Integral Equation - Classification

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 energy 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.