{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T05:25:17Z","timestamp":1780550717097,"version":"3.54.1"},"reference-count":30,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2021,1,7]],"date-time":"2021-01-07T00:00:00Z","timestamp":1609977600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,1,7]],"date-time":"2021-01-07T00:00:00Z","timestamp":1609977600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2021,2]]},"DOI":"10.1007\/s10915-020-01374-y","type":"journal-article","created":{"date-parts":[[2021,1,7]],"date-time":"2021-01-07T20:03:29Z","timestamp":1610049809000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Generalized Exponential Time Differencing Schemes for Stiff Fractional Systems with Nonsmooth Source Term"],"prefix":"10.1007","volume":"86","author":[{"given":"Ibrahim O.","family":"Sarumi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9896-1333","authenticated-orcid":false,"given":"Khaled M.","family":"Furati","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Abdul Q. M.","family":"Khaliq","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kassem","family":"Mustapha","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,1,7]]},"reference":[{"issue":"2","key":"1374_CR1","doi-asserted-by":"publisher","first-page":"362","DOI":"10.1006\/jcph.1998.6093","volume":"147","author":"G Beylkin","year":"1998","unstructured":"Beylkin, G., Keiser, J.M., Vozovoi, L.: A new class of time discretization schemes for the solution of nonlinear PDEs. J. Comput. Phys. 147(2), 362\u2013387 (1998)","journal-title":"J. Comput. Phys."},{"issue":"5","key":"1374_CR2","doi-asserted-by":"publisher","first-page":"A3070","DOI":"10.1137\/16M1070323","volume":"38","author":"W Cao","year":"2016","unstructured":"Cao, W., Zeng, F., Zhang, Z., Karniadakis, G.E.: Implicit\u2013explicit difference schemes for nonlinear fractional differential equations with nonsmooth solutions. SIAM J. Sci. Comput. 38(5), A3070\u2013A3093 (2016)","journal-title":"SIAM J. Sci. Comput."},{"key":"1374_CR3","first-page":"1","volume":"40","author":"AA Chikriy","year":"2008","unstructured":"Chikriy, A.A., Matichin, I.I.: Presentation of solutions of linear systems with fractional derivatives in the sense of Riemann\u2013Liouville, Caputo, and Miller\u2013Ross. J. Autom. Inform. Sci. 40, 1\u201311 (2008)","journal-title":"J. Autom. Inform. Sci."},{"issue":"2","key":"1374_CR4","doi-asserted-by":"publisher","first-page":"430","DOI":"10.1006\/jcph.2002.6995","volume":"176","author":"SM Cox","year":"2002","unstructured":"Cox, S.M., Matthews, P.C.: Exponential time differencing for stiff systems. J. Comput. Phys. 176(2), 430\u2013455 (2002)","journal-title":"J. Comput. Phys."},{"issue":"5","key":"1374_CR5","doi-asserted-by":"publisher","first-page":"608","DOI":"10.1016\/j.camwa.2013.03.012","volume":"66","author":"M Di Paola","year":"2013","unstructured":"Di Paola, M., Pinnola, F.P., Zingales, M.: Fractional differential equations and related exact mechanical models. Comput. Math. Appl. 66(5), 608\u2013620 (2013)","journal-title":"Comput. Math. Appl."},{"key":"1374_CR6","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-14574-2","volume-title":"The Analysis of Fractional Differential Equations","author":"K Diethelm","year":"2010","unstructured":"Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)"},{"issue":"6\u20137","key":"1374_CR7","doi-asserted-by":"publisher","first-page":"1240","DOI":"10.1080\/00207160.2017.1404037","volume":"95","author":"KM Furati","year":"2018","unstructured":"Furati, K.M., Yousuf, M., Khaliq, A.: Fourth-order methods for space fractional reaction\u2013diffusion equations with non-smooth data. Int. J. Comput. Math. 95(6\u20137), 1240\u20131256 (2018)","journal-title":"Int. J. Comput. Math."},{"key":"1374_CR8","doi-asserted-by":"publisher","first-page":"876","DOI":"10.1016\/j.camwa.2011.04.054","volume":"62","author":"R Garappa","year":"2011","unstructured":"Garappa, R., Popolizio, M.: Generalized exponential time differencing methods for fractional order problems. Comput. Math. Appl. 62, 876\u2013890 (2011)","journal-title":"Comput. Math. Appl."},{"issue":"8","key":"1374_CR9","doi-asserted-by":"publisher","first-page":"1915","DOI":"10.1140\/epjst\/e2013-01973-1","volume":"222","author":"R Garrappa","year":"2013","unstructured":"Garrappa, R.: Exponential integrators for time-fractional partial differential equations. Eur. Phys. J. Spec. Top. 222(8), 1915\u20131927 (2013)","journal-title":"Eur. Phys. J. Spec. Top."},{"issue":"3","key":"1374_CR10","doi-asserted-by":"publisher","first-page":"1350","DOI":"10.1137\/140971191","volume":"53","author":"R Garrappa","year":"2015","unstructured":"Garrappa, R.: Numerical evaluation of two and three parameter Mittag\u2013Leffler functions. SIAM J. Numer. Anal. 53(3), 1350\u20131369 (2015)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"1374_CR11","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1007\/s10915-018-0699-5","volume":"77","author":"R Garrappa","year":"2018","unstructured":"Garrappa, R., Popolizio, M.: Computing the matrix Mittag\u2013Leffler function with applications to fractional calculus. J. Sci. Comput. 77(1), 129\u2013153 (2018)","journal-title":"J. Sci. Comput."},{"issue":"5","key":"1374_CR12","doi-asserted-by":"publisher","first-page":"977","DOI":"10.1016\/j.camwa.2016.11.028","volume":"74","author":"R Garrappa","year":"2017","unstructured":"Garrappa, R., Moret, I., Popolizio, M.: On the time-fractional Schr\u00f6dinger equation: theoretical analysis and numerical solution by matrix Mittag\u2013Leffler functions. Comput. Math. Appl. 74(5), 977\u2013992 (2017)","journal-title":"Comput. Math. Appl."},{"issue":"4","key":"1374_CR13","first-page":"491","volume":"5","author":"R Gorenflo","year":"2002","unstructured":"Gorenflo, R., Loutchko, J., Luchko, Y.: Computation of the Mittag\u2013Leffler function $$E_{\\alpha,\\beta }(z)$$ and its derivative. Fract. Cal. Appl. Anal. 5(4), 491\u2013518 (2002)","journal-title":"Fract. Cal. Appl. Anal."},{"key":"1374_CR14","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1017\/S0962492910000048","volume":"19","author":"M Hochbruck","year":"2010","unstructured":"Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209\u2013286 (2010)","journal-title":"Acta Numer."},{"key":"1374_CR15","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1016\/j.cnsns.2017.04.001","volume":"51","author":"C Ionescu","year":"2017","unstructured":"Ionescu, C., Lopes, A., Copot, D., Machado, J., Bates, J.: The role of fractional calculus in modeling biological phenomena: a review. Commun. Nonlinear Sci. Numer. Simul. 51, 141\u2013159 (2017)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"issue":"8","key":"1374_CR16","doi-asserted-by":"publisher","first-page":"E403","DOI":"10.1111\/j.1750-3841.2008.00932.x","volume":"73","author":"A Kaur","year":"2008","unstructured":"Kaur, A., Takhar, P.S., Smith, D., Mann, J., Brashears, M.: Fractional differential equations based modeling of microbial survival and growth curves: model development and experimental validation. J. Food Sci. 73(8), E403\u2013E414 (2008)","journal-title":"J. Food Sci."},{"key":"1374_CR17","volume-title":"Theory and Applications of Fractional Differential Equations","author":"A Kilbas","year":"2006","unstructured":"Kilbas, A., Srivastava, H., Trujullo, J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)"},{"issue":"3","key":"1374_CR18","doi-asserted-by":"publisher","first-page":"481","DOI":"10.1007\/s00211-006-0045-y","volume":"105","author":"W McLean","year":"2007","unstructured":"McLean, W., Mustapha, K.: A second-order accurate numerical method for a fractional wave equation. Numer. Math. 105(3), 481\u2013510 (2007)","journal-title":"Numer. Math."},{"key":"1374_CR19","unstructured":"Minchev, B., Wright, W.: A review of exponential integrators for first order semi-linear problems. Technical Report 2\/05, Department of Mathematics, NTNU (2005)"},{"key":"1374_CR20","doi-asserted-by":"crossref","unstructured":"Prakash, B., Setia, A., Bose, S.: Numerical solution for a system of fractional differential equations with applications in fluid dynamics and chemical engineering. Int. J. Chem. React. Eng. 15(5), 20170093 (2017)","DOI":"10.1515\/ijcre-2017-0093"},{"key":"1374_CR21","doi-asserted-by":"crossref","first-page":"733","DOI":"10.1016\/j.amc.2018.06.037","volume":"338","author":"A Sadeghi","year":"2018","unstructured":"Sadeghi, A., Cardoso, J.R.: Some notes on properties of the matrix Mittag\u2013Leffler function. Appl. Math. Comput. 338, 733\u2013738 (2018)","journal-title":"Appl. Math. Comput."},{"key":"1374_CR22","doi-asserted-by":"crossref","unstructured":"Sarumi, I.O., Furati, K.M., Khaliq, A.Q.M.: Highly accurate global Pad\u00e9 approximations of generalized Mittag\u2013Leffler function and its inverse. J. Sci. Comput. 82, 46 (2020)","DOI":"10.1007\/s10915-020-01150-y"},{"issue":"4","key":"1374_CR23","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1299\/jsmec.42.825","volume":"42","author":"N Shimizu","year":"1999","unstructured":"Shimizu, N., Zhang, W.: Fractional calculus approach to dynamic problems of viscoelastic materials. JSME Int. J. Ser. C 42(4), 825\u2013837 (1999)","journal-title":"JSME Int. J. Ser. C"},{"issue":"1","key":"1374_CR24","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1007\/s10928-017-9547-8","volume":"45","author":"P Sopasakis","year":"2018","unstructured":"Sopasakis, P., Sarimveis, H., Macheras, P., Dokoumetzidis, A.: Fractional calculus in pharmacokinetics. J. Pharmacokinet. Pharmacodyn. 45(1), 107\u2013125 (2018)","journal-title":"J. Pharmacokinet. Pharmacodyn."},{"key":"1374_CR25","doi-asserted-by":"crossref","unstructured":"Stynes, M.: Singularities. In: Handbook of Fractional Calculus with Applications: Numerical Methods, vol. 3. De Gruyter, Berlin, pp. 287\u2013305 (2019)","DOI":"10.1515\/9783110571684-011"},{"issue":"6","key":"1374_CR26","doi-asserted-by":"crossref","first-page":"1554","DOI":"10.1515\/fca-2016-0080","volume":"19","author":"M Stynes","year":"2016","unstructured":"Stynes, M.: Too much regularity may force too much uniqueness. Fract. Cal. Appl. Anal. 19(6), 1554\u20131562 (2016)","journal-title":"Fract. Cal. Appl. Anal."},{"key":"1374_CR27","first-page":"125","volume":"5","author":"PJ Torvik","year":"1987","unstructured":"Torvik, P.J., Bagley, D.: Fractional derivatives in the description of damping materials and phenomena. Role Damp. Vib. Noise Control 5, 125\u2013135 (1987)","journal-title":"Role Damp. Vib. Noise Control"},{"key":"1374_CR28","unstructured":"Veber, V.K.: Linear equations with fractional derivatives and constant coefficients in spaces of generalized functions. In: Studies in integro-differential equations, number\u00a018, pp. 306\u2013312. Frunze, Ilim (1985) (in Russian)"},{"key":"1374_CR29","unstructured":"Veber, V.K.: On the general theory of linear systems with fractional derivatives. In: Studies in Integro-Differential Equations, number\u00a018, pp. 301\u2013305. Frunze, Ilim (1985) (in Russian)"},{"issue":"1","key":"1374_CR30","doi-asserted-by":"publisher","first-page":"A55","DOI":"10.1137\/14096390X","volume":"37","author":"F Zeng","year":"2015","unstructured":"Zeng, F., Li, C., Liu, F., Turner, I.: Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy. SIAM J. Sci. Comput. 37(1), A55\u2013A78 (2015)","journal-title":"SIAM J. Sci. Comput."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-020-01374-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-020-01374-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-020-01374-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,10]],"date-time":"2022-12-10T23:02:43Z","timestamp":1670713363000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-020-01374-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,7]]},"references-count":30,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,2]]}},"alternative-id":["1374"],"URL":"https:\/\/doi.org\/10.1007\/s10915-020-01374-y","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,7]]},"assertion":[{"value":"21 May 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 November 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 November 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 January 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"23"}}