Editing Integral equation (section)

==Integral equations as a generalization of eigenvalue equations==
{{further|Fredholm theory}}Certain homogeneous linear integral equations can be viewed as the [[continuum limit]] of [[Eigenvalue, eigenvector and eigenspace|eigenvalue equations]]. Using [[index notation]], an eigenvalue equation can be written as
:<math> \sum _j M_{i,j} v_j = \lambda v_i</math>
where {{math|1='''M''' = [''M<sub>i,j</sub>'']}} is a matrix, {{math|'''v'''}} is one of its eigenvectors, and {{mvar|λ}} is the associated eigenvalue.

Taking the continuum limit, i.e., replacing the discrete indices {{mvar|i}} and {{mvar|j}} with continuous variables {{mvar|x}} and {{mvar|y}}, yields
:<math> \int K(x,y) \, \varphi(y) \, dy = \lambda \, \varphi(x),</math>
where the sum over {{mvar|j}} has been replaced by an integral over {{mvar|y}} and the matrix {{math|'''M'''}} and the vector {{math|'''v'''}} have been replaced by the ''kernel'' {{math|''K''(''x'', ''y'')}} and the [[eigenfunction]] {{math|''φ''(''y'')}}. (The limits on the integral are fixed, analogously to the limits on the sum over {{mvar|j}}.) This gives a linear homogeneous Fredholm equation of the second type.

In general, {{math|''K''(''x'', ''y'')}} can be a [[Distribution (mathematics)|distribution]], rather than a function in the strict sense. If the distribution {{mvar|K}} has support only at the point {{math|1=''x'' = ''y''}}, then the integral equation reduces to a [[Eigenfunction|differential eigenfunction equation]].

In general, Volterra and Fredholm integral equations can arise from a single differential equation, depending on which sort of conditions are applied at the boundary of the domain of its solution.