Abstract:
|
Our goal in this work is to give an algorithmic method to find solutions of a LSS. To accomplish
this objective, we will give a geometric description of the problem and use some geometric tools
to transform the equation and the manifold (the "ambient space") into a new set of equations,
each one being an explicit ODE. When this is accomplished, one may apply Picard's theorem
to these new systems to obtain the solutions.
It is important to notice that (as many other theoretical results) Picard's theorem proves the
existence of a solution but does not give an explicit method to nd it. Usually, in problems
which require to nd explicit solutions, numerical methods are used. There also exist numerical
methods for solving LSS, so one may think that (since we will use numerical methods anyway
to nd the explicit solution) we could use them directly instead of wasting time implementing a
method that will still depend on numerical approximations. The main point of using alternative
methods for this kind of problem is (beyond the theoretical value of the algorithm) that, while
numerical methods for explicit ODEs are usually accurate (in the sense that it is easy to control
the error of the calculations), numerical approximations for an arbitrary LSS require (as it
is explained in [RR02]) numerical inversions of quasi-singular matrices; these operations are
numerically unstable. Therefore, the desingularization procedure studied here may allow to
transform a numerically unstable problem into a new stable one. |