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Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem.
== Topics in geometric function theory ==
The following are some of the most important topics in geometric function theory:
=== Conformal maps ===
A conformal ma... | Wikipedia/Geometric_function_theory |
In complex analysis, a complex-valued function
f
{\displaystyle f}
of a complex variable
z
{\displaystyle z}
:
is said to be holomorphic at a point
a
{\displaystyle a}
if it is differentia... | Wikipedia/Proof_that_holomorphic_functions_are_analytic |
In mathematics, the FBI transform or Fourier–Bros–Iagolnitzer transform is a generalization of the Fourier transform developed by the French mathematical physicists Jacques Bros and Daniel Iagolnitzer in order to characterise the local analyticity of functions (or distributions) on Rn. The transform provides an alterna... | Wikipedia/Fourier–Bros–Iagolnitzer_transform |
In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which form a necessary and sufficient condition for a complex function of a complex variable to be complex differentiable.
These equ... | Wikipedia/Cauchy–Riemann_equations |
In mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by Jaap Fabius (1966).
This function satisfies the initial condition
f
(
0
)
=
0
{\displaystyle f(0)=0}
, the sym... | Wikipedia/Fabius_function |
In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not true, as demonstrated with the counterexample below.
One of the most impo... | Wikipedia/Non-analytic_smooth_function |
In physics and chemistry, a degree of freedom is an independent physical parameter in the chosen parameterization of a physical system. More formally, given a parameterization of a physical system, the number of degrees of freedom is the smallest number
n
{\textstyle n}
of pa... | Wikipedia/Degrees_of_freedom_(physics_and_chemistry) |
In mathematics, physics and engineering, the sinc function ( SINK), denoted by sinc(x), has two forms, normalized and unnormalized.
In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by
sinc
(
x
)
=
... | Wikipedia/Sinc_function |
In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration was initially used to solve problems ... | Wikipedia/Integrable_function |
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics... | Wikipedia/Tangent_function |
In set theory, a continuous function is a sequence of ordinals such that the values assumed at limit stages are the limits (limit suprema and limit infima) of all values at previous stages. More formally, let γ be an ordinal, and
s
:=
⟨
s
... | Wikipedia/Continuous_function_(set_theory) |
In axiomatic set theory, a function f : Ord → Ord is called normal (or a normal function) if it is continuous (with respect to the order topology) and strictly monotonically increasing. This is equivalent to the following two conditions:
For every limit ordinal γ (i.e. γ is neither zero nor a successor), it is the cas... | Wikipedia/Normal_function |
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the recip... | Wikipedia/Reciprocal_function |
In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a function ... | Wikipedia/Diagram_(category_theory) |
In discrete mathematics, a direction-preserving function (or mapping) is a function on a discrete space, such as the integer grid, that (informally) does not change too drastically between two adjacent points. It can be considered a discrete analogue of a continuous function.
The concept was first defined by Iimura. So... | Wikipedia/Direction-preserving_function |
In mathematics, the Dirichlet function is the indicator function
1
Q
{\displaystyle \mathbf {1} _{\mathbb {Q} }}
of the set of rational numbers
... | Wikipedia/Dirichlet's_function |
In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is also an example of a fractal curve.
The Weierstrass function has historically served the role of a pathological function, being ... | Wikipedia/Weierstrass_function |
In mathematics, a function
f
:
R
→
R
{\displaystyle f:\mathbb {R} \to \mathbb {R} }
is symmetrically continuous at a point x if
lim
h
... | Wikipedia/Symmetrically_continuous_function |
In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry.
The most fam... | Wikipedia/Metric_topology |
In mathematics, a real function
f
{\displaystyle f}
of real numbers is said to be uniformly continuous if there is a positive real number
δ
{\displaystyle \delta }
such that function values over any function domain interval of the s... | Wikipedia/Uniformly_continuous_function |
Thomae's function is a real-valued function of a real variable that can be defined as:: 531
f
(
x
)
=
{
1
... | Wikipedia/Thomae's_function |
In mathematics, a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants C ≥ 0, α > 0, such that
|
f
(
x
)
−
f
(
y
... | Wikipedia/Hölder_continuous_function |
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air,... | Wikipedia/Dynamic_systems |
In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the con... | Wikipedia/Spin_tensor_(mechanics) |
In physics and continuum mechanics, deformation is the change in the shape or size of an object. It has dimension of length with SI unit of metre (m). It is quantified as the residual displacement of particles in a non-rigid body, from an initial configuration to a final configuration, excluding the body's average tran... | Wikipedia/Deformation_(physics) |
Contact mechanics is the study of the deformation of solids that touch each other at one or more points. This can be divided into compressive and adhesive forces in the direction perpendicular to the interface, and frictional forces in the tangential direction. Frictional contact mechanics is the study of the deformati... | Wikipedia/Frictional_contact_mechanics |
In mechanics, a displacement field is the assignment of displacement vectors for all points in a region or body that are displaced from one state to another. A displacement vector specifies the position of a point or a particle in reference to an origin or to a previous position. For example, a displacement field may b... | Wikipedia/Spatial_displacement_gradient_tensor |
The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations.
The directional deriva... | Wikipedia/Tensor_derivative_(continuum_mechanics) |
In mechanics, a displacement field is the assignment of displacement vectors for all points in a region or body that are displaced from one state to another. A displacement vector specifies the position of a point or a particle in reference to an origin or to a previous position. For example, a displacement field may b... | Wikipedia/Displacement_gradient_tensor |
Material failure theory is an interdisciplinary field of materials science and solid mechanics which attempts to predict the conditions under which solid materials fail under the action of external loads. The failure of a material is usually classified into brittle failure (fracture) or ductile failure (yield). Dependi... | Wikipedia/Material_failure_theory |
In mechanics, strain is defined as relative deformation, compared to a reference position configuration. Different equivalent choices may be made for the expression of a strain field depending on whether it is defined with respect to the initial or the final configuration of the body and on whether the metric tensor or... | Wikipedia/Strain_(mechanics) |
In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the con... | Wikipedia/Finite_strain_theory |
In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the con... | Wikipedia/Eulerian_finite_strain_tensor |
In fluid dynamics, the Hagen–Poiseuille equation, also known as the Hagen–Poiseuille law, Poiseuille law or Poiseuille equation, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section.
It can be successfu... | Wikipedia/Hagen–Poiseuille_equation |
In continuum mechanics, a compatible deformation (or strain) tensor field in a body is that unique tensor field that is obtained when the body is subjected to a continuous, single-valued, displacement field. Compatibility is the study of the conditions under which such a displacement field can be guaranteed. Compatib... | Wikipedia/Compatibility_(mechanics) |
Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to fracture.
Theoretically, the str... | Wikipedia/Fracture_mechanics |
In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decompositi... | Wikipedia/Eigenvalue_decomposition |
In mechanics, compression is the application of balanced inward ("pushing") forces to different points on a material or structure, that is, forces with no net sum or torque directed so as to reduce its size in one or more directions. It is contrasted with tension or traction, the application of balanced outward ("pulli... | Wikipedia/Dilation_(physics) |
In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the con... | Wikipedia/Lagrangian_finite_strain_tensor |
In mechanics, a displacement field is the assignment of displacement vectors for all points in a region or body that are displaced from one state to another. A displacement vector specifies the position of a point or a particle in reference to an origin or to a previous position. For example, a displacement field may b... | Wikipedia/Material_displacement_gradient_tensor |
In physics and materials science, plasticity (also known as plastic deformation) is the ability of a solid material to undergo permanent deformation, a non-reversible change of shape in response to applied forces. For example, a solid piece of metal being bent or pounded into a new shape displays plasticity as permanen... | Wikipedia/Plasticity_(physics) |
In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the con... | Wikipedia/Finite_deformation_tensor |
In mechanics and materials science, strain rate is the time derivative of strain of a material. Strain rate has dimension of inverse time and SI units of inverse second, s−1 (or its multiples).
The strain rate at some point within the material measures the rate at which the distances of adjacent parcels of the material... | Wikipedia/Strain_rate |
In physics and continuum mechanics, deformation is the change in the shape or size of an object. It has dimension of length with SI unit of metre (m). It is quantified as the residual displacement of particles in a non-rigid body, from an initial configuration to a final configuration, excluding the body's average tran... | Wikipedia/Deformation_(mechanics) |
In mathematics, a quadratic differential on a Riemann surface is a section of the symmetric square of the holomorphic cotangent bundle. If the section is holomorphic, then the quadratic differential is said to be holomorphic. The vector space of holomorphic quadratic differentials on a Riemann surface has a natural int... | Wikipedia/Quadratic_differential |
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.
For instance, the expression... | Wikipedia/Differential_1-form |
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 have many applications throughout pure mathematics and are used to model various behaviours of stochastic models such as stock p... | Wikipedia/Stochastic_differential |
Differential may refer to:
== Mathematics ==
Differential (mathematics) comprises multiple related meanings of the word, both in calculus and differential geometry, such as an infinitesimal change in the value of a function
Differential algebra
Differential calculus
Differential of a function, represents a change in ... | Wikipedia/Differential_(disambiguation) |
In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic varieties), for everywhere-regular differential 1-forms. Given a complex manifold M, a differential of the first kind ω is the... | Wikipedia/Abelian_differential |
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tan... | Wikipedia/Instantaneous_rate_of_change |
In calculus, the differential represents the principal part of the change in a function
y
=
f
(
x
)
{\displaystyle y=f(x)}
with respect to changes in the independent variable. The differential
d
y
... | Wikipedia/Total_differential |
In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calcul... | Wikipedia/Kähler_differential |
Lambda calculus is a formal mathematical system based on lambda abstraction and function application. Two definitions of the language are given here: a standard definition, and a definition using mathematical formulas.
== Standard definition ==
This formal definition was given by Alonzo Church.
=== Definition ===
L... | Wikipedia/Lambda_calculus_definition |
In computer science, a programming language is said to have first-class functions if it treats functions as first-class citizens. This means the language supports passing functions as arguments to other functions, returning them as the values from other functions, and assigning them to variables or storing them in data... | Wikipedia/First-class_function |
The Knights of the Lambda Calculus is a semi-fictional organization of expert Lisp and Scheme hackers. The name refers to the lambda calculus, a mathematical formalism invented by Alonzo Church, with which Lisp is intimately connected, and references the Knights Templar.
There is no actual organization that goes by the... | Wikipedia/Knights_of_the_Lambda_Calculus |
The SKI combinator calculus is a combinatory logic system and a computational system. It can be thought of as a computer programming language, though it is not convenient for writing software. Instead, it is important in the mathematical theory of algorithms because it is an extremely simple Turing complete language. I... | Wikipedia/SKI_combinator_calculus |
In mathematical logic and computer science, the lambda-mu calculus is an extension of the lambda calculus introduced by Michel Parigot. It introduces two new operators: the μ operator (which is completely different both from the μ operator found in computability theory and from the μ operator of modal μ-calculus) and t... | Wikipedia/Lambda-mu_calculus |
In software development, an object is an entity that has state, behavior, and identity.: 78 An object can model some part of reality or can be an invention of the design process whose collaborations with other such objects serve as the mechanisms that provide some higher-level behavior. Put another way, an object repr... | Wikipedia/Object_(computer_science) |
In rewriting, a reduction strategy or rewriting strategy is a relation specifying a rewrite for each object or term, compatible with a given reduction relation. Some authors use the term to refer to an evaluation strategy.
== Definitions ==
Formally, for an abstract rewriting system
(
... | Wikipedia/Reduction_strategy |
Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming languages... | Wikipedia/Combinator_calculus |
In a programming language, an evaluation strategy is a set of rules for evaluating expressions. The term is often used to refer to the more specific notion of a parameter-passing strategy that defines the kind of value that is passed to the function for each parameter (the binding strategy) and whether to evaluate the ... | Wikipedia/Evaluation_strategy |
Applicative computing systems, or ACS are the systems of object calculi founded on combinatory logic and lambda calculus.
The only essential notion which is under consideration in these systems is the representation of object. In combinatory logic the only metaoperator is application in a sense of applying one object ... | Wikipedia/Applicative_computing_systems |
A function pointer, also called a subroutine pointer or procedure pointer, is a pointer referencing executable code, rather than data. Dereferencing the function pointer yields the referenced function, which can be invoked and passed arguments just as in a normal function call. Such an invocation is also known as an "i... | Wikipedia/Function_pointer |
In rewriting, a reduction strategy or rewriting strategy is a relation specifying a rewrite for each object or term, compatible with a given reduction relation. Some authors use the term to refer to an evaluation strategy.
== Definitions ==
Formally, for an abstract rewriting system
(
... | Wikipedia/Reduction_strategy_(lambda_calculus) |
In computer science, graph reduction implements an efficient version of non-strict evaluation, an evaluation strategy where the arguments to a function are not immediately evaluated. This form of non-strict evaluation is also known as lazy evaluation and used in functional programming languages. The technique was first... | Wikipedia/Graph_reduction |
In computer science and computer programming, a nondeterministic algorithm is an algorithm that, even for the same input, can exhibit different behaviors on different runs, as opposed to a deterministic algorithm.
Different models of computation give rise to different reasons that an algorithm may be non-deterministic,... | Wikipedia/Nondeterministic_algorithm |
Binary combinatory logic (BCL) is a computer programming language that uses binary terms 0 and 1 to create a complete formulation of combinatory logic using only the symbols 0 and 1. Using the S and K combinators, complex boolean algebra functions can be made. BCL has applications in the theory of program-size complexi... | Wikipedia/Binary_lambda_calculus |
In programming languages, name resolution is the resolution of the tokens within program expressions to the intended program components.
== Overview ==
Expressions in computer programs reference variables, data types, functions, classes, objects, libraries, packages and other entities by name. In that context, name r... | Wikipedia/Name_resolution_(programming_languages) |
In mathematics, a fundamental solution for a linear partial differential operator L 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).
In terms of the Dirac delta "function" δ(x), ... | Wikipedia/Fundamental_solution |
In mathematics, a hyperbolic partial differential equation of order
n
{\displaystyle n}
is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first
n
−
1
{\displa... | Wikipedia/Hyperbolic_partial_differential_equation |
In mathematics, Schwartz space
S
{\displaystyle {\mathcal {S}}}
is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on... | Wikipedia/Schwartz_function |
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 in, for example, engineering science, quantum mechanics and financial mathematics. Examples include the heat equation, time-dependent Schrödinger equ... | Wikipedia/Parabolic_partial_differential_equation |
In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar, then the function's value is multiplied by some power of this scalar; the power is called the degree of homogeneity, or simply the degree. That i... | Wikipedia/Euler's_homogeneous_function_theorem |
In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function
f
{\displaystyle f}
in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be ... | Wikipedia/Eigenfunctions |
In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on
R
n
{\di... | Wikipedia/Invariant_differential_operator |
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory, e.g. in mathematical models that include ultrametric pseudo-differential equation... | Wikipedia/Pseudo-differential_operator |
In quantum mechanics, energy is defined in terms of the energy operator, acting on the wave function of the system as a consequence of time translation symmetry.
== Definition ==
It is given by:
E
^
=... | Wikipedia/Energy_operator |
In mathematical finance, the Black–Scholes equation, also called the Black–Scholes–Merton equation, is a partial differential equation (PDE) governing the price evolution of derivatives under the Black–Scholes model. Broadly speaking, the term may refer to a similar PDE that can be derived for a variety of options, or ... | Wikipedia/Black–Scholes_equation |
Regularity is a topic of the mathematical study of partial differential equations (PDE) such as Laplace's equation, about the integrability and differentiability of weak solutions. Hilbert's nineteenth problem was concerned with this concept.
The motivation for this study is as follows. It is often difficult to constru... | Wikipedia/Regularity_theory |
The homotopy analysis method (HAM) is a semi-analytical technique to solve nonlinear ordinary/partial differential equations. The homotopy analysis method employs the concept of the homotopy from topology to generate a convergent series solution for nonlinear systems. This is enabled by utilizing a homotopy-Maclaurin... | Wikipedia/Homotopy_analysis_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 Adomian, chair of the Center for Applied Mathematics at the University of Georgia.
It is further extensible to stochasti... | Wikipedia/Adomian_decomposition_method |
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. Since mass, energy, momentum, electric charge and other natural quantities ... | Wikipedia/Continuity_equation |
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium; it was devised by Ludwig Boltzmann in 1872.
The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regio... | Wikipedia/Boltzmann_equation |
In physics, the acoustic wave equation is a second-order partial differential equation that governs the propagation of acoustic waves through a material medium resp. a standing wavefield. The equation describes the evolution of acoustic pressure p or particle velocity u as a function of position x and time t. A simpli... | Wikipedia/Acoustic_wave_equation |
In mathematics, a first-order partial differential equation is a partial differential equation that involves the first derivatives of an unknown function
u
{\displaystyle u}
of
n
≥
2
{\displaystyle n\geq 2}
variables... | Wikipedia/First-order_partial_differential_equation |
The homotopy analysis method (HAM) is a semi-analytical technique to solve nonlinear ordinary/partial differential equations. The homotopy analysis method employs the concept of the homotopy from topology to generate a convergent series solution for nonlinear systems. This is enabled by utilizing a homotopy-Maclaurin... | Wikipedia/Homotopy_perturbation_method |
Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow. The equation was first introduced by Harry Bateman in 1915 and l... | Wikipedia/Burgers'_equation |
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the Turin Academy of Science in 1760 culminating in his 1788 grand opus, Mécaniq... | Wikipedia/Lagrange_equation |
A separable partial differential equation can be broken into a set of equations of lower dimensionality (fewer independent variables) by a method of separation of variables. It generally relies upon the problem having some special form or symmetry. In this way, the partial differential equation (PDE) can be solved by ... | Wikipedia/Separable_partial_differential_equation |
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. There are many different definitions of weak solution, ... | Wikipedia/Weak_solution |
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 con... | Wikipedia/Nonlinear_partial_differential_equation |
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Albert Einstein in 1915 in the form of a tensor equation which related the local spacetime curvature (expr... | Wikipedia/Einstein_equations |
The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system. The ter... | Wikipedia/Lorenz_equation |
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 neighbors. As a consequence, original extensive properties such as mass or kinetic energy are no longer assigne... | Wikipedia/Meshfree_methods |
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.
== Definition ==
A general PDAE is defined as:
0
=
F
(
... | Wikipedia/Partial_differential_algebraic_equation |
Numerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs).
In principle, specialized methods for hyperbolic, parabolic or elliptic partial differential equations exist.
== Overview of methods ==
=== Finite di... | Wikipedia/Numerical_partial_differential_equations |
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. The spectral element method was introduced in a 1984 paper by A. T. Patera. Although ... | Wikipedia/Spectral_element_method |
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 derivati... | Wikipedia/Monge–Ampère_equation |
In mathematics, a dispersive partial differential equation or dispersive PDE is a partial differential equation that is dispersive. In this context, dispersion means that waves of different wavelength propagate at different phase velocities.
== Examples ==
=== Linear equations ===
Euler–Bernoulli beam equation wi... | Wikipedia/Dispersive_partial_differential_equation |
In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the elliptic partial differential equation:
∇
2
f
=
−
k
2
... | Wikipedia/Helmholtz_equation |
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