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

${\displaystyle \phi \left(x\right)=\sum _{n=0}^{\infty }\lambda ^{n}\phi _{n}\left(x\right)}$

which, provided that ${\displaystyle \lambda }$ is small enough so that the series converges, is the unique continuous solution of the Fredholm integral equation of the second kind,

${\displaystyle f(t)=\phi (t)-\lambda \int _{a}^{b}K(t,s)\phi (s)\,ds.}$

If the nth iterated kernel is defined as n−1 nested integrals of n operators K,

${\displaystyle K_{n}\left(x,z\right)=\int \int \cdots \int K\left(x,y_{1}\right)K\left(y_{1},y_{2}\right)\cdots K\left(y_{n-1},z\right)dy_{1}dy_{2}\cdots dy_{n-1}}$

then

${\displaystyle \phi _{n}\left(x\right)=\int K_{n}\left(x,z\right)f\left(z\right)dz}$

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 for the integral operator) is then given by a schematic analog "geometric series",

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

where K₀ has been taken to be δ(x−z).

The solution of the integral equation thus becomes simply

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

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" (PDF), Acta Mathematica, 27: 365–390, doi:10.1007/bf02421317