- Liouville-Neumann series
In
mathematics , the Liouville-Neumann series is aninfinite series that corresponds to theresolvent formalism technique of solving theFredholm integral equation s inFredholm 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 equation s.
Wikimedia Foundation. 2010.