{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T10:20:13Z","timestamp":1760523613274,"version":"3.40.5"},"reference-count":21,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>A grid method for solving the first boundary value problem for ordinary and partial differential equations with the Riemann\u2013Liouville fractional derivative is justified.\nThe algorithm is based on using Green\u2019s function, the Fredholm integral equation, and the Lagrange interpolation polynomial.\nThe impact of the Dirichlet boundary condition on the accuracy of the approximate solution is revealed and quantitatively described through the weight assessment.\nAll the estimates provide clear evidence that the accuracy order of the grid method is higher near the boundary of the domain than it is in the inner nodes of the mesh set.<\/jats:p>","DOI":"10.1515\/cmam-2018-0002","type":"journal-article","created":{"date-parts":[[2018,4,12]],"date-time":"2018-04-12T07:52:21Z","timestamp":1523519541000},"page":"379-394","source":"Crossref","is-referenced-by-count":6,"title":["The Boundary Effect in the Accuracy Estimate for the Grid Solution of the Fractional Differential Equation"],"prefix":"10.1515","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4883-6574","authenticated-orcid":false,"given":"Volodymyr","family":"Makarov","sequence":"first","affiliation":[{"name":"Department of Numerical Mathematics , Institute of Mathematics of the National Academy of Sciences of Ukraine , 3 Tereshchenkivska Str., 01004 Kyiv , Ukraine"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8810-7464","authenticated-orcid":false,"given":"Nataliya","family":"Mayko","sequence":"additional","affiliation":[{"name":"Faculty of Radiophysics, Electronics and Computer Systems , Department of Mathematics and Theoretical Radiophysics , Taras Shevchenko National University of Kyiv , 64\/13 Volodymyrska Str., 01601 Kyiv , Ukraine"}]}],"member":"374","published-online":{"date-parts":[[2018,3,3]]},"reference":[{"key":"2023033110021979128_j_cmam-2018-0002_ref_001_w2aab3b7d952b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"A. A.  Alikhanov,\nA new difference scheme for the time fractional diffusion equation,\nJ. Comput. Phys. 280 (2015), 424\u2013438.","DOI":"10.1016\/j.jcp.2014.09.031"},{"key":"2023033110021979128_j_cmam-2018-0002_ref_002_w2aab3b7d952b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"V. M.  Bulavatsky,\nFractional differential analog of biparabolic evolution equation and some its applications,\nCybernet. Systems Anal. 52 (2016), no. 5, 737\u2013747.","DOI":"10.1007\/s10559-016-9875-5"},{"key":"2023033110021979128_j_cmam-2018-0002_ref_003_w2aab3b7d952b1b6b1ab2ab3Aa","doi-asserted-by":"crossref","unstructured":"I.  Demkiv, I. P.  Gavrilyuk and V. L.  Makarov,\nSuper-exponentially convergent parallel algorithm for eigenvalue problems with fractional derivatives,\nComput. Methods Appl. Math. 16 (2016), no. 4, 633\u2013652.","DOI":"10.1515\/cmam-2016-0018"},{"key":"2023033110021979128_j_cmam-2018-0002_ref_004_w2aab3b7d952b1b6b1ab2ab4Aa","unstructured":"I. S.  Gradshteyn and I. M.  Ryzhik,\nTable of Integrals, Series, and Products, 7th ed.,\nElsevier\/Academic Press, Amsterdam, 2007."},{"key":"2023033110021979128_j_cmam-2018-0002_ref_005_w2aab3b7d952b1b6b1ab2ab5Aa","doi-asserted-by":"crossref","unstructured":"B.  Jin, R.  Lazarov and P.  Vabishchevich,\nPreface: Numerical analysis of fractional differential equations,\nComput. Methods Appl. Math. 17 (2017), no. 4, 643\u2013646.","DOI":"10.1515\/cmam-2017-0036"},{"key":"2023033110021979128_j_cmam-2018-0002_ref_006_w2aab3b7d952b1b6b1ab2ab6Aa","doi-asserted-by":"crossref","unstructured":"B. S.  Jovanovi\u0107, L. G.  Vulkov and A.  Deli\u0107,\nBoundary value problems for fractional PDE and their numerical approximation,\nNumerical Analysis and its Applications,\nLecture Notes in Comput. Sci. 8236,\nSpringer, Heidelberg (2013), 38\u201349.","DOI":"10.1007\/978-3-642-41515-9_4"},{"key":"2023033110021979128_j_cmam-2018-0002_ref_007_w2aab3b7d952b1b6b1ab2ab7Aa","doi-asserted-by":"crossref","unstructured":"J. A. T.  Machado, A. M. S. F.  Galhano and J. J.  Trujillo,\nOn development of fractional calculus during the last fifty years,\nScientometrics 98 (2014), no. 1, 577\u2013582.","DOI":"10.1007\/s11192-013-1032-6"},{"key":"2023033110021979128_j_cmam-2018-0002_ref_008_w2aab3b7d952b1b6b1ab2ab8Aa","doi-asserted-by":"crossref","unstructured":"J. T.  Machado, V.  Kiryakova and F.  Mainardi,\nRecent history of fractional calculus,\nCommun. Nonlinear Sci. Numer. Simul. 16 (2011), no. 3, 1140\u20131153.","DOI":"10.1016\/j.cnsns.2010.05.027"},{"key":"2023033110021979128_j_cmam-2018-0002_ref_009_w2aab3b7d952b1b6b1ab2ab9Aa","unstructured":"V.  Makarov,\nOn a priori estimates of difference schemes giving an account of the boundary effect,\nC. R. Acad. Bulgare Sci. 42 (1989), no. 5, 41\u201344."},{"key":"2023033110021979128_j_cmam-2018-0002_ref_010_w2aab3b7d952b1b6b1ab2ac10Aa","unstructured":"V. L.  Makarov and L. I.  Demkiv,\nImproved accuracy estimates of the difference scheme for parabolic equation (in Ukrainian),\nPraci Ukr. Matem. Conhresu,\nInst. Matem. Nats. Akad. Nauk., Kyiv (2001), 31\u201342."},{"key":"2023033110021979128_j_cmam-2018-0002_ref_011_w2aab3b7d952b1b6b1ab2ac11Aa","doi-asserted-by":"crossref","unstructured":"V. L.  Makarov and L. I.  Demkiv,\nAccuracy estimates of difference schemes for quasi-linear elliptic equations with variable coefficients taking into account boundary effect,\nNumerical Analysis and its Applications,\nLecture Notes in Comput. Sci. 3401,\nSpringer, Berlin (2005), 80\u201390.","DOI":"10.1007\/978-3-540-31852-1_8"},{"key":"2023033110021979128_j_cmam-2018-0002_ref_012_w2aab3b7d952b1b6b1ab2ac12Aa","unstructured":"V. L.  Makarov and L. I.  Demkiv,\nEstimates for the accuracy of difference schemes for parabolic equations taking the initial-boundary effect into account,\nDopov. Nats. Akad. Nauk Ukr. Mat. Pryrodozn. Tekh. Nauky (2003), no. 2, 26\u201332."},{"key":"2023033110021979128_j_cmam-2018-0002_ref_013_w2aab3b7d952b1b6b1ab2ac13Aa","doi-asserted-by":"crossref","unstructured":"V. L.  Makarov and L. I.  Demkiv,\nWeight uniform accuracy estimate of finite-difference method for poisson equation taking into account boundary effect,\nNumerical Analysis and its Applications,\nLecture Notes in Comput. Sci. 5434,\nSpringer, Berlin (2009), 92\u2013103.","DOI":"10.1007\/978-3-642-00464-3_9"},{"key":"2023033110021979128_j_cmam-2018-0002_ref_014_w2aab3b7d952b1b6b1ab2ac14Aa","doi-asserted-by":"crossref","unstructured":"N. V.  Mayko,\nImproved accuracy estimates of the difference scheme for the two-dimensional parabolic equation with  regard for the effect of initial and boundary conditions,\nCybernet. Systems Anal. 53 (2017), no. 1, 99\u2013107.","DOI":"10.1007\/s10559-017-9909-7"},{"key":"2023033110021979128_j_cmam-2018-0002_ref_015_w2aab3b7d952b1b6b1ab2ac15Aa","doi-asserted-by":"crossref","unstructured":"B.  Ross,\nThe development of fractional calculus 1695\u20131900,\nHistoria Math. 4 (1977), 75\u201389.","DOI":"10.1016\/0315-0860(77)90039-8"},{"key":"2023033110021979128_j_cmam-2018-0002_ref_016_w2aab3b7d952b1b6b1ab2ac16Aa","doi-asserted-by":"crossref","unstructured":"A. A.  Samarskii,\nThe Theory of Difference Schemes,\nMonogr. Textb. Pure Appl. Math. 240,\nMarcel Dekker, New York, 2001.","DOI":"10.1201\/9780203908518"},{"key":"2023033110021979128_j_cmam-2018-0002_ref_017_w2aab3b7d952b1b6b1ab2ac17Aa","unstructured":"S. G.  Samko, A. A.  Kilbas and O. I.  Marichev,\nFractional Integrals and Derivatives: Theory and Applications,\nGordon and Breach, New York, 1993."},{"key":"2023033110021979128_j_cmam-2018-0002_ref_018_w2aab3b7d952b1b6b1ab2ac18Aa","unstructured":"V. K.  Shogenov, S. K.  Kumykova and M. K.  Shkhanukov-Lafishev,\nThe generalized transport equation and fractional derivatives (in Russian),\nDopov. Nats. Akad. Nauk Ukr. 1997 (1997), no. 12, 47\u201354."},{"key":"2023033110021979128_j_cmam-2018-0002_ref_019_w2aab3b7d952b1b6b1ab2ac19Aa","doi-asserted-by":"crossref","unstructured":"F. I.  Taukenova and M. K.  Shkhanukov-Lafishev,\nDifference methods for solving boundary value problems for fractional-order differential equations,\nComput. Math. Math. Phys. 46 (2006), no. 10, 1785\u20131795.","DOI":"10.1134\/S0965542506100149"},{"key":"2023033110021979128_j_cmam-2018-0002_ref_020_w2aab3b7d952b1b6b1ab2ac20Aa","unstructured":"V. V.  Vasiliev and L. A.  Simak,\nFractional Calculus and Approximation Methods in Modelling Dynamical Systems (in Russian),\nNational Academy of Sciences of Ukraine, Kyiv, 2008."},{"key":"2023033110021979128_j_cmam-2018-0002_ref_021_w2aab3b7d952b1b6b1ab2ac21Aa","unstructured":"N. O.  Virchenko and V. Y.  Rybak,\nThe Principles of Fractional Intergo-Differentiation,\nTOV \u201cZadruha\u201d, Kyiv, 2007."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/19\/2\/article-p379.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0002\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0002\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T10:44:25Z","timestamp":1680259465000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0002\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,3,3]]},"references-count":21,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,2,24]]},"published-print":{"date-parts":[[2019,4,1]]}},"alternative-id":["10.1515\/cmam-2018-0002"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2018-0002","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"type":"electronic","value":"1609-9389"},{"type":"print","value":"1609-4840"}],"subject":[],"published":{"date-parts":[[2018,3,3]]}}}