Liouville-Neumann series

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\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[?].