Editing Integral equation (section)

{{Short description|Equations with an unknown function under an integral sign}}
{{for2|equations of [[integer]] unknowns|Diophantine equation|the term in commutative algebra|Integral element}}

In [[mathematical analysis]], integral equations are equations in which an unknown [[Function (mathematics)|function]] appears under an [[integral]] sign.<ref name=":0" /> In mathematical notation, integral equations may thus be expressed as being of the form: <math display="block">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</math>
where <math>I^i(u)</math> is an [[integral operator]] acting on ''u.'' Hence, integral equations may be viewed as the analog to [[differential equation]]s 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:<math display="block">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</math>where <math>D^i(u)</math> may be viewed as a [[differential operator]] of order ''i''.<ref name=":0" /> 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.<ref name=":0" /> 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.<ref>{{Cite web |last=admin |date=2022-09-10 |title=Maxwell's Equations: Derivation in Integral and Differential form |url=https://oxscience.com/maxwells-equations/ |access-date=2022-12-10 |website=Ox Science |language=en-US}}</ref> See also, for example, [[Green's function]] and [[Fredholm theory]].