Liouville-Neumann series
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In mathematics, the Liouville-Neumann series is an infinite series defined as
- <math>\phi\left(x\right) = \sum^\infty_{n=0} \lambda^n \phi_n \left(x\right)<math>
which is a unique, continuous solution of a Fredholm integral equation of the second kind. If the nth iterated 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>
The 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>
hence the solution of the integral equation becomes
- <math>\phi\left(x\right) = f\left(x\right) + \lambda \int K \left( x, z;\lambda\right) f\left(z\right)dz<math>
Similar methods may be used to solve the Volterra equations.