{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T10:44:07Z","timestamp":1776768247578,"version":"3.51.2"},"reference-count":25,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,1,1]]},"abstract":"Abstract<\/jats:title>\n We consider error estimates for some time stepping methods for solving fractional diffusion problems with nonsmooth data in both homogeneous and inhomogeneous cases. McLean and Mustapha [18] established an \n \n \n \n O<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n k<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {O(k)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> convergence rate for the piecewise constant discontinuous Galerkin method with nonsmooth initial data for the homogeneous problem when the linear operator A<\/jats:italic> is assumed to be self-adjoint, positive semidefinite and densely defined in a suitable Hilbert space, where k<\/jats:italic> denotes the time step size. In this paper, we approximate the Riemann\u2013Liouville fractional derivative by Diethelm\u2019s method (or L1 scheme) and obtain the same time discretisation scheme as in McLean and Mustapha [18]. We first prove that this scheme has also convergence rate \n \n \n \n O<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n k<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {O(k)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> with nonsmooth initial data for the homogeneous problem when A<\/jats:italic> is a closed, densely defined linear operator satisfying some certain resolvent estimates. We then introduce a new time discretisation scheme for the homogeneous problem based on the convolution quadrature and prove that the convergence rate of this new scheme is \n \n \n \n O<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n k<\/m:mi>\n \n 1<\/m:mn>\n +<\/m:mo>\n \u03b1<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {O(k^{1+\\alpha})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, \n \n \n \n 0<\/m:mn>\n <<\/m:mo>\n \u03b1<\/m:mi>\n <<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n {0<\\alpha<1}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, with the nonsmooth initial data. Using this new time discretisation scheme for the homogeneous problem, we define a time stepping method for the inhomogeneous problem and prove that the convergence rate of this method is \n \n \n \n O<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n k<\/m:mi>\n \n 1<\/m:mn>\n +<\/m:mo>\n \u03b1<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {O(k^{1+\\alpha})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, \n \n \n \n 0<\/m:mn>\n <<\/m:mo>\n \u03b1<\/m:mi>\n <<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n {0<\\alpha<1}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, with the nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results.<\/jats:p>","DOI":"10.1515\/cmam-2017-0037","type":"journal-article","created":{"date-parts":[[2017,9,2]],"date-time":"2017-09-02T10:02:50Z","timestamp":1504346570000},"page":"129-146","source":"Crossref","is-referenced-by-count":16,"title":["Some Time Stepping Methods for Fractional Diffusion Problems with Nonsmooth Data"],"prefix":"10.1515","volume":"18","author":[{"given":"Yan","family":"Yang","sequence":"first","affiliation":[{"name":"Department of Mathematics , Lvliang University , Lishi 033000 , P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yubin","family":"Yan","sequence":"additional","affiliation":[{"name":"Department of Mathematics , University of Chester , Thornton Science Park, Pool Lane , Ince , CH2 4NU , United Kingdom"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Neville J.","family":"Ford","sequence":"additional","affiliation":[{"name":"Department of Mathematics , University of Chester , Thornton Science Park, Pool Lane , Ince , CH2 4NU , United Kingdom"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,9,2]]},"reference":[{"key":"2023033115122827837_j_cmam-2017-0037_ref_001_w2aab3b7e3060b1b6b1ab2b2b1Aa","doi-asserted-by":"crossref","unstructured":"C.-M. Chen, F. Liu, V. Anh and I. Turner,\nNumerical methods for solving a two-dimensional variable-order anomalous sub-diffusion equation,\nMath. 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