Integral equations

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, Green's function, Fredholm theory, and Maxwell's equations.

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.

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.

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