Liouville-Neumann series

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.


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