Liouville-Neumann series

In mathematics, the Liouville-Neumann series is an infinite series that corresponds to the resolvent formalism technique of solving the Fredholm integral equations in Fredholm theory.

[edit] Definition

The Liouville-Neumann series is defined as

$\phi\left(x\right) = \sum^\infty_{n=0} \lambda^n \phi_n \left(x\right)$

which is a unique, continuous solution of a Fredholm integral equation of the second kind:

$f(t)= \phi(t) - \lambda \int_a^bK(t,s)\phi(s)\,ds$

If the nth iterated kernel is defined as

$K_n\left(x,z\right) = \int\int\cdots\int K\left(x,y_1\right)K\left(y_1,y_2\right) \cdots K\left(y_{n-1}, z\right) dy_1 dy_2 \cdots dy_{n-1}$

then

$\phi_n\left(x\right) = \int K_n\left(x,z\right)f\left(z\right)dz$

The resolvent or solving kernel is given by

$K\left(x, z;\lambda\right) = \sum^\infty_{n=0} \lambda^n K_{n+1} \left(x, z\right)$

The solution of the integral equation becomes

$\phi\left(x\right) = \int K \left( x, z;\lambda\right) f\left(z\right)dz$

Similar methods may be used to solve the Volterra equations.