Neumann Series (Integral Equation)

$\begin{displaymath} \phi (x) = f(x)+\int^b_a K(x,t)\phi (t)\,dt \end{displaymath}$

(1)

may be solved as follows. Take

$\displaystyle \phi_0(x)$	$\textstyle \equiv$	$\displaystyle f(x)$	(2)
$\displaystyle \phi_1(x)$	$\textstyle =$	$\displaystyle f(x)+\lambda \int^b_a K(x,t)f(t)\,dt$	(3)
$\displaystyle \phi_2(x)$	$\textstyle =$	$\displaystyle f(x)+\lambda \int^b_a K(x,t_1)f(t_1)\,dt_1+\lambda^2\int^b_a \int^b_a K(x,t_1)K(t_1,t_2)f(t_2)\,dt_2\,dt_1$	(4)
$\displaystyle \phi_n(x)$	$\textstyle =$	$\displaystyle \sum_{i=0}^n \lambda^iu_i(x),$	(5)

where

$\displaystyle u_0(x)$	$\textstyle =$	$\displaystyle f(x)$	(6)
$\displaystyle u_1(x)$	$\textstyle =$	$\displaystyle \int^b_a K(x,t)f(t_1)\,dt_1$	(7)
$\displaystyle u_2(x)$	$\textstyle =$	$\displaystyle \int^b_a \int^b_a K(x,t_1)K(t_1,t_2)f(t_2)\,dt_2\,dt_1$	(8)
$\displaystyle u_n(x)$	$\textstyle =$	$\displaystyle \int^b_a \int^b_a \int^b_a K(x,t_1)K(t_1,t_2)\cdots K(t_{n-1},t_n)f(t_n)\,dt_n \cdots dt_1.$	(9)

The Neumann series solution is then

$\begin{displaymath} \phi(x) = \lim_{n\to \infty} \phi_n(x) = \lim_{n\to \infty} \sum_{i=0}^n \lambda^iu_i(x). \end{displaymath}$

(10)

References

Arfken, G. ``Neumann Series, Separable (Degenerate) Kernels.'' §16.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 879-890, 1985.