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##

4.2.6 Numerov's Method

Mostly used for a certain kind of *initial value problems* (IVPs)
that arise when *boundary value problems* (BVPs) are rewritten as IVPs.
Many BVP has the general form

with given boundary values and .

Example: one-dimensional Poisson equation

with given at and . Thus and
.
Let us assume we have, instead of and ,
the full set of initial values and
.

Divide the interval
into sub-intervals of length and at each intermediate point
expand into a power series. Adding the Taylor formulae
for and we find

Inserting
we have

The fourth derivative is approximated by

In this way we find Numerov's formula

__EXERCISE:__
Write a code that permits to solve a given second-order equation of motion
by various algorithms. Apply the program to problems of point mechanics and
explore the stabilities and accuracies of the diverse techniques.

** Next:** 4.3 Boundary Value Problems
**Up:** 4.2 Initial Value Problems
** Previous:** 4.2.5 Symplectic Algorithms
* Franz J. Vesely Oct 2005*

See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001