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

$\varphi \left(x\right)=\sum _{n=0}^{\mathrm{\infty }}{\lambda }^n{\varphi }_n\left(x\right)$

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

$f(x)=\varphi (x)-\lambda \underset{a}{\overset{b}{\int }}K(x,s)\varphi (s)ds.$

If the nth iterated kernel is defined as n−1 nested integrals of n operator kernels Template:Mvar,

$K_n(x,z)=\int \int \cdots \int K(x,y_1)K(y_1,y_2)\cdots K(y_{n-1},z)d{y}_{1}d{y}_2\cdots d{y}_{n-1}$

then

${\varphi }_n\left(x\right)=\int K_n(x,z)f\left(z\right)dz$

with

${\varphi }_0\left(x\right)=f\left(x\right),$

so K₀ may be taken to be Template:Math, 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,

$R(x,z;\lambda )=\sum _{n=0}^{\mathrm{\infty }}{\lambda }^n{K}_n(x,z),$

where K₀ is again Template:Math.

The solution of the integral equation thus becomes simply

$\varphi \left(x\right)=\int R(x,z;\lambda )f\left(z\right)dz.$

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, Template:Isbn
• Template:Citation

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