{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T18:21:53Z","timestamp":1772821313291,"version":"3.50.1"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,1,25]],"date-time":"2024-01-25T00:00:00Z","timestamp":1706140800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,1,25]],"date-time":"2024-01-25T00:00:00Z","timestamp":1706140800000},"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":[[2024,2]]},"DOI":"10.1007\/s40314-023-02573-7","type":"journal-article","created":{"date-parts":[[2024,1,25]],"date-time":"2024-01-25T20:02:06Z","timestamp":1706212926000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["An efficient iterative method for multi-order nonlinear fractional differential equations based on the integrated Bernoulli polynomials"],"prefix":"10.1007","volume":"43","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0284-084X","authenticated-orcid":false,"given":"Babak","family":"Azarnavid","sequence":"first","affiliation":[]},{"given":"Mahdi","family":"Emamjomeh","sequence":"additional","affiliation":[]},{"given":"Mohammad","family":"Nabati","sequence":"additional","affiliation":[]},{"given":"Abdollah","family":"Dinmohammadi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,1,25]]},"reference":[{"issue":"1","key":"2573_CR1","doi-asserted-by":"publisher","first-page":"74","DOI":"10.3390\/fractalfract7010074","volume":"7","author":"WM Abd-Elhameed","year":"2023","unstructured":"Abd-Elhameed WM, Alsuyuti MM (2023) Numerical treatment of multi-term fractional differential equations via new kind of generalized chebyshev polynomials. Fractal Fract 7(1):74","journal-title":"Fractal Fract"},{"key":"2573_CR2","doi-asserted-by":"publisher","first-page":"70","DOI":"10.1016\/j.cam.2017.08.004","volume":"330","author":"M Alipour","year":"2018","unstructured":"Alipour M, Agahi H (2018) New computational techniques for solving nonlinear problems using g-fractional differential operator. J Comput Appl Math 330:70\u201374","journal-title":"J Comput Appl Math"},{"key":"2573_CR3","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2022.112394","volume":"161","author":"M Alqhtani","year":"2022","unstructured":"Alqhtani M, Owolabi KM, Saad KM, Pindza E (2022) Efficient numerical techniques for computing the Riesz fractional-order reaction-diffusion models arising in biology. Chaos, Solitons & Fractals 161:112394","journal-title":"Chaos, Solitons & Fractals"},{"issue":"7","key":"2573_CR4","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1007\/s40314-022-02041-8","volume":"41","author":"R Amin","year":"2022","unstructured":"Amin R, Shah K, Mlaiki N, Y\u00fczba\u015fi \u015e, Abdeljawad T, Hussain A (2022) Existence and numerical analysis using Haar wavelet for fourth-order multi-term fractional differential equations. Comput Appl Math 41(7):329","journal-title":"Comput Appl Math"},{"issue":"1","key":"2573_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40314-022-02148-y","volume":"42","author":"B Azarnavid","year":"2023","unstructured":"Azarnavid B (2023) The Bernoulli polynomials reproducing kernel method for nonlinear Volterra integro-differential equations of fractional order with convergence analysis. Comput Appl Math 42(1):1\u201317","journal-title":"Comput Appl Math"},{"key":"2573_CR6","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2022.112159","volume":"159","author":"B Azarnavid","year":"2022","unstructured":"Azarnavid B, Emamjomeh M, Nabati M (2022) A shooting like method based on the shifted Chebyshev polynomials for solving nonlinear fractional multi-point boundary value problem. Chaos, Solitons & Fractals 159:112159","journal-title":"Chaos, Solitons & Fractals"},{"key":"2573_CR7","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1016\/j.matcom.2022.04.015","volume":"200","author":"R Bakhshandeh-Chamazkoti","year":"2022","unstructured":"Bakhshandeh-Chamazkoti R, Alipour M (2022) Lie symmetries reduction and spectral methods on the fractional two-dimensional heat equation. Math Comput Simul 200:97\u2013107","journal-title":"Math Comput Simul"},{"issue":"2","key":"2573_CR8","first-page":"482","volume":"219","author":"AH Bhrawy","year":"2012","unstructured":"Bhrawy AH, Tohidi E, Soleymani F (2012) A new Bernoulli matrix method for solving high-order linear and nonlinear Fredholm integro-differential equations with piecewise intervals. Appl Math Comput 219(2):482\u2013497","journal-title":"Appl Math Comput"},{"key":"2573_CR9","doi-asserted-by":"publisher","first-page":"1023","DOI":"10.1007\/s11071-015-2087-0","volume":"81","author":"AH Bhrawy","year":"2015","unstructured":"Bhrawy AH, Taha TM, Machado JAT (2015) A review of operational matrices and spectral techniques for fractional calculus. Nonlinear Dyn 81:1023\u20131052","journal-title":"Nonlinear Dyn"},{"issue":"3","key":"2573_CR10","first-page":"1","volume":"50","author":"X Chen","year":"2020","unstructured":"Chen X, Lei S, Wang L (2020) Bernoulli Polynomials Collocation Method for Multi-term Fractional Differential Equations. IAENG Int J Appl Math 50(3):1\u20138","journal-title":"IAENG Int J Appl Math"},{"key":"2573_CR11","doi-asserted-by":"publisher","DOI":"10.1007\/978-981-16-8802-7","volume-title":"Fractional derivative modeling in mechanics and engineering","author":"W Chen","year":"2022","unstructured":"Chen W, Sun H, Li X (2022) Fractional derivative modeling in mechanics and engineering. Springer, New York"},{"key":"2573_CR12","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1007\/s12190-019-01307-5","volume":"63","author":"E Dadkhah","year":"2020","unstructured":"Dadkhah E, Shiri B, Ghaffarzadeh H, Baleanu D (2020) Visco-elastic dampers in structural buildings and numerical solution with spline collocation methods. J Appl Math Comput 63:29\u201357","journal-title":"J Appl Math Comput"},{"key":"2573_CR13","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-14574-2","volume-title":"The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type","author":"K Diethelm","year":"2010","unstructured":"Diethelm K (2010) The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type, vol 2004. Springer, Berlin"},{"key":"2573_CR14","first-page":"621","volume":"154","author":"K Diethelm","year":"2004","unstructured":"Diethelm K, Ford NJ (2004) Multi-order fractional differential equations and their numerical solutions. Appl Math Comput 154:621\u2013640","journal-title":"Appl Math Comput"},{"issue":"1","key":"2573_CR15","doi-asserted-by":"publisher","first-page":"388","DOI":"10.1002\/mma.8517","volume":"46","author":"T Eftekhari","year":"2023","unstructured":"Eftekhari T, Rashidinia J (2023) A new operational vector approach for time-fractional subdiffusion equations of distributed order based on hybrid functions. Mathematical Methods in the Applied Sciences 46(1):388\u2013407","journal-title":"Mathematical Methods in the Applied Sciences"},{"issue":"8","key":"2573_CR16","doi-asserted-by":"publisher","first-page":"788","DOI":"10.1016\/j.apnum.2010.02.007","volume":"60","author":"AMA El-Sayed","year":"2010","unstructured":"El-Sayed AMA, El-Kalla IL, Ziada EAA (2010) Analytical and numerical solutions of multi-term nonlinear fractional orders differential equations. Appl Numer Math 60(8):788\u2013797","journal-title":"Appl Numer Math"},{"key":"2573_CR17","doi-asserted-by":"crossref","unstructured":"Erturk VS, Alomari AK, Kumar P, Murillo-Arcila M (2022) Analytic solution for the strongly nonlinear multi-order fractional version of a BVP occurring in chemical reactor theory, Discrete Dynamics in Nature and Society 2022","DOI":"10.1155\/2022\/8655340"},{"key":"2573_CR18","doi-asserted-by":"crossref","unstructured":"Gil A, Segura J, Temme NM (2007) Numerical methods for special functions. Society for Industrial and Applied Mathematics","DOI":"10.1137\/1.9780898717822"},{"issue":"1","key":"2573_CR19","doi-asserted-by":"publisher","first-page":"44","DOI":"10.1080\/00029890.1972.11992980","volume":"79","author":"WH Gould","year":"1972","unstructured":"Gould WH (1972) Explicit formulas for Bernoulli numbers. Am Math Mon 79(1):44\u201351","journal-title":"Am Math Mon"},{"issue":"8","key":"2573_CR20","doi-asserted-by":"publisher","first-page":"1260","DOI":"10.3390\/sym12081260","volume":"12","author":"M Izadi","year":"2020","unstructured":"Izadi M, Cattani C (2020) Generalized Bessel polynomial for multi-order fractional differential equations. Symmetry 12(8):1260","journal-title":"Symmetry"},{"key":"2573_CR21","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2022.106620","volume":"114","author":"AS Joujehi","year":"2022","unstructured":"Joujehi AS, Derakhshan MH, Marasi HR (2022) An efficient hybrid numerical method for multi-term time fractional partial differential equations in fluid mechanics with convergence and error analysis. Commun Nonlinear Sci Numer Simul 114:106620","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"2573_CR22","doi-asserted-by":"crossref","unstructured":"Khan NA, Ibrahim\u00a0Khalaf O, Andres Tavera Romero C, Sulaiman M., Bakar M.A. (2022) Application of intelligent paradigm through neural networks for numerical solution of multiorder fractional differential equations, Computational Intelligence and Neuroscience 2022","DOI":"10.1155\/2022\/2710576"},{"issue":"16","key":"2573_CR23","doi-asserted-by":"publisher","first-page":"10116","DOI":"10.1002\/mma.8358","volume":"45","author":"HR Khodabandelo","year":"2022","unstructured":"Khodabandelo HR, Shivanian E, Abbasbandy S (2022) A novel shifted Jacobi operational matrix method for nonlinear multi-terms delay differential equations of fractional variable-order with periodic and anti-periodic conditions. Mathematical Methods in the Applied Sciences 45(16):10116\u201310135","journal-title":"Mathematical Methods in the Applied Sciences"},{"issue":"10","key":"2573_CR24","doi-asserted-by":"publisher","first-page":"905","DOI":"10.1080\/00029890.1988.11972114","volume":"95","author":"DH Lehmer","year":"1988","unstructured":"Lehmer DH (1988) A new approach to Bernoulli polynomials. Am Math Mon 95(10):905\u2013911","journal-title":"Am Math Mon"},{"issue":"7","key":"2573_CR25","doi-asserted-by":"publisher","first-page":"1505","DOI":"10.1080\/00207160.2021.1986214","volume":"99","author":"AM Nagy","year":"2022","unstructured":"Nagy AM (2022) Numerical solutions for nonlinear multi-term fractional differential equations via Dickson operational matrix. Int J Comput Math 99(7):1505\u20131515","journal-title":"Int J Comput Math"},{"key":"2573_CR26","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1016\/j.apnum.2015.08.011","volume":"99","author":"A Napoli","year":"2016","unstructured":"Napoli A (2016) Solutions of linear second order initial value problems by using Bernoulli polynomials. Appl Numer Math 99:109\u2013120","journal-title":"Appl Numer Math"},{"key":"2573_CR27","doi-asserted-by":"publisher","first-page":"1003","DOI":"10.1016\/j.cjph.2021.08.018","volume":"77","author":"Z Odibat","year":"2022","unstructured":"Odibat Z, Baleanu D (2022) Nonlinear dynamics and chaos in fractional differential equations with a new generalized Caputo fractional derivative. Chin J Phys 77:1003\u20131014","journal-title":"Chin J Phys"},{"key":"2573_CR28","doi-asserted-by":"crossref","unstructured":"Postavaru O (2022) Generalized fractional-order hybrid of block-pulse functions and Bernoulli polynomials approach for solving fractional delay differential equations, Soft Computing: 1-13","DOI":"10.1007\/s00500-022-07693-6"},{"key":"2573_CR29","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1016\/j.matcom.2021.12.001","volume":"194","author":"O Postavaru","year":"2022","unstructured":"Postavaru O, Toma A (2022) A numerical approach based on fractional-order hybrid functions of block-pulse and Bernoulli polynomials for numerical solutions of fractional optimal control problems. Math Comput Simul 194:269\u2013284","journal-title":"Math Comput Simul"},{"key":"2573_CR30","doi-asserted-by":"crossref","unstructured":"B. Prakash, A. Setia and S. Bose, Numerical solution for a system of fractional differential equations with applications in fluid dynamics and chemical engineering, International Journal of Chemical Reactor Engineering 15(5) (2017)","DOI":"10.1515\/ijcre-2017-0093"},{"key":"2573_CR31","unstructured":"Rezabeyk S, Abbasbandy S, Shivanian E, Derili H (2023) A new approach to solve weakly singular fractional-order delay integro-differential equations using operational matrices, Journal of Mathematical Modeling"},{"key":"2573_CR32","doi-asserted-by":"publisher","first-page":"348","DOI":"10.1016\/j.matcom.2021.12.009","volume":"194","author":"J Shi","year":"2022","unstructured":"Shi J, He K, Fang H (2022) Chaos, Hopf bifurcation and control of a fractional-order delay financial system. Math Comput Simul 194:348\u2013364","journal-title":"Math Comput Simul"},{"key":"2573_CR33","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1007\/s11071-015-2326-4","volume":"83","author":"SC Shiralashetti","year":"2016","unstructured":"Shiralashetti SC, Deshi AB (2016) An efficient Haar wavelet collocation method for the numerical solution of multi-term fractional differential equations. Nonlinear Dyn 83:293\u2013303","journal-title":"Nonlinear Dyn"},{"issue":"1","key":"2573_CR34","doi-asserted-by":"publisher","first-page":"608","DOI":"10.1080\/16583655.2022.2089831","volume":"16","author":"I Talib","year":"2022","unstructured":"Talib I, Raza A, Atangana A, Riaz MB (2022) Numerical study of multi-order fractional differential equations with constant and variable coefficients. Journal of Taibah University for Science 16(1):608\u2013620","journal-title":"Journal of Taibah University for Science"},{"key":"2573_CR35","unstructured":"Verma P, Kumar M (2022) Analytical solution with existence and uniqueness conditions of non-linear initial value multi-order fractional differential equations using Caputo derivative, Engineering with Computers: 1-18"},{"key":"2573_CR36","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1007\/s40096-021-00379-4","volume":"15","author":"B Zhang","year":"2021","unstructured":"Zhang B, Tang Y, Zhang X (2021) Numerical solution of fractional differential equations using hybrid Bernoulli polynomials and block pulse functions. Mathematical Sciences 15:293\u2013304","journal-title":"Mathematical Sciences"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-023-02573-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-023-02573-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-023-02573-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,11]],"date-time":"2024-02-11T11:08:07Z","timestamp":1707649687000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-023-02573-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,25]]},"references-count":36,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,2]]}},"alternative-id":["2573"],"URL":"https:\/\/doi.org\/10.1007\/s40314-023-02573-7","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,1,25]]},"assertion":[{"value":"3 September 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 December 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 December 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 January 2024","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"None.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"68"}}