{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T18:54:37Z","timestamp":1775501677324,"version":"3.50.1"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,7,12]],"date-time":"2023-07-12T00:00:00Z","timestamp":1689120000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,7,12]],"date-time":"2023-07-12T00:00:00Z","timestamp":1689120000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11771265"],"award-info":[{"award-number":["11771265"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2024,2]]},"DOI":"10.1007\/s11075-023-01592-z","type":"journal-article","created":{"date-parts":[[2023,7,12]],"date-time":"2023-07-12T07:02:05Z","timestamp":1689145325000},"page":"859-895","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Fractional centered difference schemes and banded preconditioners for nonlinear Riesz space variable-order fractional diffusion equations"],"prefix":"10.1007","volume":"95","author":[{"given":"Qiu-Ya","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zi-Hang","family":"She","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cheng-Xue","family":"Lao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4099-6706","authenticated-orcid":false,"given":"Fu-Rong","family":"Lin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,7,12]]},"reference":[{"key":"1592_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1155\/2021\/8050017","volume":"2021","author":"A Abirami","year":"2021","unstructured":"Abirami, A., Prakash, P., Ma, Y.K.: Variable-order fractional diffusion model-based medical image denoising. Math. Probl. Eng. 2021, 1\u201310 (2021)","journal-title":"Math. Probl. Eng."},{"key":"1592_CR2","doi-asserted-by":"publisher","first-page":"1591","DOI":"10.1137\/0727093","volume":"27","author":"O Axelsson","year":"1990","unstructured":"Axelsson, O., Kolotilina, L.: Montonicity and discretization error eatimates. SIAM J. Numer. Anal. 27, 1591\u20131611 (1990)","journal-title":"SIAM J. Numer. Anal."},{"key":"1592_CR3","doi-asserted-by":"publisher","first-page":"2492","DOI":"10.1109\/TIP.2007.904971","volume":"16","author":"J Bai","year":"2007","unstructured":"Bai, J., Feng, X.C.: Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Proc. 16, 2492\u20132502 (2007)","journal-title":"IEEE Trans. Image Proc."},{"key":"1592_CR4","doi-asserted-by":"publisher","first-page":"1403","DOI":"10.1029\/2000WR900031","volume":"36","author":"DA Benson","year":"2000","unstructured":"Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection-dispersion equation. Water Resour. Res. 36, 1403\u20131412 (2000)","journal-title":"Water Resour. Res."},{"key":"1592_CR5","doi-asserted-by":"publisher","first-page":"1413","DOI":"10.1029\/2000WR900032","volume":"36","author":"DA Benson","year":"2000","unstructured":"Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: The fractional-order governing equation of L\u00e9vy motion. Water Resour. Res. 36, 1413\u20131423 (2000)","journal-title":"Water Resour. Res."},{"key":"1592_CR6","doi-asserted-by":"publisher","first-page":"1743","DOI":"10.1016\/j.jcp.2011.11.008","volume":"231","author":"C \u00c7elik","year":"2012","unstructured":"\u00c7elik, C., Duman, M.: Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J. Comput. Phys. 231, 1743\u20131750 (2012)","journal-title":"J. Comput. Phys."},{"key":"1592_CR7","doi-asserted-by":"publisher","first-page":"653797","DOI":"10.1155\/2014\/653797","volume":"2014","author":"HF Ding","year":"2014","unstructured":"Ding, H.F., Li, C.P., Chen, Y.Q.: High-order algorithms for Riesz derivative and their applications (I). Abstr. Appl. Anal. 2014, 653797 (2014). https:\/\/doi.org\/10.1155\/2014\/653797","journal-title":"Abstr. Appl. Anal."},{"key":"1592_CR8","doi-asserted-by":"publisher","first-page":"262","DOI":"10.1016\/j.jcp.2015.11.061","volume":"307","author":"M Donatelli","year":"2016","unstructured":"Donatelli, M., Mazza, M., Serra-Capizzano, S.: Spectral analysis and structure preserving preconditioners for fractional diffusion equations. J. Comput. Phys. 307, 262\u2013279 (2016)","journal-title":"J. Comput. Phys."},{"key":"1592_CR9","doi-asserted-by":"publisher","first-page":"2952","DOI":"10.1016\/j.camwa.2020.01.003","volume":"79","author":"R Du","year":"2020","unstructured":"Du, R., Alikhanov, A.A., Sun, Z.Z.: Temporal second order difference schemes for the multi-dimensional variable-order time fractional sub-diffusion equations. Comput. Math. Appl. 79, 2952\u20132972 (2020)","journal-title":"Comput. Math. Appl."},{"key":"1592_CR10","first-page":"72","volume":"1952","author":"W Feller","year":"1952","unstructured":"Feller, W.: On a generalization of Marcel Riesz\u2019 potentials and the semi-groups generated by them. Medd. Lunds Univ. Mat. Sem. 1952, 72\u201381 (1952)","journal-title":"Medd. Lunds Univ. Mat. Sem."},{"key":"1592_CR11","doi-asserted-by":"publisher","first-page":"105904","DOI":"10.1016\/j.cnsns.2021.105904","volume":"102","author":"R Garrappa","year":"2021","unstructured":"Garrappa, R., Giusti, A., Mainardi, F.: Variable-order fractional calculus: a change of perspective. Commun. Nonlinear Sci. Numer. Simul. 102, 105904 (2021). https:\/\/doi.org\/10.1016\/j.cnsns.2021.105904","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"1592_CR12","doi-asserted-by":"publisher","first-page":"893","DOI":"10.4208\/nmtma.OA-2020-0020","volume":"14","author":"XM Gu","year":"2021","unstructured":"Gu, X.M., Zhao, Y.L., Zhao, X.L., Carpentieri, B., Huang, Y.Y.: A note on parallel preconditioning for the all-at-once solution of Riesz fractional diffusion equations. Numer. Math. Theor. Meth. Appl. 14, 893\u2013919 (2021)","journal-title":"Numer. Math. Theor. Meth. Appl."},{"key":"1592_CR13","doi-asserted-by":"publisher","first-page":"126224","DOI":"10.1016\/j.amc.2021.126224","volume":"404","author":"L Guo","year":"2021","unstructured":"Guo, L., Zhao, X.L., Gu, X.M., Zhao, Y.L., Zheng, Y.B., Huang, T.Z.: Three-dimensional fractional total variation regularized tensor optimized model for image deblurring. Appl. Math. Comput. 404, 126224 (2021). https:\/\/doi.org\/10.1016\/j.amc.2021.126224","journal-title":"Appl. Math. Comput."},{"key":"1592_CR14","doi-asserted-by":"publisher","first-page":"448","DOI":"10.1016\/S0378-4371(99)00469-0","volume":"276","author":"BI Henry","year":"2000","unstructured":"Henry, B.I., Wearne, S.L.: Fractional reaction-diffusion. Physica A 276, 448\u2013455 (2000)","journal-title":"Physica A"},{"key":"1592_CR15","volume-title":"Toptics in Matrix Analysis","author":"RA Horn","year":"1994","unstructured":"Horn, R.A., Johnson, C.R.: Toptics in Matrix Analysis. Academic Press, Cambridge (1994)"},{"key":"1592_CR16","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1016\/j.cam.2019.06.008","volume":"363","author":"FR Lin","year":"2020","unstructured":"Lin, F.R., Liu, W.D.: The accuracy and stability of CN-WSGD schemes for space fractional diffusion equation. J. Comput. Appl. Math. 363, 7\u201391 (2020)","journal-title":"J. Comput. Appl. Math."},{"key":"1592_CR17","first-page":"435","volume":"212","author":"R Lin","year":"2009","unstructured":"Lin, R., Liu, F., Anh, V., Turner, I.: Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Appl. Math. Comput. 212, 435\u2013445 (2009)","journal-title":"Appl. Math. Comput."},{"key":"1592_CR18","doi-asserted-by":"publisher","first-page":"222","DOI":"10.1016\/j.apnum.2020.11.012","volume":"164","author":"FR Lin","year":"2021","unstructured":"Lin, F.R., Qiu, Y.F., She, Z.H.: IRK-WSGD methods for space fractional diffusion equations. Appl. Numer. Math. 164, 222\u2013244 (2021)","journal-title":"Appl. Numer. Math."},{"key":"1592_CR19","doi-asserted-by":"publisher","first-page":"1311","DOI":"10.1007\/s10543-021-00858-z","volume":"61","author":"FR Lin","year":"2021","unstructured":"Lin, F.R., Qu, H.D., She, Z.H.: DNT preconditioner for one-sided space fractional diffusion equations. BIT 61, 1311\u20131335 (2021)","journal-title":"BIT"},{"key":"1592_CR20","doi-asserted-by":"publisher","first-page":"601","DOI":"10.1007\/s11075-020-00980-z","volume":"87","author":"FR Lin","year":"2021","unstructured":"Lin, F.R., Wang, Q.Y., Jin, X.Q.: Crank-Nicolson-weighted-shifted-Gr\u00fcnwald difference schemes for space Riesz variable-order fractional diffusion equations. Numer. Algor. 87, 601\u2013631 (2021)","journal-title":"Numer. Algor."},{"key":"1592_CR21","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1016\/j.jcp.2013.07.040","volume":"256","author":"FR Lin","year":"2014","unstructured":"Lin, F.R., Yang, S.W., Jin, X.Q.: Preconditioned iterative methods for fractional diffusion equation. J. Comput. Phys. 256, 109\u2013117 (2014)","journal-title":"J. Comput. Phys."},{"key":"1592_CR22","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1023\/A:1016586905654","volume":"29","author":"CF Lorenzo","year":"2002","unstructured":"Lorenzo, C.F., Hartley, T.T.: Variable-order and distributed order fractional operators. Nonlinear Dyn. 29, 57\u201398 (2002)","journal-title":"Nonlinear Dyn."},{"key":"1592_CR23","volume-title":"An Introduction to the Fractional Calculus and Fractional Differential Equations","author":"KS Miller","year":"1993","unstructured":"Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley, New York (1993)"},{"key":"1592_CR24","volume-title":"The Fractional Calculus","author":"KB Oldham","year":"1974","unstructured":"Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)"},{"key":"1592_CR25","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1155\/IJMMS\/2006\/48391","volume":"2006","author":"MD Ortigueira","year":"2006","unstructured":"Ortigueira, M.D.: Riesz potential operators and inverses via fractional centred derivatives. Int. J. Math. Math. Sci. 2006, 1\u201312 (2006)","journal-title":"Int. J. Math. Math. Sci."},{"key":"1592_CR26","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1007\/s10915-021-01427-w","volume":"87","author":"HK Pang","year":"2021","unstructured":"Pang, H.K., Sun, H.W.: A fast algorithm for the variable-order spatial fractional advection-diffusion equation. J. Sci. Comput. 87, 15 (2021). https:\/\/doi.org\/10.1007\/s10915-021-01427-w","journal-title":"J. Sci. Comput."},{"key":"1592_CR27","volume-title":"Fractional Differential Equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny, I.: Fractional Differential Equations. Cambridge University Press, New York (1999)"},{"key":"1592_CR28","doi-asserted-by":"publisher","first-page":"775","DOI":"10.1081\/SAP-120030456","volume":"22","author":"MD Ruiz-Medina","year":"2004","unstructured":"Ruiz-Medina, M.D., Anh, V., Angulo, J.M.: Fractional generalized random fields of variable order. Stochastic Anal. Appl. 22, 775\u2013799 (2004)","journal-title":"Stochastic Anal. Appl."},{"key":"1592_CR29","volume-title":"Fractional Integerals and Derivatives: Theory and Applications","author":"SG Samko","year":"1993","unstructured":"Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integerals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Amsterdam (1993)"},{"key":"1592_CR30","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1080\/10652469308819027","volume":"1","author":"SG Samko","year":"1993","unstructured":"Samko, S.G., Ross, B.: Integration and differentiation to a variable fractional order. Integral Transform Spec. Funct. 1, 277\u2013300 (1993)","journal-title":"Integral Transform Spec. Funct."},{"key":"1592_CR31","doi-asserted-by":"publisher","first-page":"2165","DOI":"10.1063\/1.1587126","volume":"119","author":"K Seki","year":"2003","unstructured":"Seki, K., Wojcik, M., Tachiya, M.: Fractional reaction-diffusion equation. J. Chem. Phys. 119, 2165\u20132174 (2003)","journal-title":"J. Chem. Phys."},{"key":"1592_CR32","doi-asserted-by":"publisher","unstructured":"She, Z.H.: A class of unconditioned stable 4-point WSGD schemes and fast iteration methods for space fractional diffusion equations. J. Sci. Comput. 92, 18 (2022). https:\/\/doi.org\/10.1007\/s10915-022-01860-5","DOI":"10.1007\/s10915-022-01860-5"},{"key":"1592_CR33","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1007\/s10915-020-01398-4","volume":"86","author":"ZH She","year":"2021","unstructured":"She, Z.H., Lao, C.X., Yang, H., Lin, F.R.: Banded preconditioners for Riesz space fractional diffusion equations. J. Sci. Comput. 86, 31 (2021). https:\/\/doi.org\/10.1007\/s10915-020-01398-4","journal-title":"J. Sci. Comput."},{"key":"1592_CR34","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1016\/j.camwa.2021.02.018","volume":"89","author":"ZH She","year":"2021","unstructured":"She, Z.H., Qu, H.D., Liu, H.: Stability and convergence of finite difference method for two-sided space-fractional diffusion equations. Comput. Math. Appl. 89, 78\u201386 (2021)","journal-title":"Comput. Math. Appl."},{"key":"1592_CR35","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1063\/1.1535007","volume":"55","author":"IM Sokolov","year":"2002","unstructured":"Sokolov, I.M., Klafter, J., Blumen, A.: Fractional kinetics. Phys. Today 55, 48\u201354 (2002)","journal-title":"Phys. Today"},{"key":"1592_CR36","doi-asserted-by":"publisher","first-page":"1703","DOI":"10.1090\/S0025-5718-2015-02917-2","volume":"84","author":"W Tian","year":"2015","unstructured":"Tian, W., Zhou, H., Deng, W.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84, 1703\u20131727 (2015)","journal-title":"Math. Comput."},{"key":"1592_CR37","first-page":"241","volume":"257","author":"DL Wang","year":"2015","unstructured":"Wang, D.L., Xiao, A.G., Yang, W.: Maximum-norm error analysis of a difference scheme for the space fractional CNLS. Appl. Math. Comput. 257, 241\u2013251 (2015)","journal-title":"Appl. Math. Comput."},{"key":"1592_CR38","doi-asserted-by":"publisher","first-page":"A2710","DOI":"10.1137\/141001299","volume":"37","author":"FH Zeng","year":"2015","unstructured":"Zeng, F.H., Zhang, Z.Q., Karniadakis, G.E.: A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations. SIAM J. Sci. Comput. 37, A2710\u2013A2732 (2015)","journal-title":"SIAM J. Sci. Comput."},{"key":"1592_CR39","doi-asserted-by":"publisher","first-page":"184","DOI":"10.1016\/j.jcp.2014.08.015","volume":"293","author":"X Zhao","year":"2015","unstructured":"Zhao, X., Sun, Z.Z., Karniadakis, G.E.: Second-order approximations for variable order fractional derivatives: Algorithms and applications. J. Comput. Phys. 293, 184\u2013200 (2015)","journal-title":"J. Comput. Phys."},{"key":"1592_CR40","doi-asserted-by":"publisher","first-page":"10","DOI":"10.1007\/s10915-020-01193-1","volume":"83","author":"YL Zhao","year":"2020","unstructured":"Zhao, Y.L., Zhu, P.Y., Gu, X.M., Zhao, X.L., Jian, H.Y.: A preconditioning technique for all-at-once system from the nonlinear tempered fractional diffusion equation. J. Sci. Comput. 83, 10 (2020). https:\/\/doi.org\/10.1007\/s10915-020-01193-1","journal-title":"J. Sci. Comput."},{"key":"1592_CR41","doi-asserted-by":"publisher","first-page":"1760","DOI":"10.1137\/080730597","volume":"47","author":"P Zhuang","year":"2009","unstructured":"Zhuang, P., Liu, F., Anh, V., Turner, I.: Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 47, 1760\u20131781 (2009)","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-023-01592-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-023-01592-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-023-01592-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,1,17]],"date-time":"2024-01-17T08:12:26Z","timestamp":1705479146000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-023-01592-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,12]]},"references-count":41,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,2]]}},"alternative-id":["1592"],"URL":"https:\/\/doi.org\/10.1007\/s11075-023-01592-z","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,12]]},"assertion":[{"value":"16 June 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 May 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 July 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}