Fundamental solutionIn 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), a fundamental solution F is a solution of the inhomogeneous equation Here F is a priori only assumed to be a distribution. This concept has long been utilized for the Laplacian in two and three dimensions.
Laplace's equationIn mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as or where is the Laplace operator, is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function.
Dirichlet boundary conditionIn the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential equation, it specifies the values that a solution needs to take along the boundary of the domain. In finite element method (FEM) analysis, essential or Dirichlet boundary condition is defined by weighted-integral form of a differential equation.
Robin boundary conditionIn mathematics, the Robin boundary condition (ˈrɒbɪn; properly ʁɔbɛ̃), or third type boundary condition, is a type of boundary condition, named after Victor Gustave Robin (1855–1897). When imposed on an ordinary or a partial differential equation, it is a specification of a linear combination of the values of a function and the values of its derivative on the boundary of the domain. Other equivalent names in use are Fourier-type condition and radiation condition.
Boundary value problemIn the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems.
Neumann boundary conditionIn mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann. When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative applied at the boundary of the domain. It is possible to describe the problem using other boundary conditions: a Dirichlet boundary condition specifies the values of the solution itself (as opposed to its derivative) on the boundary, whereas the Cauchy boundary condition, mixed boundary condition and Robin boundary condition are all different types of combinations of the Neumann and Dirichlet boundary conditions.
Cauchy boundary conditionIn mathematics, a Cauchy (koʃi) boundary condition augments an ordinary differential equation or a partial differential equation with conditions that the solution must satisfy on the boundary; ideally so as to ensure that a unique solution exists. A Cauchy boundary condition specifies both the function value and normal derivative on the boundary of the domain. This corresponds to imposing both a Dirichlet and a Neumann boundary condition. It is named after the prolific 19th-century French mathematical analyst Augustin-Louis Cauchy.
DefinitionA definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitions (which try to list the objects that a term describes). Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions.
Mixed boundary conditionIn mathematics, a mixed boundary condition for a partial differential equation defines a boundary value problem in which the solution of the given equation is required to satisfy different boundary conditions on disjoint parts of the boundary of the domain where the condition is stated. Precisely, in a mixed boundary value problem, the solution is required to satisfy a Dirichlet or a Neumann boundary condition in a mutually exclusive way on disjoint parts of the boundary.
Lexical definitionThe lexical definition of a term, also known as the dictionary definition, is the definition closely matching the meaning of the term in common usage. As its other name implies, this is the sort of definition one is likely to find in the dictionary. A lexical definition is usually the type expected from a request for definition, and it is generally expected that such a definition will be stated as simply as possible in order to convey information to the widest audience.
Spherical harmonicsIn mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series.
Harmonic analysisHarmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on the real line, or by Fourier series for periodic functions. Generalizing these transforms to other domains is generally called Fourier analysis, although the term is sometimes used interchangeably with harmonic analysis.
Poisson's equationPoisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate electrostatic or gravitational (force) field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after French mathematician and physicist Siméon Denis Poisson.
Circular definitionA circular definition is a type of definition that uses the term(s) being defined as part of the description or assumes that the term(s) being described are already known. There are several kinds of circular definition, and several ways of characterising the term: pragmatic, lexicographic and linguistic. Circular definitions are related to Circular reasoning in that they both involve a self-referential approach. Circular definitions may be unhelpful if the audience must either already know the meaning of the key term, or if the term to be defined is used in the definition itself.
Green's functionIn 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 that if is the linear differential operator, then the Green's function is the solution of the equation , where is Dirac's delta function; the solution of the initial-value problem is the convolution ().
Definition of planetThe definition of planet has changed several times since the word was coined by the ancient Greeks. Greek astronomers employed the term ἀστέρες πλανῆται (), 'wandering stars', for star-like objects which apparently moved over the sky. Over the millennia, the term has included a variety of different celestial bodies, from the Sun and the Moon to satellites and asteroids. In modern astronomy, there are two primary conceptions of a 'planet'.
Geophysical definition of planetThe International Union of Geological Sciences (IUGS) is the internationally recognized body charged with fostering agreement on nomenclature and classification across geoscientific disciplines. However, they have yet to create a formal definition of the term planet. As a result, there are various geophysical definitions in use among professional geophysicists, planetary scientists, and other professionals in the geosciences. Many professionals opt to use one of several of these geophysical definitions instead of the definition voted on by the International Astronomical Union.
Poisson bracketIn mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into canonical coordinate systems.
Siméon Denis PoissonBaron Siméon Denis Poisson FRS FRSE (si.me.ɔ̃ də.ni pwa.sɔ̃; 21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electricity and magnetism, thermodynamics, elasticity, and fluid mechanics. Moreover, he predicted the Poisson spot in his attempt to disprove the wave theory of Augustin-Jean Fresnel, which was later confirmed.
Poisson manifoldIn differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics. A Poisson structure (or Poisson bracket) on a smooth manifold is a functionon the vector space of smooth functions on , making it into a Lie algebra subject to a Leibniz rule (also known as a Poisson algebra).
