Overview
The most basic type of integral equation is 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, it 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 variable, it is called a Volterra equation. Thus Volterra equations of the first and second types, respectively, would appear as:
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.
Read more about this topic: Integral Equation