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Liouville–Neumann series

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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

ϕ ( x ) = ∑ n = 0 ∞ λ n ϕ n ( x ) {\displaystyle \phi \left(x\right)=\sum _{n=0}^{\infty }\lambda ^{n}\phi _{n}\left(x\right)}

which, provided that λ {\displaystyle \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 ) = ϕ ( x ) − λ ∫ a b K ( x , s ) ϕ ( s ) d s . {\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,

K n ( x , z ) = ∫ ∫ ⋯ ∫ K ( x , y 1 ) K ( y 1 , y 2 ) ⋯ K ( y n − 1 , z ) d y 1 d y 2 ⋯ d y n − 1 {\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

ϕ n ( x ) = ∫ K n ( x , z ) f ( z ) d z {\displaystyle \phi _{n}\left(x\right)=\int K_{n}\left(x,z\right)f\left(z\right)dz}

with

ϕ 0 ( x ) = f ( x )   , {\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,

R ( x , z ; λ ) = ∑ n = 0 ∞ λ n K n ( x , z ) , {\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

ϕ ( x ) = ∫ R ( x , z ; λ ) f ( z ) d z . {\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

References


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