Liouville–Neumann series

In mathematics, the Liouville–Neumann series is a function series that results from applying the resolvent formalism to solve Fredholm integral equations in Fredholm theory.

Definition

The Liouville–Neumann 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,

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

If the nth iterated kernel is defined as n−1 nested integrals of n operator kernels 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 kernel of the identity operator.

The resolvent, also called the "solution kernel" for the integral operator, is then given by a generalization of the geometric series,

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

where K₀ is again 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 integral equations.

See also

• Neumann series

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