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Neumann Series (Integral Equation)

A Fredholm Integral Equation of the Second Kind

\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)


$\displaystyle u_0(x)$ $\textstyle =$ $\displaystyle f(x)$ (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
\phi(x) = \lim_{n\to \infty} \phi_n(x) = \lim_{n\to \infty} \sum_{i=0}^n \lambda^iu_i(x).
\end{displaymath} (10)


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

© 1996-9 Eric W. Weisstein