“integral equation”的概念、定义、翻译、参考文献-科学参考

integral equation

Physics

An equation in which the unknown function is included under an integral sign (see integration). There are many problems in physics that can be expressed as either a differential equation or an integral equation, and it is sometimes more convenient to solve them using the latter. Integrating a differential equation gives rise to an integral equation with built-in boundary conditions. See also calculus.

Mathematics

An equation that involves integration of functions. In particular, the information given in an initial-value problem such as y’(x) = y with y(0) = 1 is equivalent to the integral equation

$y (x) = 1 + \int_{t = 0}^{t = x} y (t) d t .$

This rewriting can be useful as the integral equation lends itself better to treatment by theorems of analysis, particularly to find approximate solutions.

Computer

Any equation for an unknown function f(x), a≤x≤b, involving integrals of the function. An equation of the form
$f (x) = \int_{a}^{x} K (x, y) f (y) d y + g (x)$
is a Volterra equation of the second kind. The analogous equation with constant limits
$f (x) = \int_{a}^{b} K (x, y) f (y) d y + g (x)$
is a Fredholm equation of the second kind. If the required function only appears under the integral sign it is a Volterra or Fredholm equation of the first kind; these are more difficult to treat both theoretically and numerically. The Volterra equation can be regarded as a particular case of the Fredholm equation where
$K (x, y) = 0 for y > x$
Fredholm equations of the second kind occur commonly in boundary-value problems in mathematical physics. Numerical techniques proceed by replacing the integral with a rule for numerical integration, leading to a set of linear algebraic equations determining approximations to f(x) at a set of points in a≤x≤b.

单词	integral equation
释义	integral equation Physics An equation in which the unknown function is included under an integral sign (see integration). There are many problems in physics that can be expressed as either a differential equation or an integral equation, and it is sometimes more convenient to solve them using the latter. Integrating a differential equation gives rise to an integral equation with built-in boundary conditions. See also calculus. Mathematics An equation that involves integration of functions. In particular, the information given in an initial-value problem such as y’(x) = y with y(0) = 1 is equivalent to the integral equation $y (x) = 1 + \int_{t = 0}^{t = x} y (t) d t .$ This rewriting can be useful as the integral equation lends itself better to treatment by theorems of analysis, particularly to find approximate solutions. Computer Any equation for an unknown function f(x), a≤x≤b, involving integrals of the function. An equation of the form $f (x) = \int_{a}^{x} K (x, y) f (y) d y + g (x)$ is a Volterra equation of the second kind. The analogous equation with constant limits $f (x) = \int_{a}^{b} K (x, y) f (y) d y + g (x)$ is a Fredholm equation of the second kind. If the required function only appears under the integral sign it is a Volterra or Fredholm equation of the first kind; these are more difficult to treat both theoretically and numerically. The Volterra equation can be regarded as a particular case of the Fredholm equation where $K (x, y) = 0 for y > x$ Fredholm equations of the second kind occur commonly in boundary-value problems in mathematical physics. Numerical techniques proceed by replacing the integral with a rule for numerical integration, leading to a set of linear algebraic equations determining approximations to f(x) at a set of points in a≤x≤b.
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