nonlinear partial differential equation. It is named after Yoshiki Kuramoto and Greg

Homogeneous differential equation

A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written where f and g are homogeneous functions of the

Lax pair

In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the Lax equation. Lax pairs

Liénard equation

In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation is a second order differential equation, named after the French physicist Alfred-Mari

Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those function

Method of matched asymptotic expansions

In mathematics, the method of matched asymptotic expansions is a common approach to finding an accurate approximation to the solution to an equation, or system of equations. It is particularly used wh

Fuchs relation

In mathematics, the Fuchs relation is a relation between the starting exponents of formal series solutions of certain linear differential equations, so called Fuchsian equations. It is named after Laz

Barrier certificate

A barrier certificate is an object that can serve as a proof of safety of an ordinary differential equation or hybrid dynamical system. Barrier certificates play the analogical role for safety to the

Noether identities

In mathematics, Noether identities characterize the degeneracy of a Lagrangian system. Given a Lagrangian system and its Lagrangian L, Noether identities can be defined as a differential operator whos

Non-autonomous system (mathematics)

In mathematics, an autonomous system is a dynamic equation on a smooth manifold. A non-autonomous system is a dynamic equation on a smooth fiber bundle over . For instance, this is the case of non-aut

Partial differential algebraic equation

In mathematics a partial differential algebraic equation (PDAE) set is an incomplete system of partial differential equations that is closed with a set of algebraic equations.

Jost function

In scattering theory, the Jost function is the Wronskian of the regular solution and the (irregular) Jost solution to the differential equation .It was introduced by Res Jost.

Green's function number

In mathematical heat conduction, the Green's function number is used to uniquely categorize certain fundamental solutions of the heat equation to make existing solutions easier to identify, store, and

Hyperbolic growth

When a quantity grows towards a singularity under a finite variation (a "finite-time singularity") it is said to undergo hyperbolic growth. More precisely, the reciprocal function has a hyperbola as a

Lanchester's laws

Lanchester's laws are mathematical formulae for calculating the relative strengths of military forces. The Lanchester equations are differential equations describing the time dependence of two armies'

Integro-differential equation

In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function.

Caloric polynomial

In differential equations, the mth-degree caloric polynomial (or heat polynomial) is a "parabolically m-homogeneous" polynomial Pm(x, t) that satisfies the heat equation "Parabolically m-homogeneous"

Related rates

In differential calculus, related rates problems involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known. The rate of change is

Linear differential equation

In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a0(x

Singular solution

A singular solution ys(x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have

Leaky integrator

In mathematics, a leaky integrator equation is a specific differential equation, used to describe a component or system that takes the integral of an input, but gradually leaks a small amount of input

Fuchs' theorem

In mathematics, Fuchs' theorem, named after Lazarus Fuchs, states that a second-order differential equation of the form has a solution expressible by a generalised Frobenius series when , and are anal

Rough path

In stochastic analysis, a rough path is a generalization of the notion of smooth path allowing to construct a robust solution theory for controlled differential equations driven by classically irregul

Adomian decomposition method

The Adomian decomposition method (ADM) is a semi-analytical method for solving ordinary and partial nonlinear differential equations. The method was developed from the 1970s to the 1990s by George Ado

Leray projection

The Leray projection, named after Jean Leray, is a linear operator used in the theory of partial differential equations, specifically in the fields of fluid dynamics. Informally, it can be seen as the

Hybrid automaton

In automata theory, a hybrid automaton (plural: hybrid automata or hybrid automatons) is a mathematical model for precisely describing hybrid systems, for instance systems in which digital computation

Slope field

Slope fields (also called direction fields) are a graphical representation of the solutions to a first-order differential equation of a scalar function. Solutions to a slope field are functions drawn

Laplace transform applied to differential equations

In mathematics, the Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain. The Laplace transform can be used in some cases to solve linear d

Initial condition

In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time

Vector spherical harmonics

In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the

Green's function

In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means tha

Discrete least squares meshless method

In mathematics the discrete least squares meshless (DLSM) method is a meshless method based on the least squares concept. The method is based on the minimization of a least squares functional, defined

Relativistic system (mathematics)

In mathematics, a non-autonomous system of ordinary differential equations is defined to be a dynamic equation on a smooth fiber bundle over . For instance, this is the case of non-relativistic non-au

Functional differential equation

A functional differential equation is a differential equation with deviating argument. That is, a functional differential equation is an equation that contains some function and some of its derivative

Differential Galois theory

In mathematics, differential Galois theory studies the Galois groups of differential equations.

Riemann–Hilbert correspondence

In mathematics, the term Riemann–Hilbert correspondence refers to the correspondence between regular singular flat connections on algebraic vector bundles and representations of the fundamental group,

Universal differential equation

A universal differential equation (UDE) is a non-trivial differential algebraic equation with the property that its solutions can approximate any continuous function on any interval of the real line t

Evolutionary invasion analysis

Evolutionary invasion analysis, also known as adaptive dynamics, is a set of mathematical modeling techniques that use differential equations to study the long-term evolution of traits in asexually re

Buckmaster equation

In mathematics, the Buckmaster equation is a second-order nonlinear partial differential equation, named after John D. Buckmaster, who derived the equation in 1977. The equation models the surface of

Radiodrome

In geometry, a radiodrome is the pursuit curve followed by a point that is pursuing another linearly-moving point. The term is derived from the Greek words ῥᾴδιος, rhā́idios, 'easier' and δρόμος, dróm

Smoluchowski coagulation equation

In statistical physics, the Smoluchowski coagulation equation is a population balance equation introduced by Marian Smoluchowski in a seminal 1916 publication, describing the time evolution of the num

Catenary

In physics and geometry, a catenary (US: /ˈkætənɛri/, UK: /kəˈtiːnəri/) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform

Variational bicomplex

In mathematics, the Lagrangian theory on fiber bundles is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations. For instance, this is th

Weak solution

In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless

List of equations

This is a list of equations, by Wikipedia page under appropriate bands of maths, science and engineering.

Mironenko reflecting function

In applied mathematics, the reflecting function of a differential system connects the past state of the system with the future state of the system by the formula The concept of the reflecting function

Three-wave equation

In nonlinear systems, the three-wave equations, sometimes called the three-wave resonant interaction equations or triad resonances, describe small-amplitude waves in a variety of non-linear media, inc

Zero dynamics

In mathematics, zero dynamics is known as the concept of evaluating the effect of zero on systems.

Gravity train

A gravity train is a theoretical means of transportation for purposes of commuting between two points on the surface of a sphere, by following a straight tunnel connecting the two points through the i

Partial differential equation

In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an

Inexact differential equation

An inexact differential equation is a differential equation of the form (see also: inexact differential) The solution to such equations came with the invention of the integrating factor by Leonhard Eu

Structural stability

In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact C1-

Charpit method

The Charpit method is a method for finding solutions of second-order partial differential equations in mathematics. The second-order partial differential equation is (1). (2) Then the general solution

Forward problem of electrocardiology

The forward problem of electrocardiology is a computational and mathematical approach to study the electrical activity of the heart through the body surface. The principal aim of this study is to comp

Replicator equation

In mathematics, the replicator equation is a deterministic monotone non-linear and non-innovative game dynamic used in evolutionary game theory. The replicator equation differs from other equations us

Qualitative theory of differential equations

In mathematics, the qualitative theory of differential equations studies the behavior of differential equations by means other than finding their solutions. It originated from the works of Henri Poinc

Physics-informed neural networks

Physics-informed neural networks (PINNs) are a type of universal function approximators that can embed the knowledge of any physical laws that govern a given data-set in the learning process, and can

Lie theory

In mathematics, the mathematician Sophus Lie (/liː/ LEE) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be ca

Fuzzy differential equation

Fuzzy differential equation are general concept of ordinary differential equation in mathematics defined as differential inclusion for non-uniform upper hemicontinuity convex set with compactness in f

Free motion equation

A free motion equation is a differential equation that describes a mechanical system in the absence of external forces, but in the presence only of an inertial force depending on the choice of a refer

Covariant classical field theory

In mathematical physics, covariant classical field theory represents classical fields by sections of fiber bundles, and their dynamics is phrased in the context of a finite-dimensional space of fields

Liouville's equation

In differential geometry, Liouville's equation, named after Joseph Liouville, is the nonlinear partial differential equation satisfied by the conformal factor f of a metric f2(dx2 + dy2) on a surface

Arnold–Beltrami–Childress flow

The Arnold–Beltrami–Childress (ABC) flow or Gromeka–Arnold–Beltrami–Childress (GABC) flow is a three-dimensional incompressible velocity field which is an exact solution of Euler's equation. Its repre

Pendulum (mechanics)

A pendulum is a body suspended from a fixed support so that it swings freely back and forth under the influence of gravity. When a pendulum is displaced sideways from its resting, equilibrium position

Conserved quantity

In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables, the value of which remains constant along each trajectory of the system. Not all systems have conse

Ancient solution

In mathematics, an ancient solution to a differential equation is a solution that can be extrapolated backwards to all past times, without singularities. That is, it is a solution "that is defined on

Method of mean weighted residuals

In applied mathematics, methods of mean weighted residuals (MWR) are methods for solving differential equations. The solutions of these differential equations are assumed to be well approximated by a

Degree of a differential equation

In mathematics, the degree of a differential equation is the power of its highest derivative, after the equation has been made rational and integral in all of its derivatives.

Nash–Moser theorem

In the mathematical field of analysis, the Nash–Moser theorem, discovered by mathematician John Forbes Nash and named for him and Jürgen Moser, is a generalization of the inverse function theorem on B

Differential-algebraic system of equations

In mathematics, a differential-algebraic system of equations (DAEs) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system. Suc

System of differential equations

In mathematics, a system of differential equations is a finite set of differential equations. Such a system can be either linear or non-linear. Also, such a system can be either a system of ordinary d

Euler's differential equation

In mathematics, Euler's differential equation is a first order nonlinear ordinary differential equation, named after Leonhard Euler given by This is a separable equation and the solution is given by t

Exponential response formula

In mathematics, the exponential response formula (ERF), also known as exponential response and complex replacement, is a method used to find a particular solution of a non-homogeneous linear ordinary

Liouville's theorem (differential algebra)

In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in 1833 to 1841, places an important restriction on antiderivatives that can be expressed as elementary functions. The an

Schrödinger equation

The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a signi

Floquet theory

Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form with a piecewise continuous periodi

Nahm equations

In differential geometry and gauge theory, the Nahm equations are a system of ordinary differential equations introduced by Werner Nahm in the context of the Nahm transform – an alternative to Ward's

Bidomain model

The bidomain model is a mathematical model to define the electrical activity of the heart. It consists in a continuum (volume-average) approach in which the cardiac mictrostructure is defined in terms

Logistic function

A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with equation where , the value of the sigmoid's midpoint;, the supremum of the values of the function;, the logistic g

Distinguished limit

In mathematics, a distinguished limit is an appropriately chosen scale factor used in the method of matched asymptotic expansions.

Forcing function (differential equations)

In a system of differential equations used to describe a time-dependent process, a forcing function is a function that appears in the equations and is only a function of time, and not of any of the ot

Singular perturbation

In mathematics, a singular perturbation problem is a problem containing a small parameter that cannot be approximated by setting the parameter value to zero. More precisely, the solution cannot be uni

D'Alembert's equation

In mathematics, d'Alembert's equation is a first order nonlinear ordinary differential equation, named after the French mathematician Jean le Rond d'Alembert. The equation reads as where . After diffe

Schwinger–Dyson equation

The Schwinger–Dyson equations (SDEs) or Dyson–Schwinger equations, named after Julian Schwinger and Freeman Dyson, are general relations between correlation functions in quantum field theories (QFTs).

Cylindrical harmonics

In mathematics, the cylindrical harmonics are a set of linearly independent functions that are solutions to Laplace's differential equation, , expressed in cylindrical coordinates, ρ (radial coordinat

Hybrid system

A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior – a system that can both flow (described by a differential equation) and jump (described by a state ma

Einstein–Infeld–Hoffmann equations

The Einstein–Infeld–Hoffmann equations of motion, jointly derived by Albert Einstein, Leopold Infeld and Banesh Hoffmann, are the differential equations of motion describing the approximate dynamics o

Laplace transform

In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (/ləˈplɑːs/), is an integral transform that converts a function of a real variable (usually , in the time domain)

Projected dynamical system

Projected dynamical systems is a mathematical theory investigating the behaviour of dynamical systems where solutions are restricted to a constraint set. The discipline shares connections to and appli

Monodomain model

The monodomain model is a reduction of the bidomain model of the electrical propagation in myocardial tissue.The reduction comes from assuming that the intra- and extracellular domains have equal anis

Bihari–LaSalle inequality

The Bihari–LaSalle inequality, was proved by the American mathematicianJoseph P. LaSalle (1916–1983) in 1949 and by the Hungarian mathematicianImre Bihari (1915–1998) in 1956. It is the following nonl

Group analysis of differential equations

Group analysis of differential equations is a branch of mathematics that studies the symmetry properties of differential equations with respect to various transformations of independent and dependent

Breather surface

In differential geometry, a breather surface is a one-parameter family of mathematical surfaces which correspond to breather solutions of the sine-Gordon equation, a differential equation appearing in

Differentiation of trigonometric functions

The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. For example, the der

Geometric analysis

Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential ge

Scheil equation

In metallurgy, the Scheil-Gulliver equation (or Scheil equation) describes solute redistribution during solidification of an alloy.

Bendixson–Dulac theorem

In mathematics, the Bendixson–Dulac theorem on dynamical systems states that if there exists a function (called the Dulac function) such that the expression has the same sign almost everywhere in a si

Biryukov equation

The Biryukov equation (or Biryukov oscillator), named after Vadim Biryukov (1946), is a non-linear second-order differential equation used to model damped oscillators. The equation is given by where ƒ

Wave

In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities osci

Operational calculus

Operational calculus, also known as operational analysis, is a technique by which problems in analysis, in particular differential equations, are transformed into algebraic problems, usually the probl

Tikhonov's theorem (dynamical systems)

In applied mathematics, Tikhonov's theorem on dynamical systems is a result on stability of solutions of systems of differential equations. It has applications to chemical kinetics. The theorem is nam

Loewy decomposition

In the study of differential equations, the Loewy decomposition breaks every linear ordinary differential equation (ODE) into what are called largest completely reducible components. It was introduced

Stokes operator

The Stokes operator, named after George Gabriel Stokes, is an unbounded linear operator used in the theory of partial differential equations, specifically in the fields of fluid dynamics and electroma

Autonomous system (mathematics)

In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable

Laplace–Carson transform

In mathematics, the Laplace–Carson transform, named after Pierre Simon Laplace and John Renshaw Carson, is an integral transform with significant applications in the field of physics and engineering,

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