Liouville–Neumann series - Alchetron, the free social encyclopedia

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

ϕ ( x ) = ∑ n = 0 ∞ λ n ϕ n ( x )

which is a unique, continuous solution of a Fredholm integral equation of the second kind,

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

then

ϕ n ( x ) = ∫ K n ( x , z ) f ( z ) d z

with

ϕ 0 ( x ) = f ( x ) ,

so K₀ may be taken to be δ(x−z).

The resolvent (or solving kernel) is then given by a schematic "geometric series",

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

The solution of the integral equation thus becomes simply

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

where K₀ has been taken to be δ(x−z).

Similar methods may be used to solve the Volterra equations.

References

Liouville–Neumann series Wikipedia

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