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# Partial differential equation

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. [1]

## Acoustics

Acoustics is the branch of physics that deals with the study of all mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound.

The Adomian decomposition method (ADM) is a semi-analytical method for solving ordinary and partial nonlinear differential equations.

## Aleksandr Lyapunov

Aleksandr Mikhailovich Lyapunov (Алекса́ндр Миха́йлович Ляпуно́в,; – November 3, 1918) was a Russian mathematician, mechanician and physicist.

## Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series.

## Ansatz

In physics and mathematics, an ansatz (meaning: "initial placement of a tool at a work piece", plural ansätze; or ansatzes) is an educated guessIn his book on "The Nature of Mathematical Modelling", Neil Gershenfeld introduces ansatz, with interpretation "a trial answer", to be an important technique for solving differential equations.

## Bäcklund transform

In mathematics, Bäcklund transforms or Bäcklund transformations (named after the Swedish mathematician Albert Victor Bäcklund) relate partial differential equations and their solutions.

## Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set.

## Boundary value problem

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.

## Cauchy problem

A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that are given on a hypersurface in the domain.

## Cauchy–Kowalevski theorem

In mathematics, the Cauchy–Kowalevski theorem (also written as the Cauchy–Kovalevskaya theorem) is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems.

## Change of variables (PDE)

Often a partial differential equation can be reduced to a simpler form with a known solution by a suitable change of variables.

## Classical electromagnetism

Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model.

## Computer

A computer is a device that can be instructed to carry out sequences of arithmetic or logical operations automatically via computer programming.

## Computer simulation

Computer simulation is the reproduction of the behavior of a system using a computer to simulate the outcomes of a mathematical model associated with said system.

## Configuration space (physics)

In classical mechanics, the parameters that define the configuration of a system are called generalized coordinates, and the vector space defined by these coordinates is called the configuration space of the physical system.

## Conic section

In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.

## Constant (mathematics)

In mathematics, the adjective constant means non-varying.

## Contact geometry

In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'.

## Continuous or discrete variable

In mathematics, a variable may be continuous or discrete.

## Convolution

In mathematics (and, in particular, functional analysis) convolution is a mathematical operation on two functions (f and g) to produce a third function, that is typically viewed as a modified version of one of the original functions, giving the integral of the pointwise multiplication of the two functions as a function of the amount that one of the original functions is translated.

## Del

Del, or nabla, is an operator used in mathematics, in particular in vector calculus, as a vector differential operator, usually represented by the nabla symbol &nabla;.

## Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

## Diagonal matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.

## Differential equation

A differential equation is a mathematical equation that relates some function with its derivatives.

## Differential geometry

Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.

## Dirichlet boundary condition

In mathematics, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859).

## Discontinuous Galerkin method

In applied mathematics, discontinuous Galerkin methods (DG methods) form a class of numerical methods for solving differential equations.

## Discriminant

In algebra, the discriminant of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them.

## Divergence theorem

In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface.

## Dynamical system

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space.

## Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.

## Elasticity (physics)

In physics, elasticity (from Greek ἐλαστός "ductible") is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed.

## Electrostatics

Electrostatics is a branch of physics that studies electric charges at rest.

## Elliptic partial differential equation

Second order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic.

## Euler–Tricomi equation

In mathematics, the Euler–Tricomi equation is a linear partial differential equation useful in the study of transonic flow.

## Extended finite element method

The extended finite element method (XFEM), is a numerical technique based on the generalized finite element method (GFEM) and the partition of unity method (PUM).

## Finite difference

A finite difference is a mathematical expression of the form.

## Finite difference method

In mathematics, finite-difference methods (FDM) are numerical methods for solving differential equations by approximating them with difference equations, in which finite differences approximate the derivatives.

## Finite element method

The finite element method (FEM), is a numerical method for solving problems of engineering and mathematical physics.

## Finite volume method

The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations.

## Fluid

In physics, a fluid is a substance that continually deforms (flows) under an applied shear stress.

## Fluid dynamics

In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids - liquids and gases.

## Fourier analysis

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.

## Fourier series

In mathematics, a Fourier series is a way to represent a function as the sum of simple sine waves.

## Fourier transform

The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Fundamental solution

In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not address boundary conditions).

## Green's function

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

## Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.

## Heat

In thermodynamics, heat is energy transferred from one system to another as a result of thermal interactions.

## Heat equation

The heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time.

## Heat transfer

Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems.

## Helmholtz equation

In mathematics & physics, the Helmholtz equation, named for Hermann von Helmholtz, is the partial differential equation where &nabla;2 is the Laplacian, k is the wavenumber, and A is the amplitude.

## Homogeneous polynomial

In mathematics, a homogeneous polynomial is a polynomial whose nonzero terms all have the same degree.

## Homotopy analysis method

The homotopy analysis method (HAM) is a semi-analytical technique to solve nonlinear ordinary/partial differential equations.

## Homotopy principle

In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations (PDRs).

## Hp-FEM

hp-FEM is a general version of the finite element method (FEM), a numerical method for solving partial differential equations based on piecewise-polynomial approximations that employs elements of variable size (h) and polynomial degree (p).

## Hyperbolic partial differential equation

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 derivatives.

## Impulse response

In signal processing, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse.

## Infinitesimal transformation

In mathematics, an infinitesimal transformation is a limiting form of small transformation.

## Integral transform

In mathematics, an integral transform maps an equation from its original domain into another domain where it might be manipulated and solved much more easily than in the original domain.

## 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.

## Klein–Gordon equation

The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation.

## Laplace operator

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space.

## Laplace transform applied to differential equations

The Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain.

## Laplace's equation

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties.

## 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.

## Lewy's example

In the mathematical study of partial differential equations, Lewy's example is a celebrated example, due to Hans Lewy, of a linear partial differential equation with no solutions.

## Lie algebra

In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.

## Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

## Lie theory

In mathematics, the researcher Sophus Lie initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory.

## Linear function

In mathematics, the term linear function refers to two distinct but related notions.

## List of dynamical systems and differential equations topics

This is a list of dynamical system and differential equation topics, by Wikipedia page.

## Logical consequence

Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically follows from one or more statements.

## Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Matrix differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and of its derivatives of various orders.

## Meshfree methods

In the field of numerical analysis, meshfree methods are those that do not require connection between nodes of the simulation domain, i.e. a mesh, but are rather based on interaction of each node with all its neighbours.

## Method of characteristics

In mathematics, the method of characteristics is a technique for solving partial differential equations.

## Method of quantum characteristics

Quantum characteristics are phase-space trajectories that arise in the phase space formulation of quantum mechanics through the Wigner transform of Heisenberg operators of canonical coordinates and momenta.

## Multidimensional system

In mathematical systems theory, a multidimensional system or m-D system is a system in which not only one dependent variable exists (like time), but there are several independent variables.

## Multigrid method

Multigrid (MG) methods in numerical analysis are algorithms for solving differential equations using a hierarchy of discretizations.

## Multivariable calculus

Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables, rather than just one.

## Neumann boundary condition

In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann.

## Nonlinear Schrödinger equation

In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation.

## Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).

## Numerical partial differential equations

Numerical partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs).

## Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and its derivatives.

## Overdetermined system

In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns.

## Parabolic partial differential equation

A parabolic partial differential equation is a type of partial differential equation (PDE).

## Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).

## 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.

## Perturbation theory

Perturbation theory comprises mathematical methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem.

## Phase space formulation

The phase space formulation of quantum mechanics places the position and momentum variables on equal footing, in phase space.

## Picard–Lindelöf theorem

In mathematics, in the study of differential equations, the Picard–Lindelöf theorem, Picard's existence theorem or Cauchy–Lipschitz theorem is an important theorem on existence and uniqueness of solutions to first-order equations with given initial conditions.

## Poisson's equation

In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics.

## Pressure

Pressure (symbol: p or P) is the force applied perpendicular to the surface of an object per unit area over which that force is distributed.

## Probability distribution

In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables.

## Quantum mechanics

Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

## Recurrence relation

In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms.

## Rigid body

In physics, a rigid body is a solid body in which deformation is zero or so small it can be neglected.

## Robin boundary condition

In mathematics, the Robin boundary condition (properly), or third type boundary condition, is a type of boundary condition, named after Victor Gustave Robin (1855&ndash;1897).

## Separable partial differential equation

A separable partial differential equation (PDE) 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.

## Signal processing

Signal processing concerns the analysis, synthesis, and modification of signals, which are broadly defined as functions conveying "information about the behavior or attributes of some phenomenon", such as sound, images, and biological measurements.

## Sophus Lie

Marius Sophus Lie (17 December 1842 – 18 February 1899) was a Norwegian mathematician.

## Sound

In physics, sound is a vibration that typically propagates as an audible wave of pressure, through a transmission medium such as a gas, liquid or solid.

## Spectral element method

In the numerical solution of partial differential equations, a topic in mathematics, the spectral element method (SEM) is a formulation of the finite element method (FEM) that uses high degree piecewise polynomials as basis functions.

## Split-step method

In numerical analysis, the split-step (Fourier) method is a pseudo-spectral numerical method used to solve nonlinear partial differential equations like the nonlinear Schrödinger equation.

## Stochastic partial differential equation

Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations.

## Stochastic processes and boundary value problems

In mathematics, some boundary value problems can be solved using the methods of stochastic analysis.

## Supercomputer

A supercomputer is a computer with a high level of performance compared to a general-purpose computer.

## Superposition principle

In physics and systems theory, the superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually.

## Temperature

Temperature is a physical quantity expressing hot and cold.

## Underdetermined system

In mathematics, a system of linear equations or a system of polynomial equations is considered underdetermined if there are fewer equations than unknowns (in contrast to an overdetermined system, where there are more equations than unknowns).

## Uniform convergence

In the mathematical field of analysis, uniform convergence is a type of convergence of functions stronger than pointwise convergence.

## Uniqueness quantification

In mathematics and logic, the phrase "there is one and only one" is used to indicate that exactly one object with a certain property exists.

## Wave equation

The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light waves.

## 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 deemed to satisfy the equation in some precisely defined sense.

## Well-posed problem

The mathematical term well-posed problem stems from a definition given by Jacques Hadamard.

## Wigner quasiprobability distribution

The Wigner quasiprobability distribution (also called the Wigner function or the Wigner–Ville distribution after Eugene Wigner and Jean-André Ville) is a quasiprobability distribution.

## References

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