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The Caputo time fractional derivatives are approximated by using the weighted and shifted Gr\u00fcnwald\u2013Letnikov formulae introduced in Tian et al. (Math Comput 84:2703\u20132727, 2015). After correcting a few starting steps, the proposed time stepping methods have the optimal convergence orders <jats:inline-formula><jats:alternatives><jats:tex-math>$$O(k^2)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>O<\/mml:mi><mml:mo>(<\/mml:mo><mml:msup><mml:mi>k<\/mml:mi><mml:mn>2<\/mml:mn><\/mml:msup><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$ O(k^3)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>O<\/mml:mi><mml:mo>(<\/mml:mo><mml:msup><mml:mi>k<\/mml:mi><mml:mn>3<\/mml:mn><\/mml:msup><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, respectively for any fixed time <jats:italic>t<\/jats:italic> for both smooth and nonsmooth data. The error estimates are proved by directly bounding the approximation errors of the kernel functions. Moreover, we also present briefly the applicabilities of our time stepping schemes to various other fractional evolution equations. Finally, some numerical examples are given to show that the numerical results are consistent with the proven theoretical results.<\/jats:p>","DOI":"10.1007\/s10915-020-01223-y","type":"journal-article","created":{"date-parts":[[2020,5,19]],"date-time":"2020-05-19T04:10:01Z","timestamp":1589861401000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Higher Order Time Stepping Methods for Subdiffusion Problems Based on Weighted and Shifted Gr\u00fcnwald\u2013Letnikov Formulae with Nonsmooth Data"],"prefix":"10.1007","volume":"83","author":[{"given":"Yanyong","family":"Wang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuyuan","family":"Yan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yubin","family":"Yan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Amiya K.","family":"Pani","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,5,19]]},"reference":[{"key":"1223_CR1","doi-asserted-by":"publisher","first-page":"767","DOI":"10.1515\/fca-2019-0042","volume":"22","author":"G Acosta","year":"2019","unstructured":"Acosta, G., Bersetche, F.M., Borthagaray, J.P.: Finite element approximations for fractional evolution problems. 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