# Adaptive Piecewise Poly-Sinc Methods for Ordinary Differential Equations
## 4 Adaptive Piecewise Poly-Sinc Algorithm
### Error Analysis
(b) Plot of the statistic \(\omega_{l}\). ## 6 Conclusions

In this paper, we developed an adaptive piecewise collocation method based on Poly-Sinc interpolation for the approximation of solutions to ODEs. We showed the exponential convergence in the number of iterations of the a priori error estimate obtained from the piecewise collocation method, and provided that a good estimate of the exact solution \(y(x)\) at the Sinc points exists. We used a statistical approach for partition refinement, in which we computed the fraction of a standard deviation as the ratio of the mean absolute deviation to the sample standard deviation. We demonstrated by several examples that an exponential error decay is observed for regular ODEs and ODEs whose exact solutions exhibit an interior layer, a boundary layer, and a shock layer. We showed that our adaptive algorithm can deliver results with high accuracy at the expense of the slower computation for stiff ODEs. Conceptualization, M.Y. and G.B. ; methodology, O.K., M.Y. and G.B. ; software, O.K. and G.B. ; validation, O.K., H.E.-S., M.Y. and G.B. ; formal analysis, O.K. and H.E.-S.; investigation, H.E.-S. and G.B. ; resources, not applicable; data curation, not applicable; writing--original draft preparation, O.K. ; writing--review and editing, G.B. and O.K. ; visualization, O.K. ; supervision, H.E.-S., M.Y. and G.B. ; project administration, H.E.-S.; funding acquisition, there is no funding of this article. All authors have read and agreed to the published version of the manuscript. M.Y. is funded by BMBF under the contract 05M20 VSA. Not applicable. The authors thank [PERSON] for his insightful conversations and continuous guidance. The authors declare no conflict of interest. ## References

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