Integral equation
Equations with an unknown function under an integral sign / From Wikipedia, the free encyclopedia
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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 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 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.