In mathematics, the Liouville–Neumann series is a function series that results from applying the resolvent formalism to solve Fredholm integral equations in Fredholm theory.
Definition
The Liouville–Neumann series is defined as
:<math>\phi\left(x\right) = \sum^\infty_{n=0} \lambda^n \phi_n \left(x\right)</math>
which, provided that <math>\lambda</math> is small enough so that the series converges, is the unique continuous solution of the Fredholm integral equation of the second kind,
If the nth iterated kernel is defined as n−1 nested integrals of n operator kernels ,
:<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>
with
:<math>\phi_0\left(x\right) = f\left(x\right)~,</math>
so K<sub>0</sub> may be taken to be , the kernel of the identity operator.
The resolvent, also called the "solution kernel" for the integral operator, is then given by a generalization of the geometric series,
:<math>R\left(x, z;\lambda\right) = \sum^\infty_{n=0} \lambda^n K_{n} \left(x, z\right),
</math>
where K<sub>0</sub> is again .
The solution of the integral equation thus becomes simply
:<math>\phi\left(x\right) = \int R\left( x, z;\lambda\right) f\left(z\right)dz.</math>
Similar methods may be used to solve the Volterra integral equations.
See also
- Neumann series
References
- Mathews, Jon; Walker, Robert L. (1970), Mathematical methods of physics (2nd ed.), New York: W. A. Benjamin,