{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T19:00:09Z","timestamp":1769886009214,"version":"3.49.0"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T00:00:00Z","timestamp":1760486400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T00:00:00Z","timestamp":1760486400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2026,3]]},"DOI":"10.1007\/s40314-025-03445-y","type":"journal-article","created":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T05:25:52Z","timestamp":1760505952000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Mittag\u2013Leffler time differencing for fractional oscillation models with nonlinear force"],"prefix":"10.1007","volume":"45","author":[{"given":"Aljowhara H.","family":"Honain","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9896-1333","authenticated-orcid":false,"given":"Khaled M.","family":"Furati","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ibrahim O.","family":"Sarumi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Abdul Q. M.","family":"Khaliq","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,10,15]]},"reference":[{"issue":"3","key":"3445_CR1","first-page":"1480","volume":"7","author":"N Attia","year":"2021","unstructured":"Attia N, Seba D, Akg\u00fcl A, Nour A (2021) Solving duffing-van der pol oscillator equations of fractional order by an accurate technique. J Appl Computat Mech 7(3):1480\u20131487","journal-title":"J Appl Computat Mech"},{"issue":"2","key":"3445_CR2","doi-asserted-by":"publisher","first-page":"362","DOI":"10.1006\/jcph.1998.6093","volume":"147","author":"G Beylkin","year":"1998","unstructured":"Beylkin G, Keiser JM, Vozovoi L (1998) A new class of time discretization schemes for the solution of nonlinear pdes. J Comput Phys 147(2):362\u2013387","journal-title":"J Comput Phys"},{"key":"3445_CR3","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1090\/S0025-5718-1985-0804933-3","volume":"45","author":"H Brunner","year":"1985","unstructured":"Brunner H (1985) The numerical solution of weakly singular volterra integral equations by collocation on graded meshes. Math Comput 45:417\u2013437","journal-title":"Math Comput"},{"issue":"2","key":"3445_CR4","doi-asserted-by":"publisher","first-page":"430","DOI":"10.1006\/jcph.2002.6995","volume":"176","author":"SM Cox","year":"2002","unstructured":"Cox SM, Matthews PC (2002) Exponential time differencing for stiff systems. J Comput Phys 176(2):430\u2013455","journal-title":"J Comput Phys"},{"issue":"3","key":"3445_CR5","doi-asserted-by":"publisher","first-page":"490","DOI":"10.1016\/j.cmpb.2013.08.012","volume":"112","author":"S Das","year":"2013","unstructured":"Das S, Maharatna K (2013) Fractional dynamical model for the generation of ecg like signals from filtered coupled van-der pol oscillators. Comput Methods Programs Biomed 112(3):490\u2013507","journal-title":"Comput Methods Programs Biomed"},{"issue":"1\u20134","key":"3445_CR6","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1023\/A:1016592219341","volume":"29","author":"K Diethelm","year":"2002","unstructured":"Diethelm K, Ford NJ, Freed AD (2002) A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 29(1\u20134):3\u201322","journal-title":"Nonlinear Dyn"},{"key":"3445_CR7","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1023\/B:NUMA.0000027736.85078.be","volume":"36","author":"K Diethelm","year":"2004","unstructured":"Diethelm K, Ford NJ, Freed AD (2004) Detailed error analysis for a fractional adams method. Numerical Algorithms 36:31\u201352","journal-title":"Numerical Algorithms"},{"key":"3445_CR8","doi-asserted-by":"publisher","first-page":"96","DOI":"10.1016\/j.matcom.2013.09.012","volume":"110","author":"R Garrappa","year":"2015","unstructured":"Garrappa R (2015) Trapezoidal methods for fractional differential equations: theoretical and computational aspects. Math Comput Simul 110:96\u2013112","journal-title":"Math Comput Simul"},{"key":"3445_CR9","doi-asserted-by":"publisher","first-page":"876","DOI":"10.1016\/j.camwa.2011.04.054","volume":"62","author":"R Garrappa","year":"2011","unstructured":"Garrappa R, Popolizio M (2011) Generalized exponential time differencing methods for fractional order problems. Comput Math Appl 62:876\u2013890","journal-title":"Comput Math Appl"},{"issue":"2","key":"3445_CR10","doi-asserted-by":"publisher","first-page":"1082","DOI":"10.1137\/140977217","volume":"53","author":"L Gauckler","year":"2015","unstructured":"Gauckler L (2015) Error analysis of trigonometric integrators for semilinear wave equations. SIAM J Numer Anal 53(2):1082\u20131106","journal-title":"SIAM J Numer Anal"},{"issue":"19","key":"3445_CR11","doi-asserted-by":"publisher","first-page":"5495","DOI":"10.1088\/0305-4470\/39\/19\/S10","volume":"39","author":"V Grimm","year":"2006","unstructured":"Grimm V, Hochbruck M (2006) Error analysis of exponential integrators for oscillatory second-order differential equations. J Phys A: Math Gen 39(19):5495","journal-title":"J Phys A: Math Gen"},{"issue":"1","key":"3445_CR12","doi-asserted-by":"publisher","first-page":"805","DOI":"10.4171\/owr\/2006\/14","volume":"3","author":"E Hairer","year":"2006","unstructured":"Hairer E, Hochbruck M, Iserles A, Lubich C (2006) Geometric numerical integration. Oberwolfach Rep 3(1):805\u2013882","journal-title":"Oberwolfach Rep"},{"key":"3445_CR13","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1007\/s002110050456","volume":"83","author":"M Hochbruck","year":"1999","unstructured":"Hochbruck M, Lubich C (1999) A gautschi-type method for oscillatory second-order differential equations. Numer Math 83:403\u2013426","journal-title":"Numer Math"},{"issue":"3","key":"3445_CR14","doi-asserted-by":"publisher","first-page":"1069","DOI":"10.1137\/040611434","volume":"43","author":"M Hochbruck","year":"2005","unstructured":"Hochbruck M, Ostermann A (2005) Explicit exponential runge-kutta methods for semilinear parabolic problems. SIAM J Numer Anal 43(3):1069\u20131090","journal-title":"SIAM J Numer Anal"},{"key":"3445_CR15","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1017\/S0962492910000048","volume":"19","author":"M Hochbruck","year":"2010","unstructured":"Hochbruck M, Ostermann A (2010) Exponential integrators. Acta Numer 19:209\u2013286","journal-title":"Acta Numer"},{"issue":"1","key":"3445_CR16","doi-asserted-by":"publisher","first-page":"786","DOI":"10.1137\/080717717","volume":"47","author":"M Hochbruck","year":"2009","unstructured":"Hochbruck M, Ostermann A, Schweitzer J (2009) Exponential rosenbrock-type methods. SIAM J Numer Anal 47(1):786\u2013803","journal-title":"SIAM J Numer Anal"},{"issue":"6","key":"3445_CR17","doi-asserted-by":"publisher","first-page":"319","DOI":"10.3390\/fractalfract8060319","volume":"8","author":"AH Honain","year":"2024","unstructured":"Honain AH, Furati KM, Sarumi IO, Khaliq AQM (2024) Rational approximations for the oscillatory two-parameter mittag-leffler function. Fract Fract 8(6):319","journal-title":"Fract Fract"},{"key":"3445_CR18","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2024.116456","volume":"461","author":"AH Honain","year":"2025","unstructured":"Honain AH, Furati KM, Sarumi IO, Khaliq AQM (2025) Generalized exponential time differencing for fractional oscillation models. J Comput Appl Math 461:116456","journal-title":"J Comput Appl Math"},{"key":"3445_CR19","volume-title":"Theory and applications of fractional differential equations","author":"A Kilbas","year":"2006","unstructured":"Kilbas A, Srivastava H, Trujullo J (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam"},{"issue":"11","key":"3445_CR20","doi-asserted-by":"publisher","first-page":"2063","DOI":"10.3390\/math8112063","volume":"8","author":"V Kim","year":"2020","unstructured":"Kim V, Parovik R (2020) Mathematical model of fractional duffing oscillator with variable memory. Mathematics 8(11):2063","journal-title":"Mathematics"},{"key":"3445_CR21","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898717839","volume-title":"Finite difference methods for ordinary and partial differential equations","author":"RJ LeVeque","year":"2007","unstructured":"LeVeque RJ (2007) Finite difference methods for ordinary and partial differential equations. SIAM, Philadelpedia"},{"key":"3445_CR22","doi-asserted-by":"publisher","first-page":"614","DOI":"10.1016\/j.jcp.2016.04.039","volume":"316","author":"C Li","year":"2016","unstructured":"Li C, Yi Q, Chen A (2016) Finite difference methods with non-uniform meshes for nonlinear fractional differential equations. J Comput Phys 316:614\u2013631","journal-title":"J Comput Phys"},{"issue":"4","key":"3445_CR23","doi-asserted-by":"publisher","first-page":"1195","DOI":"10.1007\/s11075-017-0419-5","volume":"78","author":"Y Liu","year":"2018","unstructured":"Liu Y, Roberts J, Yan Y (2018) Detailed error analysis for a fractional adams method with graded meshes. Numer Algorithms 78(4):1195\u20131216","journal-title":"Numer Algorithms"},{"key":"3445_CR24","doi-asserted-by":"publisher","DOI":"10.1016\/j.rinp.2022.106108","volume":"43","author":"J Lu","year":"2022","unstructured":"Lu J, Ma L (2022) Numerical analysis of a fractional nonlinear oscillator with coordinate-dependent mass. Results Phys 43:106108","journal-title":"Results Phys"},{"key":"3445_CR25","unstructured":"Minchev BV, Wright W (2005) A review of exponential integrators for first order semi-linear problems. Technical Report\u00a02, The Norwegian University of Science and Technology"},{"issue":"1","key":"3445_CR26","doi-asserted-by":"publisher","first-page":"491","DOI":"10.1137\/120880719","volume":"51","author":"K Mustapha","year":"2013","unstructured":"Mustapha K, McLean W (2013) Superconvergence of a discontinuous galerkin method for fractional diffusion and wave equations. SIAM J Numer Anal 51(1):491\u2013515","journal-title":"SIAM J Numer Anal"},{"issue":"9","key":"3445_CR27","doi-asserted-by":"publisher","first-page":"2956","DOI":"10.1016\/j.cam.2010.12.013","volume":"235","author":"Z Odibat","year":"2011","unstructured":"Odibat Z (2011) On legendre polynomial approximation with the vim or ham for numerical treatment of nonlinear fractional differential equations. J Comput Appl Math 235(9):2956\u20132968","journal-title":"J Comput Appl Math"},{"issue":"2","key":"3445_CR28","doi-asserted-by":"publisher","first-page":"1621","DOI":"10.3934\/math.2020110","volume":"5","author":"P Pirmohabbati","year":"2020","unstructured":"Pirmohabbati P, Refahi Sheikhani A, Saberi Najafi H, Abdolahzadeh Ziabari A (2020) Numerical solution of full fractional duffing equations with cubic-quintic-heptic nonlinearities. AIMS Math 5(2):1621\u20131641","journal-title":"AIMS Math"},{"key":"3445_CR29","volume-title":"Fractional differential equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny I (1999) Fractional differential equations. Academic Press, New York"},{"issue":"2","key":"3445_CR30","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40819-018-0494-x","volume":"4","author":"P Rahimkhani","year":"2018","unstructured":"Rahimkhani P, Moeti R (2018) Numerical solution of the fractional order duffing-van der pol oscillator equation by using bernoulli wavelets collocation method. Int J Appl Comput Math 4(2):1\u201318","journal-title":"Int J Appl Comput Math"},{"issue":"2","key":"3445_CR31","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1007\/s10915-020-01374-y","volume":"86","author":"IO Sarumi","year":"2021","unstructured":"Sarumi IO, Furati KM, Khaliq AQM, Mustapha K (2021) Generalized exponential time differencing schemes for stiff fractional systems with nonsmooth source term. J Sci Comput 86(2):23","journal-title":"J Sci Comput"},{"issue":"2","key":"3445_CR32","doi-asserted-by":"publisher","first-page":"1261","DOI":"10.1007\/s11075-022-01339-2","volume":"92","author":"IO Sarumi","year":"2023","unstructured":"Sarumi IO, Furati KM, Mustapha K, Khaliq AQM (2023) Efficient high-order exponential time differencing methods for nonlinear fractional differential models. Numer Algorithms 92(2):1261\u20131288","journal-title":"Numer Algorithms"},{"issue":"5","key":"3445_CR33","doi-asserted-by":"publisher","first-page":"1950075","DOI":"10.1142\/S0219887819500750","volume":"16","author":"FA Shah","year":"2019","unstructured":"Shah FA, Abass R (2019) Solution of fractional oscillator equations using ultraspherical wavelets. Int J Geom Methods Modern Phys 16(5):1950075","journal-title":"Int J Geom Methods Modern Phys"},{"key":"3445_CR34","doi-asserted-by":"publisher","first-page":"120","DOI":"10.3389\/fphy.2020.00120","volume":"8","author":"H Singh","year":"2020","unstructured":"Singh H, Srivastava H (2020) Numerical investigation of the fractional-order li\u00e9nard and duffing equations arising in oscillating circuit theory. Front Phys 8:120","journal-title":"Front Phys"},{"key":"3445_CR35","doi-asserted-by":"publisher","DOI":"10.1016\/j.rinp.2024.107403","volume":"57","author":"H Tajadodi","year":"2024","unstructured":"Tajadodi H, Khan H, Alzabut J, G\u00f3mez-Aguilar JF (2024) An optimization method for solving fractional oscillation equation. Results Phys 57:107403","journal-title":"Results Phys"},{"key":"3445_CR36","doi-asserted-by":"publisher","first-page":"220","DOI":"10.1109\/OJCAS.2020.3029254","volume":"1","author":"MS Tavazoei","year":"2020","unstructured":"Tavazoei MS, Tavakoli-Kakhki M, Bizzarri F (2020) Nonlinear fractional-order circuits and systems: motivation, a brief overview, and some future directions. IEEE Open J Circ Syst 1:220\u2013232","journal-title":"IEEE Open J Circ Syst"},{"key":"3445_CR37","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/s13662-014-0331-4","volume":"2015","author":"J Yang","year":"2015","unstructured":"Yang J, Liu A, Liu T (2015) Forced oscillation of nonlinear fractional differential equations with damping term. Adv Differ Equ 2015:1\u20137","journal-title":"Adv Differ Equ"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03445-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03445-y","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03445-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T04:46:10Z","timestamp":1769834770000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03445-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,15]]},"references-count":37,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["3445"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03445-y","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,15]]},"assertion":[{"value":"13 January 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 July 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 September 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 October 2025","order":4,"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 that they have no conflict of interest related to this work.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"53"}}