{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T16:26:50Z","timestamp":1772296010895,"version":"3.50.1"},"reference-count":21,"publisher":"Walter de Gruyter GmbH","issue":"4","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1254618"],"award-info":[{"award-number":["DMS-1254618"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,10,1]]},"abstract":"Abstract<\/jats:title>\n In this paper, we develop a numerical scheme for the space-time fractional parabolic\nequation, i.e. an equation involving a fractional time derivative and a fractional spatial operator.\nBoth the initial value problem and the non-homogeneous forcing\nproblem (with zero initial data) are considered. The solution\noperator \n \n \n \n E<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n t<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {E(t)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for the\ninitial value problem can be written as a Dunford\u2013Taylor integral\ninvolving the Mittag-Leffler function \n \n \n \n e<\/m:mi>\n \n \u03b1<\/m:mi>\n ,<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n <\/m:msub>\n <\/m:math>\n \n {e_{\\alpha,1}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and the resolvent\nof the underlying (non-fractional) spatial operator over\nan appropriate integration path in the complex plane. Here \u03b1 denotes the order of\nthe fractional time derivative.\nThe solution for\nthe non-homogeneous problem can be written as a\nconvolution involving an operator \n \n \n \n W<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n t<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {W(t)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and the forcing function\n\n \n \n \n F<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n t<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {F(t)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.\nWe develop and analyze semi-discrete methods\nbased on finite element approximation to the underlying (non-fractional)\nspatial operator in terms of analogous Dunford\u2013Taylor integrals applied to the\ndiscrete operator. The space error is of optimal order up to a logarithm of \n \n \n \n 1<\/m:mn>\n h<\/m:mi>\n <\/m:mfrac>\n <\/m:math>\n \n {\\frac{1}{h}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.\nThe fully discrete method for the initial value\nproblem is developed from the semi-discrete approximation by applying a\nsinc quadrature technique\nto approximate the Dunford\u2013Taylor integral\nof the discrete operator and is free of any time stepping. The sinc\nquadrature of step size k<\/jats:italic> involves \n \n \n \n k<\/m:mi>\n \n -<\/m:mo>\n 2<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n <\/m:math>\n \n {k^{-2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> nodes and results in an additional\n\n \n \n \n O<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n exp<\/m:mi>\n \u2061<\/m:mo>\n \n (<\/m:mo>\n \n -<\/m:mo>\n \n c<\/m:mi>\n k<\/m:mi>\n <\/m:mfrac>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {O(\\exp(-\\frac{c}{k}))}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> error.\nTo approximate the convolution appearing in the semi-discrete\napproximation to the non-homogeneous problem, we apply a pseudo-midpoint\nquadrature. This involves the average of \n \n \n \n \n W<\/m:mi>\n h<\/m:mi>\n <\/m:msub>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n s<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {W_{h}(s)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, (the semi-discrete\napproximation to \n \n \n \n W<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n s<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {W(s)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>) over the quadrature interval. This average\ncan also be written as a\nDunford\u2013Taylor integral. We first analyze the error between this\nquadrature and the semi-discrete approximation. To develop a fully\ndiscrete method, we then introduce sinc quadrature approximations to the\nDunford\u2013Taylor integrals for computing the averages.\nWe show that for\na refined grid in time with a mesh of \n \n \n \n O<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n \ud835\udca9<\/m:mi>\n \u2062<\/m:mo>\n \n log<\/m:mi>\n \u2061<\/m:mo>\n \n (<\/m:mo>\n \ud835\udca9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {O({\\mathcal{N}}\\log({\\mathcal{N}}))}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> intervals, the\nerror between the semi-discrete and fully discrete approximation\nis \n \n \n \n O<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n \n \ud835\udca9<\/m:mi>\n \n -<\/m:mo>\n 2<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n +<\/m:mo>\n \n \n log<\/m:mi>\n \u2061<\/m:mo>\n \n (<\/m:mo>\n \ud835\udca9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n \u2062<\/m:mo>\n \n exp<\/m:mi>\n \u2061<\/m:mo>\n \n (<\/m:mo>\n \n -<\/m:mo>\n \n c<\/m:mi>\n k<\/m:mi>\n <\/m:mfrac>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {O({\\mathcal{N}}^{-2}+\\log({\\mathcal{N}})\\exp(-\\frac{c}{k}))}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. We also report the results of numerical experiments\nthat are in\nagreement with the theoretical error estimates.<\/jats:p>","DOI":"10.1515\/cmam-2017-0032","type":"journal-article","created":{"date-parts":[[2017,8,27]],"date-time":"2017-08-27T10:01:20Z","timestamp":1503828080000},"page":"679-705","source":"Crossref","is-referenced-by-count":15,"title":["Numerical Approximation of Space-Time Fractional Parabolic Equations"],"prefix":"10.1515","volume":"17","author":[{"given":"Andrea","family":"Bonito","sequence":"first","affiliation":[{"name":"Department of Mathematics , Texas A&M University , College Station , TX 77843-3368 , USA"}]},{"given":"Wenyu","family":"Lei","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Texas A&M University , College Station , TX 77843-3368 , USA"}]},{"given":"Joseph E.","family":"Pasciak","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Texas A&M University , College Station , TX 77843-3368 , USA"}]}],"member":"374","published-online":{"date-parts":[[2017,8,26]]},"reference":[{"key":"2023033116270851469_j_cmam-2017-0032_ref_001_w2aab3b7b6b1b6b1ab1b6b1Aa","doi-asserted-by":"crossref","unstructured":"R. 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