Radius of convergenceIn mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or . When it is positive, the power series converges absolutely and uniformly on compact sets inside the open disk of radius equal to the radius of convergence, and it is the Taylor series of the analytic function to which it converges.
Uniform convergenceIn the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function on a set as the function domain if, given any arbitrarily small positive number , a number can be found such that each of the functions differs from by no more than at every point in .
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.
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.
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.
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.
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.
Absolute convergenceIn mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series is said to converge absolutely if for some real number Similarly, an improper integral of a function, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess – a convergent series that is not absolutely convergent is called conditionally convergent, while absolutely convergent series behave "nicely".
Convergence testsIn mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series . If the limit of the summand is undefined or nonzero, that is , then the series must diverge. In this sense, the partial sums are Cauchy only if this limit exists and is equal to zero. The test is inconclusive if the limit of the summand is zero. This is also known as the nth-term test, test for divergence, or the divergence test.
Partial differential equationIn mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similar to how x is thought of as an unknown number to be solved for in an algebraic equation like x2 − 3x + 2 = 0. However, it is usually impossible to write down explicit formulas for solutions of partial differential equations.
Outer measureIn the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer measures was first introduced by Constantin Carathéodory to provide an abstract basis for the theory of measurable sets and countably additive measures.
Measure (mathematics)In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge.
Hyperbolic partial differential equationIn mathematics, a hyperbolic partial differential equation of order is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface. Many of the equations of mechanics are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. The model hyperbolic equation is the wave equation.
Differential equationIn 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 derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
Convergent seriesIn mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence defines a series S that is denoted The nth partial sum Sn is the sum of the first n terms of the sequence; that is, A series is convergent (or converges) if the sequence of its partial sums tends to a limit; that means that, when adding one after the other in the order given by the indices, one gets partial sums that become closer and closer to a given number.
Separation of variablesIn mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation. A differential equation for the unknown will be separable if it can be written in the form where and are given functions. This is perhaps more transparent when written using as: So now as long as h(y) ≠ 0, we can rearrange terms to obtain: where the two variables x and y have been separated.
Elliptic partial differential equationSecond-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form where A, B, C, D, E, F, and G are functions of x and y and where , and similarly for . A PDE written in this form is elliptic if with this naming convention inspired by the equation for a planar ellipse.
Haar measureIn mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Haar measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory.
Ordinary differential equationIn mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equations which may be with respect to one independent variable. 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 a_0(x), .
