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.

Definition

The Liouville-Neumann series is defined as

:$phileft(x\; ight)\; =\; sum^infty\_\{n=0\}\; lambda^n\; phi\_n\; left(x\; ight)$

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 "n"th iterated kernel is defined as

:$K\_nleft(x,z\; ight)\; =\; intintcdotsint\; Kleft(x,y\_1\; ight)Kleft(y\_1,y\_2\; ight)\; cdots\; Kleft(y\_\{n-1\},\; z\; ight)\; dy\_1\; dy\_2\; cdots\; dy\_\{n-1\}$

then

:$phi\_nleft(x\; ight)\; =\; int\; K\_nleft(x,z\; ight)fleft(z\; ight)dz$

The resolvent or solving kernel is given by

:$Kleft(x,\; z;lambda\; ight)\; =\; sum^infty\_\{n=0\}\; lambda^n\; K\_\{n+1\}\; left(x,\; z\; ight)$

The solution of the integral equation becomes

:$phileft(x\; ight)\; =\; int\; K\; left(\; x,\; z;lambda\; ight)\; fleft(z\; ight)dz$

Similar methods may be used to solve the Volterra equations.