Integral equation

In mathematics, integral equations are equations in which an unknown function appears under an integral sign.^[1] In mathematical notation, integral equations may thus be expressed as being of the form:

where ${\displaystyle I^{i}(u)}$ is an integral operator acting on u.^[1] Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals.^[1] A direct comparison can be seen with the mathematical form of the general integral equation above with the general form of a differential equation which may be expressed as follows:where ${\displaystyle D^{i}(u)}$ may be viewed as a differential operator of order i.^[1] Due to this close connection between differential and integral equations, one can often convert between the two.^[1] For example, one method of solving a boundary value problem is by converting the differential equation with its boundary conditions into an integral equation and solving the integral equation.^[1] In addition, because one can convert between the two, differential equations in physics such as Maxwell's equations often have an analog integral and differential form.^[2] See also, for example, Green's function and Fredholm theory.