System of linear integral equations

Summary

A system of linear integral equations has several unknown functions $\varphi_1, \ldots, \varphi_n$, and each equation involves integrals of all of them. In vector notation it is a single linear integral equation whose kernel is an $n \times n$ matrix of functions. Much of the theory for one equation, such as Fredholm theory, carries over. The reason is that a system on an interval can be rewritten as one equation on a longer interval. Systems arise from coupled physical fields, from boundary integral methods for vector problems, and from the factorization of matrix symbols.