{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T03:43:11Z","timestamp":1771472591644,"version":"3.50.1"},"reference-count":15,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,17]],"date-time":"2022-10-17T00:00:00Z","timestamp":1665964800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper is devoted to a new first order Taylor-like formula, where the corresponding remainder is strongly reduced in comparison with the usual one, which appears in the classical Taylor\u2019s formula. To derive this new formula, we introduce a linear combination of the first derivative of the concerned function, which is computed at n+1 equally spaced points between the two points, where the function has to be evaluated. We show that an optimal choice of the weights in the linear combination leads to minimizing the corresponding remainder. Then, we analyze the Lagrange P1- interpolation error estimate and the trapezoidal quadrature error, in order to assess the gain of the accuracy we obtain using this new Taylor-like formula.<\/jats:p>","DOI":"10.3390\/axioms11100562","type":"journal-article","created":{"date-parts":[[2022,10,17]],"date-time":"2022-10-17T05:08:02Z","timestamp":1665983282000},"page":"562","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A New First Order Expansion Formula with a Reduced Remainder"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1263-5313","authenticated-orcid":false,"given":"Joel","family":"Chaskalovic","sequence":"first","affiliation":[{"name":"Jean Le Rond d\u2019Alembert, Sorbonne University, 4 Place Jussieu, CEDEX 05, 75252 Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hessam","family":"Jamshidipour","sequence":"additional","affiliation":[{"name":"Jean Le Rond d\u2019Alembert, Sorbonne University, 4 Place Jussieu, CEDEX 05, 75252 Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Chaskalovic, J. (2013). Mathematical and Numerical Methods for Partial Differential Equations, Springer.","DOI":"10.1007\/978-3-319-03563-5"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Chaskalovic, J. (2021). A probabilistic approach for solutions of determinist PDE\u2019s as well as their finite element approximations, 2020. Axioms, 10.","DOI":"10.3390\/axioms10040349"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"684","DOI":"10.3846\/mma.2021.14079","article-title":"Numerical validation of probabilistic laws to evaluate finite element error estimates","volume":"26","author":"Chaskalovic","year":"2021","journal-title":"Math. Model. Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"462","DOI":"10.1016\/j.cam.2013.12.015","article-title":"Indeterminate Constants in Numerical Approximations of PDE\u2019s: A Pilot Study Using Data Mining Techniques","volume":"270","author":"Assous","year":"2014","journal-title":"J. Comput Appl. 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Handbook of Analytic-Computational Methods in Applied Mathematics, CRC Press.","DOI":"10.1201\/9781420036053"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"163","DOI":"10.5556\/j.tkjm.37.2006.161","article-title":"Applications of Ostrowski\u2019s version of the Gr\u00fcss inequality for trapezoid type rules","volume":"37","author":"Barnett","year":"2006","journal-title":"Tamkang J. Math."},{"key":"ref_14","unstructured":"Atkinson, K.E. (1989). An Introduction to Numerical Analysis, Wiley and Sons. [2nd ed.]."},{"key":"ref_15","first-page":"29","article-title":"A note on the perturbed trapezoid inequality","volume":"3","author":"Cheng","year":"2002","journal-title":"J. Inequal. Pure and Appl. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/10\/562\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:55:27Z","timestamp":1760144127000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/10\/562"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,17]]},"references-count":15,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["axioms11100562"],"URL":"https:\/\/doi.org\/10.3390\/axioms11100562","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,17]]}}}