{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:58:17Z","timestamp":1760245097452,"version":"3.37.3"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2021,11,12]],"date-time":"2021-11-12T00:00:00Z","timestamp":1636675200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,11,12]],"date-time":"2021-11-12T00:00:00Z","timestamp":1636675200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100010909","name":"Young Scientists Fund","doi-asserted-by":"publisher","award":["11701103"],"award-info":[{"award-number":["11701103"]}],"id":[{"id":"10.13039\/501100010909","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2022,10]]},"DOI":"10.1007\/s12190-021-01661-3","type":"journal-article","created":{"date-parts":[[2021,11,12]],"date-time":"2021-11-12T16:04:48Z","timestamp":1636733088000},"page":"3199-3217","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Orthogonal spline collocation method for the two-dimensional time fractional mobile-immobile equation"],"prefix":"10.1007","volume":"68","author":[{"given":"Leijie","family":"Qiao","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Da","family":"Xu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhibo","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,11,12]]},"reference":[{"key":"1661_CR1","doi-asserted-by":"publisher","first-page":"785","DOI":"10.1016\/0362-546X(88)90039-9","volume":"12","author":"EG Yanik","year":"1988","unstructured":"Yanik, E.G., Fairweather, G.: Finite element methods for parabolic and hyperbolic partial integrodifferential equations. Nonlinear Anal. 12, 785\u2013809 (1988)","journal-title":"Nonlinear Anal."},{"key":"1661_CR2","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1007\/s11071-015-2326-4","volume":"83","author":"SC Shiralashetti","year":"2016","unstructured":"Shiralashetti, S.C., Deshi, A.B.: An efficient Haar wavelet collocation method for the numerical solution of multi-term fractional differential equations. Nonlinear Dyn. 83, 293\u2013303 (2016)","journal-title":"Nonlinear Dyn."},{"key":"1661_CR3","doi-asserted-by":"publisher","first-page":"616","DOI":"10.1016\/j.mcm.2009.11.002","volume":"51","author":"V Srivastava","year":"2010","unstructured":"Srivastava, V., Rai, K.: A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues. Math. Comput. Model. 51, 616\u2013624 (2010)","journal-title":"Math. Comput. Model."},{"key":"1661_CR4","doi-asserted-by":"publisher","first-page":"693","DOI":"10.1016\/j.camwa.2013.01.031","volume":"66","author":"H Zhang","year":"2013","unstructured":"Zhang, H., Liu, F., Phanikumar, M., Meerschaert, M.: A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model. Comput. Math. Appl. 66, 693\u2013701 (2013)","journal-title":"Comput. Math. Appl."},{"key":"1661_CR5","doi-asserted-by":"publisher","first-page":"448","DOI":"10.1016\/j.apnum.2019.11.012","volume":"151","author":"P Lyu","year":"2020","unstructured":"Lyu, P., Liang, Y., Wang, Z.: A fast linearized finite difference method for the nonlinear multi-term time-fractional wave equation. Appl. Numer. Math. 151, 448\u2013471 (2020)","journal-title":"Appl. Numer. Math."},{"key":"1661_CR6","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1016\/j.jcp.2014.10.051","volume":"281","author":"B Jin","year":"2015","unstructured":"Jin, B., Lazarov, R., Liu, Y., Zhou, Z.: The Galerkin finite element method for a multi-term time-fractional diffusion equation. J. Comput. Phys. 281, 825\u2013843 (2015)","journal-title":"J. Comput. Phys."},{"key":"1661_CR7","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1016\/j.apnum.2020.01.003","volume":"151","author":"L Qiao","year":"2020","unstructured":"Qiao, L., Wang, Z., Xu, D.: An alternating direction implicit orthogonal spline collocation method for the two-dimensional multi-term time fractional integro-differential equation. Appl. Numer. Math. 151, 199\u2013212 (2020)","journal-title":"Appl. Numer. Math."},{"key":"1661_CR8","doi-asserted-by":"crossref","unstructured":"Nigmatullin, R.: To the theoretical explanation of the universal response. Physica Status (B) Basic Res. 123, 739\u2013745 (1984)","DOI":"10.1002\/pssb.2221230241"},{"key":"1661_CR9","doi-asserted-by":"crossref","unstructured":"Nigmatullin, R.: Realization of the generalized transfer equation in a medium with fractal geometry. Physica Status (B): Basic Res. 133, 425\u2013430 (1986)","DOI":"10.1002\/pssb.2221330150"},{"key":"1661_CR10","doi-asserted-by":"publisher","first-page":"10","DOI":"10.1029\/2003WR002141","volume":"39","author":"R Schumer","year":"2003","unstructured":"Schumer, R., Benson, D., Meerschaert, M., Baeumer, B.: Fractal mobile\/immobile solute transport. Water Resour. Res. 39, 10 (2003)","journal-title":"Water Resour. Res."},{"key":"1661_CR11","doi-asserted-by":"publisher","first-page":"561","DOI":"10.1016\/j.advwatres.2009.01.008","volume":"32","author":"Y Zhang","year":"2009","unstructured":"Zhang, Y., Benson, D., Reeves, D.: Time and space nonlocalities underlying fractional-derivative models: distinction and literature review of field applications. Adv. Water Resour. 32, 561\u2013581 (2009)","journal-title":"Adv. Water Resour."},{"key":"1661_CR12","first-page":"125693","volume":"392","author":"W Qiu","year":"2021","unstructured":"Qiu, W., Xu, D., Guo, J.: Numerical solution of the fourth-order partial integro-differential equation with multi-term kernels by the Sinc-collocation method based on the double exponential transfor-mation. Appl. Math. Comput. 392, 125693 (2021)","journal-title":"Appl. Math. Comput."},{"key":"1661_CR13","doi-asserted-by":"publisher","first-page":"190","DOI":"10.1016\/j.apnum.2020.09.006","volume":"159","author":"Z Wang","year":"2021","unstructured":"Wang, Z., Cen, D., Mo, Y.: Sharp error estimate of a compact L1-ADI scheme for the two-dimensional time-fractional integro-differential equation with singular kernels. Appl. Numer. Math. 159, 190\u2013203 (2021)","journal-title":"Appl. Numer. Math."},{"key":"1661_CR14","doi-asserted-by":"publisher","first-page":"106829","DOI":"10.1016\/j.aml.2020.106829","volume":"112","author":"D Cen","year":"2021","unstructured":"Cen, D., Wang, Z., Mo, Y.: Second order difference schemes for time-fractional KdV-Burgers equation with initial singularity. Appl. Math. Lett. 112, 106829 (2021)","journal-title":"Appl. Math. Lett."},{"key":"1661_CR15","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1007\/s10915-021-01661-2","volume":"89","author":"P Lyu","year":"2021","unstructured":"Lyu, P., Vong, S.: Second-order and nonuniform time-stepping schemes for time fractional evolution equations with timeCspace dependent coefficients. J. Sci. Comput. 89, 49 (2021)","journal-title":"J. Sci. Comput."},{"key":"1661_CR16","first-page":"336","volume":"226","author":"Q Liu","year":"2014","unstructured":"Liu, Q., Liu, F., Turner, I.: A RBF meshless approach for modeling a fractal mobile\/immobile transport model. Appl. Math. Comput. 226, 336\u2013347 (2014)","journal-title":"Appl. Math. Comput."},{"key":"1661_CR17","first-page":"391","volume":"56","author":"Z Liu","year":"2018","unstructured":"Liu, Z., Li, X.: A Crank-Nicolson difference scheme for the time variable fractional mobile-immobile advection-dispersion equation. Appl. Math. Comput. 56, 391\u2013410 (2018)","journal-title":"Appl. Math. Comput."},{"key":"1661_CR18","first-page":"2975","volume":"219","author":"Z Zhao","year":"2012","unstructured":"Zhao, Z., Li, C.: Fractional difference\/finite element approximations for the time-space fractional telegraph equation. Appl. Math. Comput. 219, 2975\u20132988 (2012)","journal-title":"Appl. Math. Comput."},{"key":"1661_CR19","first-page":"124799","volume":"368","author":"B Yin","year":"2020","unstructured":"Yin, B., Liu, Y., Hong, L.: A class of shifted high-order numerical methods for the fractional mobile\/immobile transport equations. Appl. Math. Comput. 368, 124799 (2020)","journal-title":"Appl. Math. Comput."},{"key":"1661_CR20","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1007\/s11075-019-00801-y","volume":"85","author":"W Qiu","year":"2020","unstructured":"Qiu, W., Xu, D., Guo, J., Zhou, J.: A time two-grid algorithm based on finite difference method for the two-dimensional nonlinear time-fractional mobile\/immobile transport model. Numer. Algor. 85, 39\u201358 (2020)","journal-title":"Numer. Algor."},{"key":"1661_CR21","doi-asserted-by":"publisher","first-page":"360","DOI":"10.1186\/s13662-020-02786-8","volume":"2020","author":"J Zhao","year":"2020","unstructured":"Zhao, J., Fang, Z., Li, H., Liu, Y.: Finite volume element method with the WSGD formula for nonlinear fractional mobile\/immobile transport equations. Adv. Differ. Equ. 2020, 360 (2020)","journal-title":"Adv. Differ. Equ."},{"key":"1661_CR22","doi-asserted-by":"publisher","first-page":"1364","DOI":"10.1016\/j.jmaa.2007.06.023","volume":"338","author":"J Chen","year":"2008","unstructured":"Chen, J., Liu, F., Anh, V.: Analytical solution for the time-fractional telegraph equation by the method of separating variables. J. Math. Anal. Appl. 338, 1364\u20131377 (2008)","journal-title":"J. Math. Anal. Appl."},{"key":"1661_CR23","doi-asserted-by":"publisher","first-page":"248","DOI":"10.1093\/imanum\/drp024","volume":"30","author":"A Pani","year":"2010","unstructured":"Pani, A., Fairweather, G., Fernandes, R.: Orthogonal spline collocation methods for partial integro-differential equations. SIAM J. Numer. Anal. 30, 248\u2013276 (2010)","journal-title":"SIAM J. Numer. Anal."},{"key":"1661_CR24","first-page":"1689","volume":"38B","author":"X Yang","year":"2018","unstructured":"Yang, X., Zhang, H., Xu, D.: Alternatting direction implicit OSC scheme for the two-dimensional fractional evolution equation with a weakly singular kernel. Acta Math. Sci. 38B, 1689\u20131711 (2018)","journal-title":"Acta Math. Sci."},{"key":"1661_CR25","doi-asserted-by":"publisher","first-page":"3807","DOI":"10.1016\/j.camwa.2019.06.002","volume":"78","author":"L Qiao","year":"2019","unstructured":"Qiao, L., Xu, D.: BDF ADI orthogonal spline collocation scheme for the fractional integro-differential equation with two weakly singular kernels. Comput. Math. Appl. 78, 3807\u20133820 (2019)","journal-title":"Comput. Math. Appl."},{"key":"1661_CR26","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1016\/j.camwa.2021.10.014","volume":"102","author":"L Qiao","year":"2021","unstructured":"Qiao, L., Qiu, W., Xu, D.: A second-order ADI difference scheme based on non-uniform meshes for the three-dimensional nonlocal evolution problem. Comput. Math. Appl. 102, 137\u2013145 (2021)","journal-title":"Comput. Math. Appl."},{"key":"1661_CR27","doi-asserted-by":"publisher","first-page":"64","DOI":"10.1007\/s10444-021-09884-5","volume":"47","author":"L Qiao","year":"2021","unstructured":"Qiao, L., Xu, D.: A fast ADI orthogonal spline collocation method with graded meshes for the two-dimensional fractional integro-differential equation. Adv. Comput. Math. 47, 64 (2021)","journal-title":"Adv. Comput. Math."},{"key":"1661_CR28","doi-asserted-by":"crossref","unstructured":"Atangana, A., Baleanu, D.: Numerical solution of a kind of fractional parabolic equations via two difference schemes. Abstr. Appl. Anal. 141467 (2013)","DOI":"10.1155\/2013\/828764"},{"key":"1661_CR29","doi-asserted-by":"publisher","first-page":"755","DOI":"10.1137\/0729047","volume":"29","author":"Y Yan","year":"1992","unstructured":"Yan, Y., Fairweather, G.: Orthogonal spline collocation methods for some partial integro-differential equations. SIAM J. Numer. Anal. 29, 755\u2013768 (1992)","journal-title":"SIAM J. Numer. Anal."},{"key":"1661_CR30","doi-asserted-by":"publisher","first-page":"1533","DOI":"10.1016\/j.jcp.2007.02.001","volume":"225","author":"Y Lin","year":"2007","unstructured":"Lin, Y., Xu, C.: Finite difference\/spectral approximations for the time fractional diffusion equation. J. Comput. Phys. 225, 1533\u20131552 (2007)","journal-title":"J. Comput. Phys."},{"key":"1661_CR31","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1016\/j.apnum.2005.03.003","volume":"56","author":"Z Sun","year":"2006","unstructured":"Sun, Z., Wu, X.: A fully discrete scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193\u2013209 (2006)","journal-title":"Appl. Numer. Math."},{"key":"1661_CR32","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1137\/S003614450240506X","volume":"46","author":"G Fairweather","year":"2004","unstructured":"Fairweather, G., Gladwell, I.: Algorithms for almost block diagonal linear systems. SIAM Rev. 46, 49\u201358 (2004)","journal-title":"SIAM Rev."},{"key":"1661_CR33","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1002\/num.1690090207","volume":"9","author":"R Fernandes","year":"1993","unstructured":"Fernandes, R., Fairweather, G.: Analysis of alternating direction collocation methods for parabolic and hyperbolic problems in two space variables. Numer. Math. Part. Differ. Equ. 9, 191\u2013211 (1993)","journal-title":"Numer. Math. Part. Differ. Equ."},{"key":"1661_CR34","doi-asserted-by":"publisher","first-page":"344","DOI":"10.1137\/050634967","volume":"46","author":"A Pani","year":"2008","unstructured":"Pani, A., Fairweather, G., Fernandes, R.: Alternating direction implicit orthogonal spline collocation methods for an evolution equation with a positive-type memory term. SIAM J. Numer. Anal. 46, 344\u2013364 (2008)","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-021-01661-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-021-01661-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-021-01661-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,24]],"date-time":"2022-09-24T02:16:01Z","timestamp":1663985761000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-021-01661-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,12]]},"references-count":34,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2022,10]]}},"alternative-id":["1661"],"URL":"https:\/\/doi.org\/10.1007\/s12190-021-01661-3","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"type":"print","value":"1598-5865"},{"type":"electronic","value":"1865-2085"}],"subject":[],"published":{"date-parts":[[2021,11,12]]},"assertion":[{"value":"6 September 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 October 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 October 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 November 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}