Fredholm theory in the context of Integral equations

In mathematical analysis, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: $${\displaystyle f(x_{1},x_{2},x_{3},\ldots ,x_{n};u(x_{1},x_{2},x_{3},\ldots ,x_{n});I^{1}(u),I^{2}(u),I^{3}(u),\ldots ,I^{m}(u))=0}$$where ${\displaystyle I^{i}(u)}$ is an integral operator acting on u. Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals. 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:$${\displaystyle f(x_{1},x_{2},x_{3},\ldots ,x_{n};u(x_{1},x_{2},x_{3},\ldots ,x_{n});D^{1}(u),D^{2}(u),D^{3}(u),\ldots ,D^{m}(u))=0}$$where ${\displaystyle D^{i}(u)}$ may be viewed as a differential operator of order i. Due to this close connection between differential and integral equations, one can often convert between the two. 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. 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. See also, for example, Green's function and Fredholm theory.