{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,24]],"date-time":"2025-07-24T11:58:36Z","timestamp":1753358316913},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"15","license":[{"start":{"date-parts":[[2023,6,1]],"date-time":"2023-06-01T00:00:00Z","timestamp":1685577600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,6,1]],"date-time":"2023-06-01T00:00:00Z","timestamp":1685577600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"National Natural Science Foundation of Guangdong Province","award":["2020A1515010812","2021A1515011594"],"award-info":[{"award-number":["2020A1515010812","2021A1515011594"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Soft Comput"],"published-print":{"date-parts":[[2023,8]]},"DOI":"10.1007\/s00500-023-08549-3","type":"journal-article","created":{"date-parts":[[2023,6,1]],"date-time":"2023-06-01T11:01:55Z","timestamp":1685617315000},"page":"10457-10467","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Granular fuzzy fractional descriptor linear systems under granular caputo fuzzy fractional derivative"],"prefix":"10.1007","volume":"27","author":[{"given":"Marzieh","family":"Najariyan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yi","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,6,1]]},"reference":[{"key":"8549_CR1","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1016\/j.fss.2014.03.006","volume":"245","author":"R Alikhani","year":"2014","unstructured":"Alikhani R, Bahrami F, Bhaskar TG (2014) Existence and uniqueness results for fuzzy linear differential-algebraic equations. Fuzzy Sets Syst 245:30\u201342","journal-title":"Fuzzy Sets Syst"},{"key":"8549_CR2","doi-asserted-by":"publisher","first-page":"1525","DOI":"10.1109\/TCSI.2004.832741","volume":"51","author":"S Ayasun","year":"2004","unstructured":"Ayasun S, Nwankpa CO, Kwatny HG (2004) Computation of singular and singularity induced bifurcation points of differential-algebraic power system model. IEEE Trans Circ Sys I 51:1525\u20131538","journal-title":"IEEE Trans Circ Sys I"},{"key":"8549_CR3","doi-asserted-by":"publisher","first-page":"186","DOI":"10.1016\/j.cam.2006.02.024","volume":"202","author":"SL Campbell","year":"2007","unstructured":"Campbell SL, Marz R (2007) Direct transcription solution of high index optimal control problems and regular Euler-Lagrange equations. J Comput Appl Math 202:186\u2013202","journal-title":"J Comput Appl Math"},{"key":"8549_CR4","doi-asserted-by":"crossref","unstructured":"Dai L (1989) Singular control systems, Thoma M and Wyner A (eds) Lecture notes in control and information sciences, Berlin, Springer","DOI":"10.1007\/BFb0002475"},{"key":"8549_CR5","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2020.112919","volume":"378","author":"I Dassios","year":"2020","unstructured":"Dassios I, Tzounas G, Milano F (2020) Generalized fractional controller for singular systems of differential equations. J Comput Appl Math 378:112919","journal-title":"J Comput Appl Math"},{"key":"8549_CR6","doi-asserted-by":"crossref","unstructured":"Eich-Soellner E, Fuhrer C (1998) Numerical methods in multibody dynamics. Teubner, B. G","DOI":"10.1007\/978-3-663-09828-7"},{"key":"8549_CR7","doi-asserted-by":"publisher","first-page":"1527","DOI":"10.1137\/0727089","volume":"27","author":"CW Gear","year":"1990","unstructured":"Gear CW (1990) Differential algebraic equations, indices and integral algebraic equations. SIAM J Numer Anal 27:1527\u20131534","journal-title":"SIAM J Numer Anal"},{"key":"8549_CR8","unstructured":"Jiang Y, Qiu D (2022) The relationship of three difference operations for fuzzy numbers to three kinds of derivative, 1 Jan. 1-15"},{"key":"8549_CR9","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-84628-605-6","volume-title":"Polynomial and rational matrices: applications in dynamical systems theory","author":"T Kaczorek","year":"2007","unstructured":"Kaczorek T (2007) Polynomial and rational matrices: applications in dynamical systems theory. Springer-Verlag, London Limited"},{"key":"8549_CR10","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-20502-6","volume-title":"Selected problems of fractional systems theory","author":"T Kaczorek","year":"2011","unstructured":"Kaczorek T (2011) Selected problems of fractional systems theory. Springer, Berlin Heidelberg"},{"key":"8549_CR11","first-page":"409","volume":"62","author":"T Kaczorek","year":"2014","unstructured":"Kaczorek T (2014) Drazin inverse matrix method for fractional descriptor continuous-time linear systems. Bull Pol Acad Sci Tech Sci 62:409\u2013418","journal-title":"Bull Pol Acad Sci Tech Sci"},{"key":"8549_CR12","doi-asserted-by":"crossref","unstructured":"Kaczorek T, Rogowski K (2015) Fractional linear systems and electrical circuits, Springer International Publishing Switzerland","DOI":"10.1007\/978-3-319-11361-6"},{"key":"8549_CR13","first-page":"1","volume":"7","author":"G Karamali","year":"2018","unstructured":"Karamali G, Shiri B (2018) Numerical solution of higher index DAEs using their IAE\u2019s structure: trajectory-prescribed path control problem and simple pendulum, Caspian. J Math Sci 7:1\u201315","journal-title":"J Math Sci"},{"key":"8549_CR14","doi-asserted-by":"crossref","unstructured":"Kumar A, Daoutidis P (1999) Control of nonlinear differential algebraic equation systems with applications to chemical processes, Chapman and Hall","DOI":"10.1007\/978-94-011-5094-1_11"},{"key":"8549_CR15","doi-asserted-by":"crossref","unstructured":"Leimkuhler B, Reich S (2004) Simulating Hamiltonian dynamics. Cambridge University Press","DOI":"10.1017\/CBO9780511614118"},{"key":"8549_CR16","doi-asserted-by":"publisher","first-page":"312","DOI":"10.1109\/TAC.1977.1101502","volume":"22","author":"DG Luenberger","year":"1977","unstructured":"Luenberger DG (1977) Dynamic equations in descriptor form. IEEE Trans Autom Control 22:312\u2013321","journal-title":"IEEE Trans Autom Control"},{"key":"8549_CR17","doi-asserted-by":"publisher","first-page":"312","DOI":"10.1109\/TPWRS.2004.841244","volume":"20","author":"W Marszalek","year":"2005","unstructured":"Marszalek W, Trzaska ZW (2005) Singularity-induced bifurcations in electrical power systems. IEEE Trans Power Syst 20:312\u2013320","journal-title":"IEEE Trans Power Syst"},{"key":"8549_CR18","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.isatra.2018.02.001","volume":"76","author":"M Mazandarani","year":"2018","unstructured":"Mazandarani M, Pariz N (2018) Sub-optimal control of fuzzy linear dynamical systems under granular differentiability concept. ISA Trans 76:1\u201317","journal-title":"ISA Trans"},{"key":"8549_CR19","doi-asserted-by":"publisher","first-page":"62195","DOI":"10.1109\/ACCESS.2021.3074245","volume":"9","author":"M Mazandarani","year":"2021","unstructured":"Mazandarani M, Xiu L (2021) A review on fuzzy differential equations. IEEE Access 9:62195\u201362211","journal-title":"IEEE Access"},{"key":"8549_CR20","doi-asserted-by":"publisher","first-page":"462","DOI":"10.1109\/TFUZZ.2019.2908131","volume":"28","author":"M Mazandarani","year":"2020","unstructured":"Mazandarani M, Zhao Y (2020) Z-differential equations. J IEEE Trans Fuzzy Syst 28:462\u2013473","journal-title":"J IEEE Trans Fuzzy Syst"},{"key":"8549_CR21","doi-asserted-by":"publisher","first-page":"310","DOI":"10.1109\/TFUZZ.2017.2659731","volume":"26","author":"M Mazandarani","year":"2018","unstructured":"Mazandarani M, Pariz N, Kamyad AV (2018) Granular differentiability of fuzzy-number-valued functions. J IEEE Trans Fuzzy Syst 26:310\u2013323","journal-title":"J IEEE Trans Fuzzy Syst"},{"key":"8549_CR22","doi-asserted-by":"crossref","unstructured":"Mustafa AM, Gong Z, Osman M (2020) The solution of fuzzy variational problem and fuzzy optimal control problem under granular differentiability concept, 1495-1520","DOI":"10.1080\/00207160.2020.1823974"},{"key":"8549_CR23","doi-asserted-by":"publisher","unstructured":"Najariyan M, Pariz N (2022) Stability and controllability of fuzzy singular dynamical systems. J Franklin Inst. https:\/\/doi.org\/10.1016\/j.jfranklin.2022.07.035","DOI":"10.1016\/j.jfranklin.2022.07.035"},{"key":"8549_CR24","doi-asserted-by":"publisher","first-page":"2273","DOI":"10.1109\/TFUZZ.2017.2783895","volume":"26","author":"M Najariyan","year":"2018","unstructured":"Najariyan M, Zhao Y (2018) Fuzzy fractional quadratic regulator problem under granular fuzzy fractional derivatives. IEEE Trans Fuzzy Syst 26:2273\u20132288","journal-title":"IEEE Trans Fuzzy Syst"},{"key":"8549_CR25","doi-asserted-by":"publisher","first-page":"11251","DOI":"10.1007\/s00500-020-05055-8","volume":"24","author":"M Najariyan","year":"2020","unstructured":"Najariyan M, Zhao Y (2020) The explicit solution of fuzzy singular differential equations using fuzzy Drazin inverse matrix. Soft Comput 24:11251\u201311264","journal-title":"Soft Comput"},{"key":"8549_CR26","doi-asserted-by":"publisher","first-page":"5502","DOI":"10.1016\/j.jfranklin.2020.02.023","volume":"357","author":"M Najariyan","year":"2020","unstructured":"Najariyan M, Zhao Y (2020) On the stability of fuzzy linear dynamical systems. J Franklin Inst 357:5502\u20135522","journal-title":"J Franklin Inst"},{"key":"8549_CR27","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1016\/j.fss.2021.01.003","volume":"429","author":"M Najariyan","year":"2022","unstructured":"Najariyan M, Pariz N, Vu H (2022) Fuzzy linear singular differential equations under granular differentiability concept. Fuzzy Sets Syst 429:169\u2013187","journal-title":"Fuzzy Sets Syst"},{"key":"8549_CR28","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1007\/s40815-015-0013-8","volume":"17","author":"A Piegat","year":"2015","unstructured":"Piegat A, Landowski M (2015) Horizontal membership function and examples of its applications. Int J Fuzzy Syst 17:22\u201330","journal-title":"Int J Fuzzy Syst"},{"key":"8549_CR29","doi-asserted-by":"publisher","DOI":"10.1155\/2015\/367214","author":"A Piegat","year":"2015","unstructured":"Piegat A, Plucinski M (2015) Fuzzy number addition with the application of horizontal membership functions. Sci World J. https:\/\/doi.org\/10.1155\/2015\/367214","journal-title":"Sci World J"},{"key":"8549_CR30","doi-asserted-by":"crossref","unstructured":"Piegat A, Landowski M (2016) On fuzzy RDM-arithmetic. Hard and soft computing for artificial intelligence, multimedia and security 534:3\u201316","DOI":"10.1007\/978-3-319-48429-7_1"},{"key":"8549_CR31","doi-asserted-by":"crossref","unstructured":"Piegat A, Landowski M (2017) Fuzzy arithmetic type 1 with horizontal membership functions, uncertainty modeling, vol 683 of the series Studies in Computational Intelligence, 233-250","DOI":"10.1007\/978-3-319-51052-1_14"},{"key":"8549_CR32","doi-asserted-by":"crossref","unstructured":"Rabier PJ, Rheinboldt WC (2000) Nonholonomic motion of rigid mechanical systems from a DAE viewpoint, SIAM","DOI":"10.1137\/1.9780898719536"},{"key":"8549_CR33","doi-asserted-by":"crossref","unstructured":"Riaza R (2008) Differential-algebraic systems-analytical aspects and circuit applications, World Scientific Publishing Co. Pte. Ltd","DOI":"10.1142\/6746"},{"key":"8549_CR34","doi-asserted-by":"crossref","unstructured":"Sajewski L (2017) Descriptor fractional continuous-time linear system and its solution\u2014comparison of three different methods. In: Szewczyk R., Zieli\u0144ski C., Kaliczy\u0144ska M. (eds) Automation 2017. ICA 2017. Advances in Intelligent Systems and Computing, vol 550. Springer, Cham","DOI":"10.1007\/978-3-319-54042-9_4"},{"key":"8549_CR35","doi-asserted-by":"publisher","first-page":"661","DOI":"10.1090\/S0025-5718-99-01010-8","volume":"68","author":"WM Seiler","year":"1999","unstructured":"Seiler WM (1999) Numerical integration of constrained Hamiltonian systems using Dirac brackets. Math Comp 68:661\u2013681","journal-title":"Math Comp"},{"key":"8549_CR36","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1016\/j.chaos.2019.01.028","volume":"120","author":"B Shiri","year":"2019","unstructured":"Shiri B, Baleanu D (2019) System of fractional differential algebraic equations with applications. Chaos Solitons Fractals 120:203\u2013212","journal-title":"Chaos Solitons Fractals"},{"key":"8549_CR37","first-page":"63","volume":"342","author":"K Tomaszewska","year":"2015","unstructured":"Tomaszewska K, Piegat A (2015) Application of the horizontal membership function to the uncertain displacement calculation of a composite massless rod under a tensile load. Soft Comput Comput Inf Sci 342:63\u201372","journal-title":"Soft Comput Comput Inf Sci"},{"key":"8549_CR38","doi-asserted-by":"publisher","first-page":"286","DOI":"10.3390\/fractalfract6060286","volume":"6","author":"J Zhang","year":"2022","unstructured":"Zhang J, Wang Y, Zhang S (2022) A new homotopy transformation method for solving the fuzzy fractional black-scholes european option pricing equations under the concept of granular differentiability. Fractal Fract 6:286. https:\/\/doi.org\/10.3390\/fractalfract6060286","journal-title":"Fractal Fract"}],"container-title":["Soft Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-023-08549-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00500-023-08549-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-023-08549-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,16]],"date-time":"2023-06-16T16:11:47Z","timestamp":1686931907000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00500-023-08549-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,1]]},"references-count":38,"journal-issue":{"issue":"15","published-print":{"date-parts":[[2023,8]]}},"alternative-id":["8549"],"URL":"https:\/\/doi.org\/10.1007\/s00500-023-08549-3","relation":{},"ISSN":["1432-7643","1433-7479"],"issn-type":[{"value":"1432-7643","type":"print"},{"value":"1433-7479","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,1]]},"assertion":[{"value":"1 May 2023","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 June 2023","order":2,"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.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"This article does not contain any studies with human participants or animals performed by any of the authors.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}},{"value":"Informed consent was obtained from all individual participants included in the study","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Informed consent"}}]}}