{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T04:47:18Z","timestamp":1747198038589,"version":"3.40.5"},"reference-count":46,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,10,1]]},"abstract":"Abstract<\/jats:title>\n In this paper, the distributed-order time fractional sub-diffusion equation on the bounded domains is studied by using the finite-point-type meshless method. The finite point method is a point collocation based method which is truly meshless and computationally efficient. To construct the shape functions of the finite point method, the moving least square reproducing kernel approximation is employed. Two implicit discretisation of order \n \n \n \n O<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u03c4<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {O(\\tau)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n \n \n O<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n \u03c4<\/m:mi>\n \n 1<\/m:mn>\n +<\/m:mo>\n \n \n 1<\/m:mn>\n 2<\/m:mn>\n <\/m:mfrac>\n \u2062<\/m:mo>\n \u03c3<\/m:mi>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {O(\\tau^{1+\\frac{1}{2}\\sigma})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> are derived, respectively. Stability and \n \n \n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n {L^{2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> norm convergence of the obtained difference schemes are proved. Numerical examples are provided to confirm the theoretical results.<\/jats:p>","DOI":"10.1515\/cmam-2018-0009","type":"journal-article","created":{"date-parts":[[2018,4,18]],"date-time":"2018-04-18T22:16:03Z","timestamp":1524089763000},"page":"813-831","source":"Crossref","is-referenced-by-count":0,"title":["Two Implicit Meshless Finite Point Schemes for the Two-Dimensional Distributed-Order Fractional Equation"],"prefix":"10.1515","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7351-9223","authenticated-orcid":false,"given":"Rezvan","family":"Salehi","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics , Faculty of Mathematical Sciences , Tarbiat Modares University , P.O. Box 14115-134 , Tehran , Iran"}]}],"member":"374","published-online":{"date-parts":[[2018,4,18]]},"reference":[{"key":"2023033110105331281_j_cmam-2018-0009_ref_001_w2aab3b7d978b1b6b1ab2b1b1Aa","doi-asserted-by":"crossref","unstructured":"E. E. Adams and L. W. Gelhar,\nField study of dispersion in a heterogeneous aquifer: 2. Spatial movement analysis,\nWater Resour. Res. 28 (1992), no. 2, 3293\u20133307.","DOI":"10.1029\/92WR01757"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_002_w2aab3b7d978b1b6b1ab2b1b2Aa","doi-asserted-by":"crossref","unstructured":"A. Angulo, L. P\u00e9rez Pozo and F. Perazzo,\nA posteriori error estimator and an adaptive technique in meshless finite points method,\nEng. Anal. Bound. Elem. 33 (2009), no. 11, 1322\u20131338.","DOI":"10.1016\/j.enganabound.2009.06.004"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_003_w2aab3b7d978b1b6b1ab2b1b3Aa","doi-asserted-by":"crossref","unstructured":"T. M. Atanackovic, S. Pilipovic and D. Zorica,\nExistence and calculation of the solution to the time distributed order diffusion equation,\nPhys. Scr. 2009 (2009), Article ID 014012.","DOI":"10.1088\/0031-8949\/2009\/T136\/014012"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_004_w2aab3b7d978b1b6b1ab2b1b4Aa","unstructured":"K. Atkinson and W. Han,\nTheoretical Numerical Analysis: A Functional Analysis Framework, 3rd ed.,\nTexts Appl. Math. 39,\nSpringer, Dordrecht, 2009."},{"key":"2023033110105331281_j_cmam-2018-0009_ref_005_w2aab3b7d978b1b6b1ab2b1b5Aa","doi-asserted-by":"crossref","unstructured":"S. N. Atluri and T. Zhu,\nA new meshless local Petrov\u2013Galerkin (MLPG) approach in computational mechanics,\nComput. Mech. 22 (1998), no. 2, 117\u2013127.","DOI":"10.1007\/s004660050346"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_006_w2aab3b7d978b1b6b1ab2b1b6Aa","doi-asserted-by":"crossref","unstructured":"T. Belytschko, Y. Y. Lu and L. Gu,\nElement-free Galerkin methods,\nInternat. J. Numer. Methods Engrg. 37 (1994), no. 2, 229\u2013256.","DOI":"10.1002\/nme.1620370205"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_007_w2aab3b7d978b1b6b1ab2b1b7Aa","doi-asserted-by":"crossref","unstructured":"W. Bu, A. Xiao and W. Zeng,\nFinite difference\/finite element methods for distributed-order time fractional diffusion equations,\nJ. Sci. Comput. 72 (2017), no. 1, 422\u2013441.","DOI":"10.1007\/s10915-017-0360-8"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_008_w2aab3b7d978b1b6b1ab2b1b8Aa","doi-asserted-by":"crossref","unstructured":"M. Caputo,\nLinear models of dissipation whose Q is almost frequency independent. II,\nGeophys. J. Roy. Astronom. Soc. 13 (1967), no. 5, 529\u2013539.","DOI":"10.1111\/j.1365-246X.1967.tb02303.x"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_009_w2aab3b7d978b1b6b1ab2b1b9Aa","unstructured":"M. Caputo,\nElasticit\u00e0 e dissipazione,\nZanichelli, Bologna, 1969."},{"key":"2023033110105331281_j_cmam-2018-0009_ref_010_w2aab3b7d978b1b6b1ab2b1c10Aa","unstructured":"M. Caputo,\nDistributed order differential equations modelling dielectric induction and diffusion,\nFract. Calc. Appl. Anal. 4 (2001), no. 4, 421\u2013442."},{"key":"2023033110105331281_j_cmam-2018-0009_ref_011_w2aab3b7d978b1b6b1ab2b1c11Aa","doi-asserted-by":"crossref","unstructured":"A. V. Chechkin, R. Gorenflo and I. M. Sokolov,\nRetarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations,\nPhys. Rev. E. 66 (2002), no. 4, Article ID 046129.","DOI":"10.1103\/PhysRevE.66.046129"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_012_w2aab3b7d978b1b6b1ab2b1c12Aa","unstructured":"A. V. Chechkin, R. Gorenflo, I. M. Sokolov and V. Y. Gonchar,\nDistributed order time fractional diffusion equation,\nFract. Calc. Appl. Anal. 6 (2003), no. 3, 259\u2013279."},{"key":"2023033110105331281_j_cmam-2018-0009_ref_013_w2aab3b7d978b1b6b1ab2b1c13Aa","doi-asserted-by":"crossref","unstructured":"R. Cheng and Y. Cheng,\nError estimates for the finite point method,\nAppl. Numer. Math. 58 (2008), no. 6, 884\u2013898.","DOI":"10.1016\/j.apnum.2007.04.003"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_014_w2aab3b7d978b1b6b1ab2b1c14Aa","doi-asserted-by":"crossref","unstructured":"K. Diethelm and N. J. Ford,\nNumerical analysis for distributed-order differential equations,\nJ. Comput. Appl. Math. 225 (2009), no. 1, 96\u2013104.","DOI":"10.1016\/j.cam.2008.07.018"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_015_w2aab3b7d978b1b6b1ab2b1c15Aa","doi-asserted-by":"crossref","unstructured":"N. J. Ford and M. L. Morgado,\nDistributed order equations as boundary value problems,\nComput. Math. Appl. 64 (2012), no. 10, 2973\u20132981.","DOI":"10.1016\/j.camwa.2012.01.053"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_016_w2aab3b7d978b1b6b1ab2b1c16Aa","doi-asserted-by":"crossref","unstructured":"G.-H. Gao, H.-W. Sun and Z.-Z. Sun,\nSome high-order difference schemes for the distributed-order differential equations,\nJ. Comput. Phys. 298 (2015), 337\u2013359.","DOI":"10.1016\/j.jcp.2015.05.047"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_017_w2aab3b7d978b1b6b1ab2b1c17Aa","doi-asserted-by":"crossref","unstructured":"G.-H. Gao and Z.-Z. Sun,\nTwo alternating direction implicit difference schemes for two-dimensional distributed-order fractional diffusion equations,\nJ. Sci. Comput. 66 (2016), no. 3, 1281\u20131312.","DOI":"10.1007\/s10915-015-0064-x"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_018_w2aab3b7d978b1b6b1ab2b1c18Aa","doi-asserted-by":"crossref","unstructured":"G.-H. Gao and Z.-Z. Sun,\nTwo unconditionally stable and convergent difference schemes with the extrapolation method for the one-dimensional distributed-order differential equations,\nNumer. Methods Partial Differential Equations 32 (2016), no. 2, 591\u2013615.","DOI":"10.1002\/num.22020"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_019_w2aab3b7d978b1b6b1ab2b1c19Aa","doi-asserted-by":"crossref","unstructured":"R. A. Gingold and J. J. Monaghan,\nSmoothed particle hydrodynamics: Theory and application to non-spherical stars,\nMon. Not. R. Astron. Soc. 181 (1977), no. 3, 375\u2013389.","DOI":"10.1093\/mnras\/181.3.375"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_020_w2aab3b7d978b1b6b1ab2b1c20Aa","doi-asserted-by":"crossref","unstructured":"A. Hanyga,\nAnomalous diffusion without scale invariance,\nJ. Phys. A 40 (2007), no. 21, 5551\u20135563.","DOI":"10.1088\/1751-8113\/40\/21\/007"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_021_w2aab3b7d978b1b6b1ab2b1c21Aa","doi-asserted-by":"crossref","unstructured":"Z. Jiao, Y. Chen and I. Podlubny,\nDistributed-Order Dynamic Systems,\nSpringer Briefs Electr. Comput. Eng.,\nSpringer, London, 2012.","DOI":"10.1007\/978-1-4471-2852-6"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_022_w2aab3b7d978b1b6b1ab2b1c22Aa","doi-asserted-by":"crossref","unstructured":"B. Jin, R. Lazarov, D. Sheen and Z. Zhou,\nError estimates for approximations of distributed order time fractional diffusion with nonsmooth data,\nFract. Calc. Appl. Anal. 19 (2016), no. 1, 69\u201393.","DOI":"10.1515\/fca-2016-0005"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_023_w2aab3b7d978b1b6b1ab2b1c23Aa","doi-asserted-by":"crossref","unstructured":"J. T. Katsikadelis,\nNumerical solution of distributed order fractional differential equations,\nJ. Comput. Phys. 259 (2014), 11\u201322.","DOI":"10.1016\/j.jcp.2013.11.013"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_024_w2aab3b7d978b1b6b1ab2b1c24Aa","unstructured":"A. A. Kilbas, H. M. Srivastava and J. J. Trujillo,\nTheory and Applications of Fractional Differential Equations,\nNorth-Holland Math. Stud. 204,\nElsevier Science, Amsterdam, 2006."},{"key":"2023033110105331281_j_cmam-2018-0009_ref_025_w2aab3b7d978b1b6b1ab2b1c25Aa","doi-asserted-by":"crossref","unstructured":"A. N. Kochubei,\nDistributed order calculus and equations of ultraslow diffusion,\nJ. Math. Anal. Appl. 340 (2008), no. 1, 252\u2013281.","DOI":"10.1016\/j.jmaa.2007.08.024"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_026_w2aab3b7d978b1b6b1ab2b1c26Aa","doi-asserted-by":"crossref","unstructured":"S. Li and W. K. Liu,\nMoving least-square reproducing kernel method. II. Fourier analysis,\nComput. Methods Appl. Mech. Engrg. 139 (1996), no. 1\u20134, 159\u2013193.","DOI":"10.1016\/S0045-7825(96)01082-1"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_027_w2aab3b7d978b1b6b1ab2b1c27Aa","doi-asserted-by":"crossref","unstructured":"G. R. Liu and Y. T. Gu,\nA point interpolation method for two-dimensional solids,\nInt. J. Numer. Methods Eng. 50 (2001), no. 4, 937\u2013951.","DOI":"10.1002\/1097-0207(20010210)50:4<937::AID-NME62>3.0.CO;2-X"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_028_w2aab3b7d978b1b6b1ab2b1c28Aa","doi-asserted-by":"crossref","unstructured":"W. K. Liu, Y. Chen, S. Jun, J. S. Chen, T. Belytschko, C. Pan, R. A. Uras and C. T. Chang,\nOverview and applications of the reproducing kernel particle methods,\nArch. Comput. Methods Eng. 3 (1996), no. 1, 3\u201380.","DOI":"10.1007\/BF02736130"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_029_w2aab3b7d978b1b6b1ab2b1c29Aa","doi-asserted-by":"crossref","unstructured":"W. K. Liu, S. Jun and Y. F. Zhang,\nReproducing kernel particle methods,\nInternat. J. Numer. Methods Fluids 20 (1995), no. 8\u20139, 1081\u20131106.","DOI":"10.1002\/fld.1650200824"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_030_w2aab3b7d978b1b6b1ab2b1c30Aa","doi-asserted-by":"crossref","unstructured":"W.-K. Liu, S. Li and T. Belytschko,\nMoving least-square reproducing kernel methods. I. Methodology and convergence,\nComput. Methods Appl. Mech. Engrg. 143 (1997), no. 1\u20132, 113\u2013154.","DOI":"10.1016\/S0045-7825(96)01132-2"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_031_w2aab3b7d978b1b6b1ab2b1c31Aa","unstructured":"C. F. Lorenzo and T. T. Hartley,\nVariable order and distributed order fractional operators,\nNonlinear Dynam. 29 (2002), no. 1\u20134, 57\u201398."},{"key":"2023033110105331281_j_cmam-2018-0009_ref_032_w2aab3b7d978b1b6b1ab2b1c32Aa","unstructured":"Y. Luchko,\nBoundary value problems for the generalized time-fractional diffusion equation of distributed order,\nFract. Calc. Appl. Anal. 12 (2009), no. 4, 409\u2013422."},{"key":"2023033110105331281_j_cmam-2018-0009_ref_033_w2aab3b7d978b1b6b1ab2b1c33Aa","doi-asserted-by":"crossref","unstructured":"F. Mainardi, A. Mura, G. Pagnini and R. Gorenflo,\nTime-fractional diffusion of distributed order,\nJ. Vib. Control 14 (2008), no. 9\u201310, 1267\u20131290.","DOI":"10.1177\/1077546307087452"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_034_w2aab3b7d978b1b6b1ab2b1c34Aa","doi-asserted-by":"crossref","unstructured":"F. Mainardi, G. Pagnini and R. Gorenflo,\nSome aspects of fractional diffusion equations of single and distributed order,\nAppl. Math. Comput. 187 (2007), no. 1, 295\u2013305.","DOI":"10.1016\/j.amc.2006.08.126"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_035_w2aab3b7d978b1b6b1ab2b1c35Aa","doi-asserted-by":"crossref","unstructured":"S. Mashayekhi and M. Razzaghi,\nNumerical solution of distributed order fractional differential equations by hybrid functions,\nJ. Comput. Phys. 315 (2016), 169\u2013181.","DOI":"10.1016\/j.jcp.2016.01.041"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_036_w2aab3b7d978b1b6b1ab2b1c36Aa","doi-asserted-by":"crossref","unstructured":"M. M. Meerschaert, E. Nane and P. Vellaisamy,\nDistributed-order fractional diffusions on bounded domains,\nJ. Math. Anal. Appl. 379 (2011), no. 1, 216\u2013228.","DOI":"10.1016\/j.jmaa.2010.12.056"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_037_w2aab3b7d978b1b6b1ab2b1c37Aa","doi-asserted-by":"crossref","unstructured":"J. M. Melenk and I. Babu\u0161ka,\nThe partition of unity finite element method: basic theory and applications,\nComput. Methods Appl. Mech. Engrg. 139 (1996), no. 1\u20134, 289\u2013314.","DOI":"10.1016\/S0045-7825(96)01087-0"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_038_w2aab3b7d978b1b6b1ab2b1c38Aa","doi-asserted-by":"crossref","unstructured":"B. Nayroles, G. Touzot and P. Villon,\nGeneralizing the finite element method: Diffuse approximation and diffuse elements,\nComput. Mech. 10 (1992), no. 5, 307\u2013315.","DOI":"10.1007\/BF00364252"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_039_w2aab3b7d978b1b6b1ab2b1c39Aa","doi-asserted-by":"crossref","unstructured":"R. Nigmatulin,\nThe realization of the generalized transfer equation in a medium with fractal geometry,\nPhys. Stat. Sol. B. 133 (1986), no. 1, 425\u2013430.","DOI":"10.1002\/pssb.2221330150"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_040_w2aab3b7d978b1b6b1ab2b1c40Aa","doi-asserted-by":"crossref","unstructured":"E. O\u00f1ate, S. Idelsohn, O. C. Zienkiewicz and R. L. Taylor,\nA finite point method in computational mechanics. Applications to convective transport and fluid flow,\nInternat. J. Numer. Methods Engrg. 39 (1996), no. 22, 3839\u20133866.","DOI":"10.1002\/(SICI)1097-0207(19961130)39:22<3839::AID-NME27>3.0.CO;2-R"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_041_w2aab3b7d978b1b6b1ab2b1c41Aa","doi-asserted-by":"crossref","unstructured":"E. O\u00f1ate, S. Idelsohn, O. C. Zienkiewicz, R. L. Taylor and C. Sacco,\nA stabilized finite point method for analysis of fluid mechanics problems,\nComput. Methods Appl. Mech. Engrg. 139 (1996), no. 1\u20134, 315\u2013346.","DOI":"10.1016\/S0045-7825(96)01088-2"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_042_w2aab3b7d978b1b6b1ab2b1c42Aa","doi-asserted-by":"crossref","unstructured":"I. Podlubny, T. Skovranek, B. M. Vinagre Jara, I. Petras, V. Verbitsky and Y. Chen,\nMatrix approach to discrete fractional calculus III: Non-equidistant grids, variable step length and distributed orders,\nPhilos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 371 (2013), no. 1990, Article iD 20120153.","DOI":"10.1098\/rsta.2012.0153"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_043_w2aab3b7d978b1b6b1ab2b1c43Aa","doi-asserted-by":"crossref","unstructured":"A. Quarteroni and A. Valli,\nNumerical Approximation of Partial Differential Equations,\nSpringer Ser. Comput. Math. 23,\nSpringer, Berlin, 1994.","DOI":"10.1007\/978-3-540-85268-1"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_044_w2aab3b7d978b1b6b1ab2b1c44Aa","doi-asserted-by":"crossref","unstructured":"Z.-Z. Sun and X. Wu,\nA fully discrete difference scheme for a diffusion-wave system,\nAppl. Numer. Math. 56 (2006), no. 2, 193\u2013209.","DOI":"10.1016\/j.apnum.2005.03.003"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_045_w2aab3b7d978b1b6b1ab2b1c45Aa","doi-asserted-by":"crossref","unstructured":"S. Umarov and R. Gorenflo,\nCauchy and nonlocal multi-point problems for distributed order pseudo-differential equations. I,\nZ. Anal. Anwend. 24 (2005), no. 3, 449\u2013466.","DOI":"10.4171\/ZAA\/1250"},{"key":"2023033110105331281_j_cmam-2018-0009_ref_046_w2aab3b7d978b1b6b1ab2b1c46Aa","doi-asserted-by":"crossref","unstructured":"X. X. Zhang and L. Mouchao,\nPersistence of anomalous dispersion in uniform porous media demonstrated by pore-scale simulations,\nWater. Resour. Res. 43 (2007), no. 7, 407\u2013437.","DOI":"10.1029\/2006WR005557"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/19\/4\/article-p813.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0009\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0009\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T11:22:38Z","timestamp":1680261758000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0009\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,4,18]]},"references-count":46,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2018,4,10]]},"published-print":{"date-parts":[[2019,10,1]]}},"alternative-id":["10.1515\/cmam-2018-0009"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2018-0009","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"type":"electronic","value":"1609-9389"},{"type":"print","value":"1609-4840"}],"subject":[],"published":{"date-parts":[[2018,4,18]]}}}