{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T20:58:25Z","timestamp":1773435505870,"version":"3.50.1"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2020,6,15]],"date-time":"2020-06-15T00:00:00Z","timestamp":1592179200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,6,15]],"date-time":"2020-06-15T00:00:00Z","timestamp":1592179200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Engineering with Computers"],"published-print":{"date-parts":[[2022,2]]},"DOI":"10.1007\/s00366-020-01061-4","type":"journal-article","created":{"date-parts":[[2020,6,17]],"date-time":"2020-06-17T17:05:52Z","timestamp":1592413552000},"page":"661-678","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["Analytical solution with existence and uniqueness conditions of non-linear initial value multi-order fractional differential equations using Caputo derivative"],"prefix":"10.1007","volume":"38","author":[{"given":"Pratibha","family":"Verma","sequence":"first","affiliation":[]},{"given":"Manoj","family":"Kumar","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,6,15]]},"reference":[{"key":"1061_CR1","volume-title":"Fractional differential equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny I (1999) Fractional differential equations. Academic Press, New York"},{"key":"1061_CR2","first-page":"3413","volume":"54","author":"D Lokenath","year":"2003","unstructured":"Lokenath D (2003) Recent applications of fractional calculus to science and engineering. Int J Math Math Sci 54:3413\u20133442","journal-title":"Int J Math Math Sci"},{"key":"1061_CR3","doi-asserted-by":"publisher","DOI":"10.1007\/s00366-019-00861-7","author":"M Abbaszadeh","year":"2019","unstructured":"Abbaszadeh M, Dehghan M (2019) Meshless upwind local radial basis function-finite difference technique to simulate the time-fractional distributed-order advection-diffusion equation. Eng Comput. https:\/\/doi.org\/10.1007\/s00366-019-00861-7","journal-title":"Eng Comput"},{"issue":"1","key":"1061_CR4","doi-asserted-by":"publisher","first-page":"262","DOI":"10.1016\/j.nonrwa.2010.06.013","volume":"12","author":"W JinRong","year":"2011","unstructured":"JinRong W, Yong Z (2011) A class of fractional evolution equations and optimal controls. Nonlinear Anal Real World Appl 12(1):262\u2013272","journal-title":"Nonlinear Anal Real World Appl"},{"issue":"10","key":"1061_CR5","doi-asserted-by":"publisher","first-page":"3337","DOI":"10.1016\/j.na.2007.09.025","volume":"69","author":"V Lakshmikantham","year":"2008","unstructured":"Lakshmikantham V (2008) Theory of fractional functional differential equations. Nonlinear Anal Real World Appl 69(10):3337\u20133343","journal-title":"Nonlinear Anal Real World Appl"},{"issue":"5","key":"1061_CR6","doi-asserted-by":"publisher","first-page":"4465","DOI":"10.1016\/j.nonrwa.2010.05.029","volume":"11","author":"Z Yong","year":"2010","unstructured":"Yong Z, Feng J (2010) Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal Real World Appl 11(5):4465\u20134475","journal-title":"Nonlinear Anal Real World Appl"},{"key":"1061_CR7","doi-asserted-by":"publisher","DOI":"10.1007\/s00366-020-00968-2","author":"F Fakhar-Izadi","year":"2020","unstructured":"Fakhar-Izadi F (2020) Fully Petrov-Galerkin spectral method for the distributed-order time-fractional fourth-order partial differential equation. Eng Comput. https:\/\/doi.org\/10.1007\/s00366-020-00968-2","journal-title":"Eng Comput"},{"issue":"6","key":"1061_CR8","doi-asserted-by":"publisher","first-page":"3642","DOI":"10.1016\/j.nonrwa.2011.06.021","volume":"12","author":"W JinRong","year":"2011","unstructured":"JinRong W, Yong Z (2011) Existence and controllability results for fractional semilinear differential inclusions. Nonlinear Anal Real World Appl 12(6):3642\u20133653","journal-title":"Nonlinear Anal Real World Appl"},{"key":"1061_CR9","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1016\/j.apnum.2018.08.005","volume":"135","author":"D Hengfei","year":"2019","unstructured":"Hengfei D (2019) A high-order numerical algorithm for two-dimensional time-space tempered fractional diffusion-wave equation. Appl Numer Math 135:30\u201346","journal-title":"Appl Numer Math"},{"issue":"5","key":"1061_CR10","doi-asserted-by":"publisher","first-page":"2006","DOI":"10.1016\/j.cnsns.2008.05.008","volume":"14","author":"H Jafari","year":"2009","unstructured":"Jafari H, Seifi S (2009) Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation. Commun Nonlinear Sci Numer Simul 14(5):2006\u20132012","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"2","key":"1061_CR11","doi-asserted-by":"publisher","first-page":"56","DOI":"10.14419\/ijamr.v7i2.10116","volume":"7","author":"I Mousa","year":"2018","unstructured":"Mousa I, Jafar B, Zainab A (2018) General solution of second order fractional differential equations. Int J Appl Math Res 7(2):56\u201361","journal-title":"Int J Appl Math Res"},{"key":"1061_CR12","doi-asserted-by":"publisher","DOI":"10.1007\/s00366-020-00975-3","author":"AS Hendy","year":"2020","unstructured":"Hendy AS (2020) Numerical treatment for after-effected multi-term time-space fractional advection-diffusion equations. Eng Comput. https:\/\/doi.org\/10.1007\/s00366-020-00975-3","journal-title":"Eng Comput"},{"key":"1061_CR13","doi-asserted-by":"crossref","unstructured":"Albadarneh Ramzi B, Batiha Iqbal M, Mohammad Z (2016) Numerical solutions for linear fractional differential equations of order 1 $$\\le \\alpha \\le 2$$ using finite difference method (FFDM). J Math Comput Sci 16:103\u2013111","DOI":"10.22436\/jmcs.016.01.11"},{"issue":"1","key":"1061_CR14","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1080\/00207160701405477","volume":"85","author":"O Galip","year":"2008","unstructured":"Galip O, Aydin K, Yildiray K (2008) A new analytical approximate method for the solution of fractional differential equations. Int J Comput Math 85(1):131\u2013142","journal-title":"Int J Comput Math"},{"key":"1061_CR15","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1007\/s00366-018-0584-8","volume":"35","author":"O Oruc","year":"2019","unstructured":"Oruc O, Esen A, Bulut F (2019) A haar wavelet approximation for two-dimensional time fractional reaction-subdiffusion equation. Eng Comput 35:75\u201386","journal-title":"Eng Comput"},{"issue":"2","key":"1061_CR16","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1016\/0022-247X(88)90170-9","volume":"135","author":"G Adomian","year":"1988","unstructured":"Adomian G (1988) A review of the decomposition method in applied mathematics. J Math Anal Appl 135(2):501\u2013544","journal-title":"J Math Anal Appl"},{"issue":"2","key":"1061_CR17","first-page":"147","volume":"6","author":"T Veyis","year":"2013","unstructured":"Veyis T, Nuran G (2013) On solving partial differential equations of fractional order by using the variational iteration method and multivariate $$Pad\\acute{e}$$ approximations. Eur J Pure Appl Math 6(2):147\u2013171","journal-title":"Eur J Pure Appl Math"},{"issue":"11\u201312","key":"1061_CR18","first-page":"2199","volume":"58","author":"O Zaid","year":"2009","unstructured":"Zaid O, Shaher M (2009) The variational iteration method an efficient scheme for handling fractional partial differential equations in fluid mechanics. Comput Math Appl 58(11\u201312):2199\u20132208","journal-title":"Comput Math Appl"},{"issue":"1","key":"1061_CR19","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1016\/j.apm.2006.10.025","volume":"32","author":"O Zaid","year":"2008","unstructured":"Zaid O, Shaher M (2008) Numerical method for nonlinear partial differential equation of fractional order. Appl Math Model 32(1):28\u201339","journal-title":"Appl Math Model"},{"issue":"4","key":"1061_CR20","doi-asserted-by":"publisher","first-page":"1295","DOI":"10.1016\/j.cnsns.2008.01.010","volume":"14","author":"S Ray Santanu","year":"2009","unstructured":"Ray Santanu S (2009) Analytical solution for the space fractional diffusion equation by two-step Adomian Decomposition Method. Commun Non-linear Sci Numer Simul 14(4):1295\u20131306","journal-title":"Commun Non-linear Sci Numer Simul"},{"issue":"6","key":"1061_CR21","doi-asserted-by":"publisher","first-page":"2122","DOI":"10.1016\/S0252-9602(11)60388-3","volume":"31","author":"A Bashir","year":"2011","unstructured":"Bashir A, Nieto Juan J, Ahmed A (2011) Existence and uniqueness of solutions for nonlinear fractional differential equations with non-separated type integral boundary conditions. Acta Mathematica Scientia 31(6):2122\u20132130","journal-title":"Acta Mathematica Scientia"},{"issue":"11","key":"1061_CR22","doi-asserted-by":"publisher","first-page":"4037","DOI":"10.1016\/j.cnsns.2011.11.036","volume":"17","author":"K Karthikeyan","year":"2012","unstructured":"Karthikeyan K, Trujillo JJ (2012) Existence and uniqueness results for fractional integrodifferential equations with boundary value conditions. Commun Nonlinear Sci Numer Simul 17(11):4037\u20134043","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"3","key":"1061_CR23","doi-asserted-by":"publisher","first-page":"758","DOI":"10.1016\/S0252-9602(13)60036-3","volume":"33","author":"N Dimplekumar","year":"2013","unstructured":"Dimplekumar N, Chalishajar K (2013) Karthikeyan, Existence and uniqueness results for boundary value problems of higher order fractional integro-differential equations involving gronwall\u2019s inequality in banach spaces. Acta Mathematica Scientia 33(3):758\u2013772","journal-title":"Acta Mathematica Scientia"},{"issue":"12","key":"1061_CR24","doi-asserted-by":"publisher","first-page":"2587","DOI":"10.1016\/j.camwa.2013.01.034","volume":"66","author":"A Anguraj","year":"2014","unstructured":"Anguraj A, Karthikeyan P, Rivero M, Trujillo JJ (2014) On new existence results for fractional integro-differential equations with impulsive and integral conditions. Appl Math Comput 66(12):2587\u20132594","journal-title":"Appl Math Comput"},{"issue":"4","key":"1061_CR25","doi-asserted-by":"publisher","first-page":"821","DOI":"10.1016\/j.cnsns.2013.07.025","volume":"19","author":"C Archana","year":"2014","unstructured":"Archana C, Jaydev D (2014) Local and global existence of mild solution to an impulsive fractional functional integro-differential equation with nonlocal condition. Commun Nonlinear Sci Numer Simul 19(4):821\u2013829","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"1061_CR26","doi-asserted-by":"publisher","first-page":"429","DOI":"10.1016\/j.apnum.2019.05.008","volume":"145","author":"A Zaky Mahmoud","year":"2019","unstructured":"Zaky Mahmoud A (2019) Existence, uniqueness and numerical analysis of solutions of tempered fractional boundary value problems. Appl Numer Math 145:429\u2013457","journal-title":"Appl Numer Math"},{"issue":"1","key":"1061_CR27","first-page":"570","volume":"170","author":"L Xing-Guo","year":"2005","unstructured":"Xing-Guo L (2005) A Two-step Adomian decomposition method. Appl Math Comput 170(1):570\u2013583","journal-title":"Appl Math Comput"},{"issue":"2","key":"1061_CR28","doi-asserted-by":"publisher","first-page":"508","DOI":"10.1016\/j.jmaa.2004.07.039","volume":"301","author":"D-G Varsha","year":"2005","unstructured":"Varsha D-G, Hossein J (2005) Adomian decomposition: a tool for solving a system of fractional differential equations. J Math Anal Appl 301(2):508\u2013518","journal-title":"J Math Anal Appl"},{"key":"1061_CR29","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1155\/2011\/298628","volume":"2011","author":"HJ Haubold","year":"2011","unstructured":"Haubold HJ, Mathai AM, Saxena RK (2011) Mittag-Leffler functions and their applications. J Appl Math 2011:51","journal-title":"J Appl Math"},{"key":"1061_CR30","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1016\/j.cam.2016.11.013","volume":"315","author":"K Kamlesh","year":"2017","unstructured":"Kamlesh K, Pandey Rajesh K, Shiva S (2017) Comparative study of three numerical schemes for fractional integro-differential equations. J Comput Appl Math 315:287\u2013302","journal-title":"J Comput Appl Math"},{"issue":"2","key":"1061_CR31","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1007\/s11784-007-0041-6","volume":"2","author":"S Richard","year":"2007","unstructured":"Richard S (2007) Palais, A simple proof of the Banach contraction principle. J Fixed Point Theory Appl 2(2):221\u2013223","journal-title":"J Fixed Point Theory Appl"},{"issue":"5","key":"1061_CR32","doi-asserted-by":"publisher","first-page":"3436","DOI":"10.1016\/j.jde.2011.11.012","volume":"252","author":"J Garcia-Falset","year":"2012","unstructured":"Garcia-Falset J, Latrach K, Moreno-G\u00e0lvez E, Taoudi M-A (2012) Schaefer-Krasnoselskii fixed point theorems using a usual measure of weak noncompactness. J Differ Equ 252(5):3436\u20133452","journal-title":"J Differ Equ"},{"issue":"4","key":"1061_CR33","first-page":"199","volume":"34","author":"JW Green","year":"2018","unstructured":"Green JW, Valentine FA (2018) On the Arzel$$\\grave{a}$$-Ascoli Theorem. Math Mag 34(4):199\u2013202","journal-title":"Math Mag"},{"issue":"6","key":"1061_CR34","first-page":"1","volume":"3","author":"SA Murad","year":"2012","unstructured":"Murad SA, Hadid S (2012) Existence and uniqueness theorem for fractional differential equation with integral boundary condition. J Fract Calculus Appl 3(6):1\u20139","journal-title":"J Fract Calculus Appl"},{"issue":"2","key":"1061_CR35","first-page":"279","volume":"32","author":"S Abdo Mohammed","year":"2019","unstructured":"Abdo Mohammed S, Saeed Abdulkafi M, Panchal Satish K (2019) Caputo fractional integro-differential equation with non-local condition in banach space. Int J Appl Math 32(2):279\u2013288","journal-title":"Int J Appl Math"},{"issue":"1","key":"1061_CR36","first-page":"77","volume":"102","author":"W Abdul-Majid","year":"1999","unstructured":"Abdul-Majid W (1999) A reliable modification of Adomian decomposition method. Appl Math Comput 102(1):77\u201386","journal-title":"Appl Math Comput"},{"issue":"6","key":"1061_CR37","doi-asserted-by":"publisher","first-page":"1017","DOI":"10.1007\/s13226-014-0102-7","volume":"45","author":"E Babolian","year":"2014","unstructured":"Babolian E, Vahidi AR, Shoja A (2014) An efficient Method for non-linear fractional differential equations: Combination of the Adomian decomposition method and spectral method. Indian J Pure Appl Math 45(6):1017\u20131028","journal-title":"Indian J Pure Appl Math"},{"issue":"2","key":"1061_CR38","first-page":"442","volume":"10","author":"B Sohrab","year":"2020","unstructured":"Sohrab B, Alireza H (2020) The alternative Legendre Tau method for solving nonlinear multi-order fractional differential equations. J Appl Anal Comput 10(2):442\u2013456","journal-title":"J Appl Anal Comput"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-020-01061-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-020-01061-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-020-01061-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,2,22]],"date-time":"2022-02-22T01:17:57Z","timestamp":1645492677000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-020-01061-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,15]]},"references-count":38,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,2]]}},"alternative-id":["1061"],"URL":"https:\/\/doi.org\/10.1007\/s00366-020-01061-4","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,6,15]]},"assertion":[{"value":"25 April 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 May 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 June 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}