Integral equation

In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way. See, for example, Maxwell's equations.

Overview

The most basic type of integral equation is a "Fredholm equation of the first type":

:$f(x)\; =\; int\_a^b\; K(x,t),varphi(t),dt.$

The notation follows Arfken. Here φ is an unknown function, "f" is a known function,and "K" is another known function of two variables,often called the kernel function.Note that the limits of integration are constant; this is what characterizes a Fredholm equation.

If the unknown function occurs both inside and outside of the integral, it is known as a "Fredholm equation of the second type":

:$varphi(x)\; =\; f(x)+\; lambda\; int\_a^b\; K(x,t),varphi(t),dt.$

The parameter λ is an unknown factor,which plays the same role as the eigenvalue in linear algebra.

If one limit of integration is variable, it is called a Volterra equation. Thus "Volterra equations of the first and second types", respectively, would appear as:

:$f(x)\; =\; int\_a^x\; K(x,t),varphi(t),dt$:$varphi(x)\; =\; f(x)\; +\; lambda\; int\_a^x\; K(x,t),varphi(t),dt.$

In all of the above, if the known function "f" is identically zero, it is called a "homogeneous integral equation". If "f" is nonzero, it is called an "inhomogeneous integral equation".

In summary, integral equations are classified according to three different dichotomies, creating eight different kinds:

;Limits of integration: both fixed: Fredholm equation: one variable: Volterra equation;Placement of unknown function: only inside integral: first kind: both inside and outside integral: second kind;Nature of known function "f": identically zero: homogeneous: not identically zero: inhomogeneous

Integral equations are important in many applications. Problems in which integral equations are encountered include radiative energy transfer and the oscillation of a string, membrane, or axle. Oscillation problems may also be solved as differential equations.

ee also

*Hilbert-Schmidt operator

References

* George Arfken and Hans Weber. "Mathematical Methods for Physicists". Harcourt/Academic Press, 2000.
* Andrei D. Polyanin and Alexander V. Manzhirov "Handbook of Integral Equations". CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4.
* E. T. Whittaker and G. N. Watson. "A Course of Modern Analysis" Cambridge Mathematical Library.

External links

* [http://eqworld.ipmnet.ru/en/solutions/ie.htm Integral Equations: Exact Solutions] at EqWorld: The World of Mathematical Equations.
* [http://eqworld.ipmnet.ru/en/solutions/eqindex/eqindex-ie.htm Integral Equations: Index] at EqWorld: The World of Mathematical Equations.
* [http://www.exampleproblems.com/wiki/index.php/Integral_Equations Integral equations] at exampleproblems.com