{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T06:55:00Z","timestamp":1762066500037,"version":"build-2065373602"},"reference-count":67,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,8]],"date-time":"2022-09-08T00:00:00Z","timestamp":1662595200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"BMBF","award":["05M20VSA"],"award-info":[{"award-number":["05M20VSA"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>We propose a new method of adaptive piecewise approximation based on Sinc points for ordinary differential equations. The adaptive method is a piecewise collocation method which utilizes Poly-Sinc interpolation to reach a preset level of accuracy for the approximation. Our work extends the adaptive piecewise Poly-Sinc method to function approximation, for which we derived an a priori error estimate for our adaptive method and showed its exponential convergence in the number of iterations. In this work, we show the exponential convergence in the number of iterations of the a priori error estimate obtained from the piecewise collocation method, provided that a good estimate of the exact solution of the ordinary differential equation at the Sinc points exists. We use a statistical approach for partition refinement. The adaptive greedy piecewise Poly-Sinc algorithm is validated on regular and stiff ordinary differential equations.<\/jats:p>","DOI":"10.3390\/a15090320","type":"journal-article","created":{"date-parts":[[2022,9,8]],"date-time":"2022-09-08T20:50:27Z","timestamp":1662670227000},"page":"320","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Adaptive Piecewise Poly-Sinc Methods for Ordinary Differential Equations"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5704-0340","authenticated-orcid":false,"given":"Omar","family":"Khalil","sequence":"first","affiliation":[{"name":"Mathematics Department, German University in Cairo, New Cairo City 11835, Egypt"}]},{"given":"Hany","family":"El-Sharkawy","sequence":"additional","affiliation":[{"name":"Mathematics Department, German University in Cairo, New Cairo City 11835, Egypt"},{"name":"Department of Mathematics, Faculty of Science, Ain Shams University, Abbassia 11566, Egypt"}]},{"given":"Maha","family":"Youssef","sequence":"additional","affiliation":[{"name":"Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1642-2462","authenticated-orcid":false,"given":"Gerd","family":"Baumann","sequence":"additional","affiliation":[{"name":"Mathematics Department, German University in Cairo, New Cairo City 11835, Egypt"},{"name":"Faculty of Natural Science, University of Ulm, Albert\u2013Einstein\u2013Allee 11, D-89069 Ulm, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,8]]},"reference":[{"key":"ref_1","unstructured":"Hairer, E., N\u00f8rsett, S.P., and Wanner, G. 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