In mathematics, **integral equations** are equations 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, Green's function, Fredholm theory, and Maxwell's equations.

## Overview

The most basic type of integral equation is called a *Fredholm equation of the first type*,

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, the equation is known as a *Fredholm equation of the second type*,

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

If one limit of integration is a variable, the equation is called a Volterra equation. The following are called *Volterra equations of the first and second types*, respectively,

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

## Numerical solution

It is worth noting that integral equations often do not have an analytical solution, and must be solved numerically. An example of this is evaluating the Electric-Field Integral Equation (EFIE) or Magnetic-Field Integral Equation (MFIE) over an arbitrarily shaped object in an electromagnetic scattering problem.

One method to solve numerically requires discretizing variables and replacing integral by a quadrature rule

Then we have a system with n equations and n variables. By solving it we get the value of the n variables

## Classification

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 transfer, and the oscillation of a string, membrane, or axle. Oscillation problems may also be solved as differential equations.

Both Fredholm and Volterra equations are linear integral equations, due to the linear behaviour of *φ*(*x*) under the integral. A nonlinear Volterra integral equation has the general form:

where F is a known function.

## Wiener–Hopf integral equations

## Power series solution for integral equations

In many cases, if the Kernel of the integral equation is of the form *K*(*xt*) and the Mellin transform of *K*(*t*) exists, we can find the solution of the integral equation

in the form of a power series

where

are the Z-transform of the function *g*(*s*), and *M*(*n* + 1) is the Mellin transform of the Kernel.

## Integral equations as a generalization of eigenvalue equations

Certain homogeneous linear integral equations can be viewed as the continuum limit of eigenvalue equations. Using index notation, an eigenvalue equation can be written as

where **M** = [*M _{i,j}*] is a matrix,

**v**is one of its eigenvectors, and λ is the associated eigenvalue.

Taking the continuum limit, i.e., replacing the discrete indices i and j with continuous variables x and y, yields

where the sum over j has been replaced by an integral over y and the matrix **M** and the vector **v** have been replaced by the *kernel* *K*(*x*, *y*) and the eigenfunction *φ*(*y*). (The limits on the integral are fixed, analogously to the limits on the sum over j.) This gives a linear homogeneous Fredholm equation of the second type.

In general, *K*(*x*, *y*) can be a distribution, rather than a function in the strict sense. If the distribution K has support only at the point *x* = *y*, then the integral equation reduces to a differential eigenfunction equation.

In general, Volterra and Fredholm integral equations can arise from a single differential equation, depending on which sort of conditions are applied at the boundary of the domain of its solution.

## Applications

- Actuarial science (ruin theory
^{[1]}) - Computational electromagnetics
- Inverse problems
- Options pricing under jump-diffusion
^{[2]} - Radiative transfer
- Viscoelasticity

## See also

## References

**^**"Lecture Notes on Risk Theory" (PDF). 2010.**^**Sachs, E. W.; Strauss, A. K. (2008-11-01). "Efficient solution of a partial integro-differential equation in finance".*Applied Numerical Mathematics*.**58**(11): 1687–1703. doi:10.1016/j.apnum.2007.11.002. ISSN 0168-9274.

## Further reading

- Kendall E. Atkinson
*The Numerical Solution of Integral Equations of the Second Kind*. Cambridge Monographs on Applied and Computational Mathematics, 1997. - George Arfken and Hans Weber.
*Mathematical Methods for Physicists*. Harcourt/Academic Press, 2000. - Harry Bateman (1910) History and Present State of the Theory of Integral Equations,
*Report*of the British Association. - 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. - M. Krasnov, A. Kiselev, G. Makarenko,
*Problems and Exercises in Integral Equations*, Mir Publishers, Moscow, 1971 - Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Chapter 19. Integral Equations and Inverse Theory".
*Numerical Recipes: The Art of Scientific Computing*(3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.

## External links

- Integral Equations: Exact Solutions at EqWorld: The World of Mathematical Equations.
- Integral Equations: Index at EqWorld: The World of Mathematical Equations.
- "Integral equation",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Integral Equations (MIT OpenCourseWare)