{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T00:35:52Z","timestamp":1768782952536,"version":"3.49.0"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2020,2,11]],"date-time":"2020-02-11T00:00:00Z","timestamp":1581379200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,2,11]],"date-time":"2020-02-11T00:00:00Z","timestamp":1581379200000},"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":["Comp. Appl. Math."],"published-print":{"date-parts":[[2020,5]]},"DOI":"10.1007\/s40314-020-1068-1","type":"journal-article","created":{"date-parts":[[2020,2,11]],"date-time":"2020-02-11T11:03:01Z","timestamp":1581418981000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Structure preserving computational technique for fractional order Schnakenberg model"],"prefix":"10.1007","volume":"39","author":[{"given":"Zafar","family":"Iqbal","sequence":"first","affiliation":[]},{"given":"Nauman","family":"Ahmed","sequence":"additional","affiliation":[]},{"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[]},{"given":"Muhammad","family":"Rafiq","sequence":"additional","affiliation":[]},{"given":"Muhammad Sajid","family":"Iqbal","sequence":"additional","affiliation":[]},{"given":"Muhammad Aziz-ur","family":"Rehman","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,2,11]]},"reference":[{"key":"1068_CR1","first-page":"110","volume":"64","author":"N Ahmed","year":"2019","unstructured":"Ahmed N, Rafiq M, Baleanu D, Rehman MA (2019) Spatio-temporal numerical modeling of auto-catalytic Brusselator model. Rom J Phys 64:110","journal-title":"Rom J Phys"},{"key":"1068_CR2","doi-asserted-by":"publisher","first-page":"015205","DOI":"10.1063\/1.5070093","volume":"9","author":"N Ahmed","year":"2019","unstructured":"Ahmed N, Rafiq M, Rehman MA, Iqbal MS, Ali M (2019) Numerical modelling of three dimensional Brusselator reaction diffusion system. AIP Adv 9:015205","journal-title":"AIP Adv"},{"key":"1068_CR3","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1515\/math-2019-0027","volume":"17","author":"N Ahmed","year":"2019","unstructured":"Ahmed N, Tahira SS, Rafiq M, Rehman MA, Ali M, Ahmad MO (2019) Positivity preserving operator splitting nonstandard finite difference methods for SEIR reaction diffusion model. Open Math 17:313\u2013330","journal-title":"Open Math"},{"key":"1068_CR4","doi-asserted-by":"publisher","first-page":"358","DOI":"10.2298\/AADM170428002A","volume":"11","author":"R Almeida","year":"2017","unstructured":"Almeida R (2017) What is the best fractional derivative to fil data? Appl Anal Discrete Math 11:358\u2013368","journal-title":"Appl Anal Discrete Math"},{"key":"1068_CR5","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1016\/j.apm.2016.10.054","volume":"43","author":"I Ameen","year":"2017","unstructured":"Ameen I, Novati P (2017) The solution of fractional order epidemic model by implicit Adams methods. Appl Math Model 43:78\u201384","journal-title":"Appl Math Model"},{"key":"1068_CR6","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1016\/j.matcom.2015.09.001","volume":"21","author":"AJ Arenas","year":"2016","unstructured":"Arenas AJ, Gonzalez G, Chen Charpentier B (2016) Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order. Math Comput Simul 21:48\u201363","journal-title":"Math Comput Simul"},{"issue":"9","key":"1068_CR7","doi-asserted-by":"publisher","first-page":"830","DOI":"10.3390\/math7090830","volume":"7","author":"D Baleanu","year":"2019","unstructured":"Baleanu D, Fernandez A (2019) On fractional operators and their classification. Mathematics 7(9):830","journal-title":"Mathematics"},{"key":"1068_CR8","doi-asserted-by":"publisher","DOI":"10.1142\/9413","volume-title":"Asymptotic integration and stability: for ordinary, functional and discrete differential equations of fractional order. Series on complexity, nonlinearity and Chaos","author":"D Baleanu","year":"2015","unstructured":"Baleanu D, Mustafa OG (2015) Asymptotic integration and stability: for ordinary, functional and discrete differential equations of fractional order. Series on complexity, nonlinearity and Chaos. World Scientific Publishing Company, Singapore"},{"key":"1068_CR9","volume-title":"New trends in nanotechnology and fractional calculus applications","author":"D Baleanu","year":"2009","unstructured":"Baleanu D, Machado JAT, Guven\u00e7 ZB (2009) New trends in nanotechnology and fractional calculus applications. Springer, Dordrecht"},{"key":"1068_CR10","volume-title":"Fractional dynamics and control","author":"D Baleanu","year":"2011","unstructured":"Baleanu D, Machado JAT, Luo ACJ (2011) Fractional dynamics and control. Springer, New York"},{"issue":"2","key":"1068_CR11","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1088\/0253-6102\/61\/2\/13","volume":"61","author":"D Baleanu","year":"2014","unstructured":"Baleanu D, Asad J, Petras I (2014) Fractional Bateman\u2013Feshbach Tikochinsky oscillator. Commun Theor Phys 61(2):221\u2013225","journal-title":"Commun Theor Phys"},{"issue":"1\u20133","key":"1068_CR12","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1016\/j.cnsns.2015.01.004","volume":"25","author":"D Baleanu","year":"2015","unstructured":"Baleanu D, Magin R, Bhalekar S, Daftardar-Gejji V (2015) Chaos in the fractional order nonlinear Bloch equation with delay. Commun Nonlinear Sci Numer Simul 25(1\u20133):41\u201349","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"1068_CR13","doi-asserted-by":"publisher","DOI":"10.1142\/10044","volume-title":"Fractional calculus: models and numerical methods","author":"D Baleanu","year":"2016","unstructured":"Baleanu D, Diethelm K, Scalas E, Trujillo JJ (2016) Fractional calculus: models and numerical methods. World Scientific, Singapore"},{"issue":"8","key":"1068_CR14","doi-asserted-by":"publisher","first-page":"083127","DOI":"10.1063\/1.5096159","volume":"29","author":"D Baleanu","year":"2019","unstructured":"Baleanu D, Jajarmi A, Sajjadi SS, Mozyrska D (2019) A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator. Chaos Interdiscip J Nonlinear Sci 29(8):083127","journal-title":"Chaos Interdiscip J Nonlinear Sci"},{"issue":"5","key":"1068_CR15","doi-asserted-by":"publisher","first-page":"529","DOI":"10.1111\/j.1365-246X.1967.tb02303.x","volume":"13","author":"M Caputo","year":"1967","unstructured":"Caputo M (1967) Linear model of dissipation whose Q is almost frequency independent\u2014II. Geophys J Int 13(5):529\u2013539","journal-title":"Geophys J Int"},{"key":"1068_CR16","first-page":"245","volume-title":"The role of memory in modeling social and economic cycles of extreme events. A handbook of alternative theories of public economics","author":"M Caputo","year":"2014","unstructured":"Caputo M (2014) The role of memory in modeling social and economic cycles of extreme events. A handbook of alternative theories of public economics. Edward Elgar Publishing, Cheltenham, pp 245\u2013259"},{"key":"1068_CR17","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1071\/EG03051","volume":"34","author":"G Cooper","year":"2003","unstructured":"Cooper G, Cowan D (2003) The application of fractional calculus to potential field data. Explor Geophys 34:51\u201356","journal-title":"Explor Geophys"},{"key":"1068_CR18","doi-asserted-by":"publisher","first-page":"e00631","DOI":"10.1016\/j.heliyon.2018.e00631","volume":"4","author":"U Fatima","year":"2018","unstructured":"Fatima U, Ali M, Ahmed N (2018) Numerical modeling of susceptible latent breaking-out quarantine computer virus epidemic dynamics. Heliyon 4:e00631","journal-title":"Heliyon"},{"key":"1068_CR19","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/j.amc.2009.04.060","volume":"215","author":"M Francisco Fernandez","year":"2009","unstructured":"Francisco Fernandez M (2009) On some approximate methods for nonlinear models. Appl Math Comput 215:168\u2013174","journal-title":"Appl Math Comput"},{"issue":"2","key":"1068_CR20","doi-asserted-by":"publisher","first-page":"021013","DOI":"10.1115\/1.4038444","volume":"13","author":"M Hajipour","year":"2018","unstructured":"Hajipour M, Jajarmi A, Baleanu D (2018) An efficient non-standard finite difference scheme for a class of fractional chaotic systems. J Comput Nonlinear Dynam 13(2):021013","journal-title":"J Comput Nonlinear Dynam"},{"key":"1068_CR21","doi-asserted-by":"crossref","unstructured":"Hammouch Z, Mekkaoui T, Belgacem FBM (2014) Numerical simulations for a variable order fractional Schnakenberg model. In: AIP conference proceeding, pp 1450\u20131637","DOI":"10.1063\/1.4907312"},{"key":"1068_CR22","doi-asserted-by":"crossref","unstructured":"Haq F, Shah K, ur Rahman G, Li Y, Shahzad M (2018) Computational analysis of complex population dynamical model with arbitrary order. Complexity 1\u20138","DOI":"10.1155\/2018\/8918541"},{"issue":"3","key":"1068_CR23","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1051\/mmnp\/2018074","volume":"14","author":"MA Khan","year":"2019","unstructured":"Khan MA, Hammouch Z, Baleanu D (2019) Modeling the dynamics of hepatitis E via the Caputo\u2013Fabrizio derivative. Math Model Nat Phenom 14(3):311","journal-title":"Math Model Nat Phenom"},{"key":"1068_CR24","volume-title":"Nonstandard finite difference models of differential equations","author":"RE Mickens","year":"1994","unstructured":"Mickens RE (1994) Nonstandard finite difference models of differential equations. World Scientific, Singapore"},{"key":"1068_CR25","doi-asserted-by":"publisher","unstructured":"Ongun M, Arslan D, Garrappa R (2013) Nonstandard finite difference schemes for a fractional-order Brusselator system. Adv Differ Equ 2013:102. https:\/\/doi.org\/10.1186\/1687-1847-2013-102","DOI":"10.1186\/1687-1847-2013-102"},{"issue":"1","key":"1068_CR26","doi-asserted-by":"publisher","first-page":"3","DOI":"10.3390\/fractalfract1010003","volume":"1","author":"M Ortigueira","year":"2017","unstructured":"Ortigueira M, Machado J (2017) Which derivative? Fractal Fract 1(1):3","journal-title":"Fractal Fract"},{"key":"1068_CR27","volume-title":"Fractional differential equations, mathematics in science and engineering","author":"I Podlubny","year":"1999","unstructured":"Podlubny I (1999) Fractional differential equations, mathematics in science and engineering. Academic Press, New York"},{"key":"1068_CR28","doi-asserted-by":"publisher","first-page":"902","DOI":"10.1016\/j.camwa.2011.03.054","volume":"62","author":"R Scherer","year":"2011","unstructured":"Scherer R, Kalla S, Tang Y, Huang J (2011) The Grunwald\u2013Letnikov method for fractional differential equations. Comput Math Appl 62:902\u2013917","journal-title":"Comput Math Appl"},{"key":"1068_CR29","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1016\/0022-5193(79)90042-0","volume":"81","author":"J Schnakenberg","year":"1979","unstructured":"Schnakenberg J (1979) Simple chemical reaction systems with limit cycle behaviour. J Theor Biol 81:389\u2013400","journal-title":"J Theor Biol"},{"key":"1068_CR30","doi-asserted-by":"crossref","unstructured":"Suryanto A, Darti I (2017) Stability analysis and nonstandard Gr\u00fcnwald\u2013Letnikov scheme for a fractional order predator-prey model with ratio-dependent functional response. In: AIP conference proceedings, vol 1913, p 020011","DOI":"10.1063\/1.5016645"},{"issue":"1","key":"1068_CR31","first-page":"197","volume":"10","author":"NH Sweilam","year":"2019","unstructured":"Sweilam NH, Nagy AM, Elpahri LE (2019) Nonstandard finite difference scheme for the fractional order Salmonella transmission model. J Fract Calc Appl 10(1):197\u2013212","journal-title":"J Fract Calc Appl"},{"key":"1068_CR32","doi-asserted-by":"publisher","unstructured":"Veeresha P, Prakasha DG, Baskonus HM (2019) Solving smoking epidemic model of fractional order using a modified homotopy analysis transform method. Math Sci. https:\/\/doi.org\/10.1007\/s40096-019-0284-6","DOI":"10.1007\/s40096-019-0284-6"},{"key":"1068_CR33","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1016\/j.ijnonlinmec.2017.03.017","volume":"92","author":"W Xie","year":"2017","unstructured":"Xie W, Xu J, Cai L, Lin Z (2017) Dynamic preserving method with changeable memory length of fractional-order chaotic system. Int J Nonlinear Mech 92:59\u201365","journal-title":"Int J Nonlinear Mech"},{"key":"1068_CR34","first-page":"113","volume":"69","author":"XJ Yang","year":"2017","unstructured":"Yang XJ, Gao F, Srivastava HM (2017) New rheological models within local fractional derivative. Rom Rep Phys 69:113","journal-title":"Rom Rep Phys"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-020-1068-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s40314-020-1068-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-020-1068-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,10]],"date-time":"2021-02-10T06:46:46Z","timestamp":1612939606000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s40314-020-1068-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,11]]},"references-count":34,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,5]]}},"alternative-id":["1068"],"URL":"https:\/\/doi.org\/10.1007\/s40314-020-1068-1","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,11]]},"assertion":[{"value":"18 July 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 November 2019","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 December 2019","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 February 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"61"}}