Polynomial interpolation: Difference between revisions

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-{{Unreferenced|date=December 2009}}
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-In [[mathematics]], the '''Liouville–Neumann series''' is an [[infinite series]] that corresponds to the [[resolvent formalism]] technique of solving the [[Fredholm integral equation]]s in [[Fredholm theory]].
-==Definition==
-The Liouville–Neumann series is defined as
-:<math>\phi\left(x\right) = \sum^\infty_{n=0} \lambda^n \phi_n \left(x\right)</math>
-which is a unique, [[continuous function|continuous]] solution of a [[Fredholm integral equation]] of the second kind:
-:<math>f(t)= \phi(t) - \lambda \int_a^bK(t,s)\phi(s)\,ds.</math>
-If the ''n''th iterated [[Kernel (integral operator)|kernel]] is defined as
-:<math>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}</math>
-then
-:<math>\phi_n\left(x\right) = \int K_n\left(x,z\right)f\left(z\right)dz</math>
-with
-:<math>\phi_0\left(x\right) = f\left(x\right).</math>
-The [[Resolvent formalism|resolvent]] or solving kernel is given by
-:<math>K\left(x, z;\lambda\right) = \sum^\infty_{n=0} \lambda^n K_{n+1} \left(x, z\right).
-</math>
-The solution of the integral equation becomes
-:<math>\phi\left(x\right) = \int K \left( x, z;\lambda\right) f\left(z\right)dz.</math>
-Similar methods may be used to solve the [[Volterra equation]]s.
-{{DEFAULTSORT:Liouville-Neumann Series}}
-[[Category:Fredholm theory]]
-[[Category:Mathematical series]]

Polynomial interpolation: Difference between revisions

Revision as of 13:36, 28 February 2014

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