{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,26]],"date-time":"2025-08-26T07:08:38Z","timestamp":1756192118368,"version":"3.37.3"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2021,10,9]],"date-time":"2021-10-09T00:00:00Z","timestamp":1633737600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,10,9]],"date-time":"2021-10-09T00:00:00Z","timestamp":1633737600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11502028","11832001"],"award-info":[{"award-number":["11502028","11832001"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11890681","11988102"],"award-info":[{"award-number":["11890681","11988102"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2021,11]]},"DOI":"10.1007\/s10915-021-01655-0","type":"journal-article","created":{"date-parts":[[2021,10,10]],"date-time":"2021-10-10T14:57:10Z","timestamp":1633877830000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Accurate Boundary Treatment for Riesz Space Fractional Diffusion Equations"],"prefix":"10.1007","volume":"89","author":[{"given":"Shaoqiang","family":"Tang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gang","family":"Pang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,10,9]]},"reference":[{"key":"1655_CR1","doi-asserted-by":"publisher","DOI":"10.1142\/9543","volume-title":"Fractional Partial Differential Equations and Their Numerical Solutions","author":"B Guo","year":"2015","unstructured":"Guo, B., Pu, X., Huang, F.: Fractional Partial Differential Equations and Their Numerical Solutions. World Scientific, Singapore (2015)"},{"key":"1655_CR2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jcp.2015.04.007","volume":"293","author":"G Karniadakis","year":"2015","unstructured":"Karniadakis, G., Hesthaven, J., Podlubny, I.: Special issue on fractional PDEs: theory, numerics, and applications. J. Comput. Phys. 293, 1\u20133 (2015)","journal-title":"J. Comput. Phys."},{"key":"1655_CR3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BFb0067096","volume":"457","author":"B Ross","year":"1975","unstructured":"Ross, B.: A brief history and exposition of the fundamental theory of fractional calculus. Lect. Notes Math. 457, 1\u201336 (1975)","journal-title":"Lect. Notes Math."},{"key":"1655_CR4","doi-asserted-by":"publisher","first-page":"298","DOI":"10.1016\/S0375-9601(00)00201-2","volume":"268","author":"N Laskin","year":"2000","unstructured":"Laskin, N.: Fractional quantum mechanics and L\u00e9vy path integrals. Phys. Lett. A 268, 298\u2013305 (2000)","journal-title":"Phys. Lett. A"},{"key":"1655_CR5","first-page":"218","volume":"25","author":"J Zhang","year":"2019","unstructured":"Zhang, J., Li, D., Antoine, X.: Efficient numerical computation of time-fractional nonlinear Schr\u00f6dinger equations in unbounded domain. Commun. Comput. Phys. 25, 218\u2013243 (2019)","journal-title":"Commun. Comput. Phys."},{"key":"1655_CR6","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1016\/j.jcp.2019.03.008","volume":"388","author":"G Teodoro","year":"2019","unstructured":"Teodoro, G., Machado, J., Oliveira, E.: A review of definitions of fractional derivatives and other operators. J. Comput. Phys. 388, 195\u2013208 (2019)","journal-title":"J. Comput. Phys."},{"key":"1655_CR7","doi-asserted-by":"publisher","first-page":"204","DOI":"10.1137\/080714130","volume":"47","author":"W Deng","year":"2008","unstructured":"Deng, W.: Finite element method for the space and time fractional Fokker\u2013Planck equation. SIAM J. Numer. Anal. 47, 204\u2013226 (2008)","journal-title":"SIAM J. Numer. Anal."},{"key":"1655_CR8","doi-asserted-by":"publisher","first-page":"264","DOI":"10.1016\/j.jcp.2014.06.031","volume":"293","author":"W Bu","year":"2015","unstructured":"Bu, W., Tang, Y., Wu, Y., Yang, J.: Finite difference\/finite element method for two-dimensional space and time fractional Bloch\u2013Torrey equations. J. Comput. Phys. 293, 264\u2013279 (2015)","journal-title":"J. Comput. Phys."},{"key":"1655_CR9","doi-asserted-by":"publisher","first-page":"1320","DOI":"10.1137\/16M1078197","volume":"39","author":"F Song","year":"2017","unstructured":"Song, F., Xu, C., Karniadakis, G.: Computing fractional Laplacians on complex-geometry domains: algorithms and simulations. SIAM J. Sci. Comput. 39, 1320\u20131344 (2017)","journal-title":"SIAM J. Sci. Comput."},{"key":"1655_CR10","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1016\/j.chaos.2017.03.038","volume":"102","author":"Y Hu","year":"2017","unstructured":"Hu, Y., Li, C., Li, H.: The finite difference method for Caputo-type parabolic equation with fractional Laplacian: one-dimension case. Chaos Soliton Fract. 102, 319\u2013326 (2017)","journal-title":"Chaos Soliton Fract."},{"key":"1655_CR11","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1016\/j.mechmat.2018.07.012","volume":"127","author":"J Long","year":"2018","unstructured":"Long, J., Xiao, R., Chen, W.: Fractional viscoelastic models with non-singular kernels. Mech. Mater. 127, 55\u201364 (2018)","journal-title":"Mech. Mater."},{"key":"1655_CR12","doi-asserted-by":"publisher","first-page":"388","DOI":"10.1016\/j.cma.2016.06.013","volume":"309","author":"Y Lian","year":"2016","unstructured":"Lian, Y., Ying, Y., Tang, S., Lin, S., Wagner, G., Liu, W.: A Petrov-Galerkin finite element method for the fractional advection\u2013diffusion equation. Comput. Methods Appl. Mech. Eng. 309, 388\u2013410 (2016)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1655_CR13","doi-asserted-by":"publisher","first-page":"879","DOI":"10.1007\/s00466-016-1320-0","volume":"58","author":"S Tang","year":"2016","unstructured":"Tang, S., Ying, Y., Lian, Y., Lin, S., Yang, Y., Wagner, G., Liu, W.: Differential operator multiplication method for fractional differential equations. Comput. Mech. 58, 879\u2013888 (2016)","journal-title":"Comput. Mech."},{"key":"1655_CR14","doi-asserted-by":"publisher","first-page":"515","DOI":"10.1007\/s10409-017-0742-z","volume":"34","author":"Y Ying","year":"2018","unstructured":"Ying, Y., Lian, Y., Tang, S., Liu, W.: Enriched reproducing kernel particle method for fractional advection\u2013diffusion equation. Acta. Mech. Sin. 34, 515\u2013527 (2018)","journal-title":"Acta. Mech. Sin."},{"key":"1655_CR15","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1016\/j.cma.2016.12.008","volume":"317","author":"Y Ying","year":"2017","unstructured":"Ying, Y., Lian, Y., Tang, S., Liu, W.: High-order central difference scheme for Caputo fractional derivative. Comput. Methods Appl. Mech. Eng. 317, 42\u201354 (2017)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1655_CR16","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1016\/j.jcp.2017.11.011","volume":"355","author":"S Duo","year":"2018","unstructured":"Duo, S., Wyk, H., Zhang, Y.: A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem. J. Comput. Phys. 355, 233\u2013252 (2018)","journal-title":"J. Comput. Phys."},{"key":"1655_CR17","doi-asserted-by":"publisher","first-page":"639","DOI":"10.1016\/j.cma.2019.06.016","volume":"355","author":"S Duo","year":"2019","unstructured":"Duo, S., Zhang, Y.: Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications. Comput. Methods Appl. Mech. Eng. 355, 639\u2013662 (2019)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1655_CR18","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1007\/s00466-020-01853-x","volume":"66","author":"Z Lin","year":"2020","unstructured":"Lin, Z., Wang, D., Qi, D., Deng, L.: A Petrov-Galerkin finite element-meshfree formulation for multi-dimensional fractional diffusion equations. Comput. Mech. 66, 323\u2013350 (2020)","journal-title":"Comput. Mech."},{"key":"1655_CR19","doi-asserted-by":"publisher","first-page":"84","DOI":"10.1016\/j.cpc.2017.09.019","volume":"22","author":"S Ji","year":"2018","unstructured":"Ji, S., Yang, Y., Pang, G., Antoine, X.: Accurate artificial boundary conditions for the semi-discretized linear Schr\u00f6dinger and heat equations on rectangular domains. Comput. Phys. Commun. 22, 84\u201393 (2018)","journal-title":"Comput. Phys. Commun."},{"key":"1655_CR20","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1007\/s00466-011-0615-4","volume":"48","author":"G Pang","year":"2011","unstructured":"Pang, G., Tang, S.: Time history kernel functions for square lattice. Comput. Mech. 48, 699\u2013711 (2011)","journal-title":"Comput. Mech."},{"key":"1655_CR21","first-page":"294","volume":"43","author":"A Arnold","year":"2002","unstructured":"Arnold, A., Ehrhardt, M., Sofronov, I.: Discrete transparent boundary conditions for the Schr\u00f6dinger equation: fast calculation, approximation, and stability. Math. Comput. Model. 43, 294\u2013309 (2002)","journal-title":"Math. Comput. Model."},{"key":"1655_CR22","doi-asserted-by":"publisher","first-page":"1430","DOI":"10.1137\/16M1102896","volume":"40","author":"Q Du","year":"2018","unstructured":"Du, Q., Han, H., Zhang, J., Zheng, C.: Numerical solution of a two-dimensional nonlocal wave equation on unbounded domains. SIAM J. Sci. Comput. 40, 1430\u20131445 (2018)","journal-title":"SIAM J. Sci. Comput."},{"key":"1655_CR23","doi-asserted-by":"publisher","first-page":"310","DOI":"10.1016\/j.cma.2016.05.002","volume":"308","author":"X Li","year":"2016","unstructured":"Li, X., Lu, J.: Traction boundary conditions for molecular static simulations. Comput. Methods Appl. Mech. Eng. 308, 310\u2013329 (2016)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1655_CR24","doi-asserted-by":"publisher","first-page":"813","DOI":"10.1007\/s00466-010-0515-z","volume":"46","author":"X Wang","year":"2010","unstructured":"Wang, X., Tang, S.: Matching boundary conditions for diatomic chains. Comput. Mech. 46, 813\u2013826 (2010)","journal-title":"Comput. Mech."},{"key":"1655_CR25","doi-asserted-by":"publisher","first-page":"026709","DOI":"10.1103\/PhysRevE.78.026709","volume":"78","author":"J Zhang","year":"2008","unstructured":"Zhang, J., Xu, Z., Wu, X.: Unified approach to split absorbing boundary conditions for nonlinear Schrodinger equations. Phys. Rev. E 78, 026709 (2008)","journal-title":"Phys. Rev. E"},{"issue":"12","key":"1655_CR26","doi-asserted-by":"publisher","first-page":"1255","DOI":"10.1002\/nme.4426","volume":"93","author":"X Wang","year":"2013","unstructured":"Wang, X., Tang, S.: Matching boundary conditions for lattice dynamics. Int. J. Numer. Methods Eng. 93(12), 1255\u20131285 (2013)","journal-title":"Int. J. Numer. Methods Eng."},{"key":"1655_CR27","doi-asserted-by":"publisher","first-page":"760","DOI":"10.1103\/PhysRevB.81.094301","volume":"81","author":"R Jones","year":"2010","unstructured":"Jones, R., Kimmer, C.: Efficient non-reflecting boundary condition constructed via optimization of damped layers. Phys. Rev. B 81, 760\u2013762 (2010)","journal-title":"Phys. Rev. B"},{"key":"1655_CR28","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1006\/jcph.1994.1159","volume":"114","author":"J Berenger","year":"1994","unstructured":"Berenger, J.: A perfect matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185\u2013200 (1994)","journal-title":"J. Comput. Phys."},{"key":"1655_CR29","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1016\/j.wavemoti.2003.12.004","volume":"39","author":"D Givoli","year":"2004","unstructured":"Givoli, D.: High-order local non-reflecting boundary conditions: a review. Wave Motion 39, 319\u2013326 (2004)","journal-title":"Wave Motion"},{"key":"1655_CR30","doi-asserted-by":"publisher","first-page":"3322","DOI":"10.1016\/j.jcp.2007.11.040","volume":"227","author":"T Hagstrom","year":"2008","unstructured":"Hagstrom, T., Mar-Or, A., Givoli, D.: High-order local absorbing conditions for the wave equation: extensions and improvements. J. Comput. Phys. 227, 3322\u20133357 (2008)","journal-title":"J. Comput. Phys."},{"key":"1655_CR31","doi-asserted-by":"publisher","first-page":"64","DOI":"10.1137\/0731004","volume":"31","author":"R Higdon","year":"1994","unstructured":"Higdon, R.: Radiation boundary conditions for dispersive waves. SIAM J. Numer. Anal. 31, 64\u2013100 (1994)","journal-title":"SIAM J. Numer. Anal."},{"key":"1655_CR32","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1016\/j.jcp.2019.04.025","volume":"391","author":"X Antoine","year":"2019","unstructured":"Antoine, X., Lorin, E.: Towards perfectly matched layers for time-dependent space fractional PDEs. J. Comput. Phys. 391, 59\u201390 (2019)","journal-title":"J. Comput. Phys."},{"key":"1655_CR33","doi-asserted-by":"publisher","first-page":"112623","DOI":"10.1016\/j.cma.2019.112623","volume":"358","author":"G Kissasa","year":"2020","unstructured":"Kissasa, G., Yang, Y., Hwuang, E., Witscheyc, W., Detred, J., Perdikaris, P.: Machine learning in cardiovascular flows modeling: predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks. Comput. Methods Appl. Mech. Eng. 358, 112623 (2020)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1655_CR34","doi-asserted-by":"publisher","first-page":"13","DOI":"10.3901\/JME.2006.04.013","volume":"42","author":"Q Zhang","year":"2020","unstructured":"Zhang, Q., Qiao, D., Tang, S.: Designing artificial boundary conditions for atomic chains by machine learning. Mech. Eng. 42, 13\u201316 (2020). (in Chinese)","journal-title":"Mech. Eng."},{"key":"1655_CR35","unstructured":"Erd\u00e9lyi, A.: Higher Transcendental Functions, Volume 1, Chapter 1, Nature. Mcgraw-hill book company, inc. New York (1955)"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01655-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-021-01655-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01655-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,10,24]],"date-time":"2021-10-24T05:04:38Z","timestamp":1635051878000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-021-01655-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,9]]},"references-count":35,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,11]]}},"alternative-id":["1655"],"URL":"https:\/\/doi.org\/10.1007\/s10915-021-01655-0","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"type":"print","value":"0885-7474"},{"type":"electronic","value":"1573-7691"}],"subject":[],"published":{"date-parts":[[2021,10,9]]},"assertion":[{"value":"23 October 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 September 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 September 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 October 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"42"}}