Liouville–Neumann series explained

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

\phi\left(x\right)=

λⁿ\phi_n\left(x\right)

which, provided that

is small enough so that the series converges, is the unique continuous solution of the Fredholm integral equation of the second kind,

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

K_{n\left(x,z\right)}=\int\int … \intK\left(x,y_{1\right)K\left(y}_1,y_2\right) … K\left(y_n-1,z\right)dy₁dy₂ … dy_n-1

then

\phi_{n\left(x\right)}=\intK_{n\left(x,z\right)f\left(z\right)dz}

with

\phi_{0\left(x\right)}=f\left(x\right)~,

so K₀ may be taken to be, 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\left(x,z;λ\right)=

λⁿK_n\left(x,z\right),

where K₀ is again .

The solution of the integral equation thus becomes simply

\phi\left(x\right)=\intR\left(x,z;λ\right)f\left(z\right)dz.

Similar methods may be used to solve the Volterra integral equations.

Mathews, Jon; Walker, Robert L. (1970), Mathematical methods of physics (2nd ed.), New York: W. A. Benjamin,