{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,5]],"date-time":"2026-01-05T22:09:47Z","timestamp":1767650987558},"reference-count":39,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2021,7,2]],"date-time":"2021-07-02T00:00:00Z","timestamp":1625184000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"name":"Centre for Mathematics of the University of Coimbra","award":["UIDB\/00324\/2020"],"award-info":[{"award-number":["UIDB\/00324\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,7,22]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The Gr\u00fcnwald\u2013Letnikov approximation is a well-known discretization to approximate a Riemann\u2013Liouville derivative of order $\\alpha&amp;gt;0$. This approximation has been proved to be a consistent approximation, of order $1 $, when the domain is the real line, using Fourier transforms. However, in recent years, this approximation has been applied frequently to solve fractional differential equations in bounded domains and the result proved for the real line has been assumed to be true in general. In this work we show that when assuming a bounded domain, the Gr\u00fcnwald\u2013Letnikov approximation can be inconsistent, for a large number of cases, and when it is consistent we have mostly an order of $n-\\alpha $, for $n-1&amp;lt;\\alpha &amp;lt;n$.<\/jats:p>","DOI":"10.1093\/imanum\/drab051","type":"journal-article","created":{"date-parts":[[2021,5,28]],"date-time":"2021-05-28T19:52:20Z","timestamp":1622231540000},"page":"2771-2793","source":"Crossref","is-referenced-by-count":7,"title":["Consistency analysis of the Gr\u00fcnwald\u2013Letnikov approximation in a bounded domain"],"prefix":"10.1093","volume":"42","author":[{"given":"Erc\u00edlia","family":"Sousa","sequence":"first","affiliation":[{"name":"CMUC, Department of Mathematics, University of Coimbra , 3001-501 Coimbra, Portugal"}]}],"member":"286","published-online":{"date-parts":[[2021,7,2]]},"reference":[{"key":"2022071909091816100_ref1","volume-title":"Handbook of Mathematical Functions: With Formulas, Graphs and Mathematical Tables","author":"Abramowitz","year":"1965"},{"key":"2022071909091816100_ref2","volume-title":"Special Functions","author":"Andrews","year":"2000"},{"key":"2022071909091816100_ref4","doi-asserted-by":"crossref","first-page":"813","DOI":"10.1090\/S0002-9947-2014-05887-X","article-title":"Higher order Gr\u00fcnwald approximations of fractional derivatives and fractional powers of operators","volume":"367","author":"Baeumer","year":"2015","journal-title":"Trans. Amer. Math. Soc."},{"key":"2022071909091816100_ref3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10543-018-0727-8","article-title":"On banded $M$-splitting iteration methods for solving discretized spatial fractional diffusion equation","volume":"59","author":"Bai","year":"2019","journal-title":"BIT"},{"key":"2022071909091816100_ref5","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/j.apnum.2019.01.004","article-title":"An analysis of the Gr\u00fcnwald\u2013Letnikov scheme for initial-value problems with weakly singular solutions","volume":"139","author":"Chen","year":"2019","journal-title":"Appl. Numer. Math."},{"key":"2022071909091816100_ref6","volume-title":"Interpolation and Approximation","author":"Davis","year":"1975"},{"key":"2022071909091816100_ref7","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1093\/imanum\/13.3.445","article-title":"An asymptotic analysis of two algorithms for certain Hadamard finite-part integrals","volume":"13","author":"Elliott","year":"1993","journal-title":"IMA J. Numer. Anal."},{"key":"2022071909091816100_ref8","doi-asserted-by":"crossref","first-page":"133","DOI":"10.2140\/pjm.1951.1.133","article-title":"The asymptotic expansion of a ratio of Gamma functions","volume":"1","author":"Erd\u00e9lyi","year":"1951","journal-title":"Pacific J. Math."},{"key":"2022071909091816100_ref9","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.cam.2013.02.009","article-title":"Numerical analysis of a two-parameter fractional telegraph equation","volume":"249","author":"Ford","year":"2013","journal-title":"J. Comput. Appl. Math."},{"key":"2022071909091816100_ref10","first-page":"77","article-title":"Some elementary inequalities relating to the Gamma and incomplete Gamma function","volume":"38","author":"Gautschi","year":"1959","journal-title":"Stud. Appl. Math."},{"key":"2022071909091816100_ref11","doi-asserted-by":"crossref","first-page":"871","DOI":"10.1016\/j.cam.2005.12.043","article-title":"Convergence of the Gr\u00fcnwald\u2013Letnikov scheme for time-fractional diffusion","volume":"205","author":"Gorenflo","year":"2007","journal-title":"J. Comput. Appl. Math."},{"key":"2022071909091816100_ref12","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1515\/fca-2015-0027","article-title":"Formal consistency versus actual convergence rates of difference schemes for fractional derivative boundary value problems","volume":"18","author":"Gracia","year":"2015","journal-title":"Fract. Calc. Appl. Anal."},{"key":"2022071909091816100_ref13","doi-asserted-by":"crossref","first-page":"A2485","DOI":"10.1137\/18M1204991","article-title":"Machine learning of space-fractional differential equations","volume":"41","author":"Gulian","year":"2019","journal-title":"SIAM J. Sci. Comput."},{"key":"2022071909091816100_ref14","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1007\/s11075-015-0051-1","article-title":"An implicit numerical method of a new time distributed-order and two-sided space-fractional advection-dispersion equation","volume":"72","author":"Hu","year":"2016","journal-title":"Numer. Algorithms"},{"key":"2022071909091816100_ref15","doi-asserted-by":"crossref","first-page":"1217","DOI":"10.1093\/imanum\/drz006","article-title":"A new analysis of a numerical method for the time-fractional Fokker\u2013Planck equation with general forcing","volume":"40","author":"Huang","year":"2020","journal-title":"IMA J. Numer. Anal."},{"key":"2022071909091816100_ref16","volume-title":"Theory and Applications of Fractional Differential Equations","author":"Kilbas","year":"2006"},{"key":"2022071909091816100_ref17","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1137\/1018042","article-title":"Fractional derivatives and special functions","volume":"18","author":"Lavoie","year":"1976","journal-title":"SIAM Rev."},{"key":"2022071909091816100_ref18","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1007\/s10543-018-0699-8","article-title":"Efficient preconditioner of one-sided space fractional diffusion equation","volume":"58","author":"Lin","year":"2018","journal-title":"BIT"},{"key":"2022071909091816100_ref19","doi-asserted-by":"crossref","first-page":"704","DOI":"10.1137\/0517050","article-title":"Discretized fractional calculus","volume":"17","author":"Lubich","year":"1986","journal-title":"SIAM J. Math. Anal."},{"key":"2022071909091816100_ref20","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1093\/imanum\/7.1.97","article-title":"Fractional linear multistep methods for Abel\u2013Volterra integral equations of the first kind","volume":"7","author":"Lubich","year":"1987","journal-title":"IMA J. Numer. Anal."},{"key":"2022071909091816100_ref21","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/j.jcp.2015.04.048","article-title":"Efficient computation of the Gr\u00fcnwald\u2013Letnikov fractional diffusion derivative using adaptive time step memory","volume":"297","author":"MacDonald","year":"2015","journal-title":"J. Comput. Phys."},{"key":"2022071909091816100_ref22","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2019.109043","article-title":"A parallelized computational model for multidimensional systems of coupled nonlinear fractional hyperbolic equations","volume":"402","author":"Mac\u00edas-D\u00edaz","year":"2020","journal-title":"J. Comput. Phys."},{"key":"2022071909091816100_ref23","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.cam.2004.01.033","article-title":"Finite difference approximations for fractional advection\u2013dispersion flow equations","volume":"172","author":"Meerschaert","year":"2004","journal-title":"J. Comput. Appl. Math."},{"key":"2022071909091816100_ref24","doi-asserted-by":"crossref","first-page":"R161","DOI":"10.1088\/0305-4470\/37\/31\/R01","article-title":"The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics","volume":"37","author":"Metzler","year":"2004","journal-title":"J. Phys. A: Math. Gen."},{"key":"2022071909091816100_ref25","volume-title":"The Fractional Calculus","author":"Oldham","year":"1974"},{"key":"2022071909091816100_ref26","doi-asserted-by":"crossref","first-page":"275002","DOI":"10.1088\/1751-8121\/ab9030","article-title":"First passage time moments of asymmetric L\u00e9vy flights","volume":"53","author":"Padash","year":"2020","journal-title":"J. Phys. A Math. Theor."},{"key":"2022071909091816100_ref27","doi-asserted-by":"crossref","first-page":"A2698","DOI":"10.1137\/130931795","article-title":"Preconditioning techniques for diagonal times Toeplitz matrices in fractional diffusion equations","volume":"36","author":"Pan","year":"2014","journal-title":"SIAM J. Sci. Comput."},{"key":"2022071909091816100_ref28","volume-title":"Fractional Differential Equations","author":"Podlubny","year":"1999"},{"key":"2022071909091816100_ref29","volume-title":"Difference Methods for Initial Value Problems","author":"Richtmyer","year":"1967"},{"key":"2022071909091816100_ref30","volume-title":"Fractional Integrals and Derivatives","author":"Samko","year":"1987"},{"key":"2022071909091816100_ref31","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1515\/fca-2018-0002","article-title":"From continuous time random walks to the generalized diffusion equation","volume":"21","author":"Sandev","year":"2018","journal-title":"Fract. Calc. Appl. Anal."},{"key":"2022071909091816100_ref32","doi-asserted-by":"crossref","first-page":"4038","DOI":"10.1016\/j.jcp.2009.02.011","article-title":"Finite difference approximations for a fractional advection diffusion problem","volume":"228","author":"Sousa","year":"2009","journal-title":"J. Comput. Phys."},{"key":"2022071909091816100_ref33","doi-asserted-by":"crossref","first-page":"1250075","DOI":"10.1142\/S0218127412500757","article-title":"How to approximate the fractional derivative of order $1$","volume":"22","author":"Sousa","year":"2012","journal-title":"Int. J. Bifurcat. Chaos"},{"key":"2022071909091816100_ref34","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/j.jcp.2005.08.008","article-title":"A second-order accurate numerical approximation for the fractional diffusion equation","volume":"213","author":"Tadjeran","year":"2006","journal-title":"J. Comput. Phys."},{"key":"2022071909091816100_ref35","doi-asserted-by":"crossref","first-page":"1703","DOI":"10.1090\/S0025-5718-2015-02917-2","article-title":"A class of second order difference approximations for solving space fractional diffusion equations","volume":"84","author":"Tian","year":"2015","journal-title":"Math. Comp."},{"key":"2022071909091816100_ref36","doi-asserted-by":"crossref","first-page":"646","DOI":"10.1002\/zamm.19950750826","article-title":"Extrapolation to the limit for numerical fractional differentiation","volume":"75","author":"Tuan","year":"1995","journal-title":"Z. Agnew. Math. Mech."},{"key":"2022071909091816100_ref37","doi-asserted-by":"crossref","first-page":"1862","DOI":"10.1137\/030602666","article-title":"An explicit finite difference method and a new von Neumann type stability analysis for fractional diffusion equations","volume":"42","author":"Yuste","year":"2005","journal-title":"SIAM J. Numer. Anal."},{"key":"2022071909091816100_ref38","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1016\/S0370-1573(02)00331-9","article-title":"Chaos, fractional kinetics, and anomalous transport","volume":"371","author":"Zaslavsky","year":"2002","journal-title":"Phys. Rep"},{"key":"2022071909091816100_ref39","doi-asserted-by":"crossref","first-page":"478","DOI":"10.1016\/j.cma.2017.08.029","article-title":"Second-order numerical methods for multi-term fractional differential equations: smooth and non-smooth solutions","volume":"327","author":"Zeng","year":"2017","journal-title":"Comput. Methods Appl. Mech. Engrg."}],"container-title":["IMA Journal of Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/imajna\/article-pdf\/42\/3\/2771\/44955810\/drab051.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/imajna\/article-pdf\/42\/3\/2771\/44955810\/drab051.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,7,19]],"date-time":"2022-07-19T09:11:52Z","timestamp":1658221912000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/imajna\/article\/42\/3\/2771\/6310602"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,2]]},"references-count":39,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,7,2]]},"published-print":{"date-parts":[[2022,7,22]]}},"URL":"https:\/\/doi.org\/10.1093\/imanum\/drab051","relation":{},"ISSN":["0272-4979","1464-3642"],"issn-type":[{"value":"0272-4979","type":"print"},{"value":"1464-3642","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2022,7,16]]},"published":{"date-parts":[[2021,7,2]]}}}