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 (iterative) 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 n−1 nested integrals of n operators K,

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

then

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

with

${\displaystyle \phi _{0}\left(x\right)=f\left(x\right)~,}$

so K₀ may be taken to be δ(x−z).

The resolvent (or solving kernel) is then given by a schematic "geometric series",

${\displaystyle R\left(x,z;\lambda \right)=\sum _{n=0}^{\infty }\lambda ^{n}K_{n}\left(x,z\right).}$

The solution of the integral equation thus becomes simply

${\displaystyle \phi \left(x\right)=\int R\left(x,z;\lambda \right)f\left(z\right)dz}$

where K₀ has been taken to be δ(x−z).

Similar methods may be used to solve the Volterra equations.

References

• Mathews, Jon; Walker, Robert L. (1970), Mathematical methods of physics (2nd ed.), New York: W. A. Benjamin, ISBN 0-8053-7002-1
• Fredholm, Erik I. (1903), "Sur une classe d'equations fonctionnelles", Acta Mathematica, 27: 365–390, doi:10.1007/bf02421317