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__Optimal adaptive approximation of a class of non-autonomous IVPs with unknown singularities__

P. Przybyłowicz and B. Kacewicz

**Abstract**

The numerical solution of singular initial-value problems in ODEs leads to many interesting theoretical questions. For example, there arise problem of finding strict bounds on error and information cost for algorithms that solve such problems. Optimality of used algorithms is also of interest.
We present recent theoretical results concerning the solution of a class of scalar non-autonomous equations with separated variables and unknown singularities. In the case when the right-hand side function has two singularities (which leads to four unknown 'events' in the two-dimensional space), we show the construction of an algorithm that automatically takes care of the singularities. It adaptively modifies the mesh points to preserve the optimal error known for the regular case. Lower bounds in the case of more than two singularities will also be discussed.

**Bibliography**

[1]
B. Kacewicz and P. Przyby\l owicz,
Optimal adaptive solution of initial-value problems
with unknown singularities, J. Complexity, vol 24 (2008), pp. 455-476.

[2]
B. Kacewicz and P. Przyby\l owicz,
Optimal solution of a class of non-autonomous initial-value problems with unknown singularities, submitted.