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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.

Integral equations are classified in three ways. There are thus eight kinds of integral equations.

If the limits of integration are fixed, it is called a Fredholm equation. If one limit of integration is variable, it is called a Volterra equation .
If the unknown function appears only under the integral sign, it is called an integral equation of the first kind. If the unknown function appears both inside and outside the integral sign, it is called an integral equation of the second kind.
If the known function f (see below) is identically zero, it is called a homogeneous integral equation. If f is nonzero, it is called an inhomogeneous integral equation.

As examples of four types of integral equations (lumping homogeneous and inhomogeneous equations together) we have the following. The notation follows Arfken. Here φ is the unknown function, f is a known function, and K is another known function of two variables, often called the kernel function. The parameter λ is an unknown factor, which plays the same role as the eigenvalue in linear algebra.

A Fredholm equation of the first type:

$f(x) = \int_a^b K(x,t)\,\phi(t)\,dt$

A Fredholm equation of the second type:

$\phi(x) = f(x) + \lambda \int_a^b K(x,t)\,\phi(t)\,dt$

A Volterra equation of the first type:

$f(x) = \int_a^x K(x,t)\,\phi(t)\,dt$

A Volterra equation of the second type:

$\phi(x) = f(x) + \lambda \int_a^x K(x,t)\,\phi(t)\,dt$

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

References

George Arfken and Hans Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000.

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