Liouville–Neumann series

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 Template:Mvar,

<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 K₀ may be taken to be Template:Math, 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 K₀ is again Template:Math.

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, Template:Isbn
• Template:Citation

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