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In mathematics, integral equations are equations in which an unknown function appears under an integral sign. [1] In mathematical notation, integral equations may thus be expressed as being of the form: where is an integral operator acting on u. Hence, integral equations may be viewed as the analog to differential equations where instead of the ...
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations.The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the ...
Calculus. In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, [1] is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can loosely be thought of as using the chain rule "backwards."
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, [a] the other being differentiation. Integration was initially used to solve problems in mathematics and ...
More detail may be found on the following pages for the lists of integrals: Gradshteyn, Ryzhik, Geronimus, Tseytlin, Jeffrey, Zwillinger, and Moll 's (GR) Table of Integrals, Series, and Products contains a large collection of results. An even larger, multivolume table is the Integrals and Series by Prudnikov, Brychkov, and Marichev (with ...
In mathematics, the Volterra integral equations are a special type of integral equations. [1] They are divided into two groups referred to as the first and the second kind. A linear Volterra equation of the first kind is. where f is a given function and x is an unknown function to be solved for. A linear Volterra equation of the second kind is.
Nyström method. In mathematics numerical analysis, the Nyström method[1] or quadrature method seeks the numerical solution of an integral equation by replacing the integral with a representative weighted sum. The continuous problem is broken into discrete intervals; quadrature or numerical integration determines the weights and locations of ...
Mercer's theorem. In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in (Mercer 1909), is one of the most notable results of the work of James Mercer (1883–1932).