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Integral Equation Neumann Series

An integral equation Neumann series is the specialization of a Neumann series to a Fredholm integral equation of the second kind of the form

phi(x)=f(x)+lambdaint_a^bK(x,t)phi(t)dt
(1)

It is obtained by taking

phi_0(x)=f(x)
(2)
phi_1(x)=f(x)+lambdaint_a^bK(x,t)f(t)dt
(3)
phi_2(x)=f(x)+lambdaint_a^bK(x,t_1)f(t_1)dt_1+lambda^2int_a^bint_a^bK(x,t_1)K(t_1,t_2)f(t_2)dt_2dt_1
(4)
phi_n(x)=sum_(i=0)^(n)lambda^iu_i(x),
(5)

where

u_0(x)=f(x)
(6)
u_1(x)=int_a^bK(x,t_1)f(t_1)dt_1
(7)
u_2(x)=int_a^bint_a^bK(x,t_1)K(t_1,t_2)f(t_2)dt_2dt_1
(8)
u_n(x)=int_a^b...int_a^bK(x,t_1)K(t_1,t_2)...K(t_(n-1),t_n)f(t_n)dt_n...dt_1.
(9)

Let T be the bounded operator on a Banach space of functions defined by (Tpsi)(x)=int_a^bK(x,t)psi(t)dt. If |lambda|||T||<1, where ||T|| is the operator norm, then the Neumann series converges in norm and the unique solution is

phi(x)=lim_(n->infty)phi_n(x)=lim_(n->infty)sum_(i=0)^nlambda^iu_i(x).
(10)

See also

Banach Space, Fredholm Integral Equation of the Second Kind, Neumann Series, Operator Norm

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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.

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Integral Equation Neumann Series

Cite this as:

Weisstein, Eric W. "Integral Equation Neumann Series." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/IntegralEquationNeumannSeries.html

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