{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T15:23:16Z","timestamp":1775575396061,"version":"3.50.1"},"reference-count":51,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2017,4,22]],"date-time":"2017-04-22T00:00:00Z","timestamp":1492819200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2017,4,22]],"date-time":"2017-04-22T00:00:00Z","timestamp":1492819200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1513239"],"award-info":[{"award-number":["1513239"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Neural Comput & Applic"],"published-print":{"date-parts":[[2019,1]]},"DOI":"10.1007\/s00521-017-2982-z","type":"journal-article","created":{"date-parts":[[2017,4,22]],"date-time":"2017-04-22T05:45:00Z","timestamp":1492839900000},"page":"101-115","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Heuristic computational intelligence approach to solve nonlinear multiple singularity problem of sixth Painlev\u0301e equation"],"prefix":"10.1007","volume":"31","author":[{"given":"Iftikhar","family":"Ahmad","sequence":"first","affiliation":[]},{"given":"Abdul","family":"Rehman","sequence":"additional","affiliation":[]},{"given":"Fayyaz","family":"Ahmad","sequence":"additional","affiliation":[]},{"given":"Muhammad Asif","family":"Zahoor Raja","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2017,4,22]]},"reference":[{"key":"2982_CR1","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1007\/BF02419026","volume":"25","author":"P Painlev","year":"1902","unstructured":"Painlev P (1902) Sur les quations diffrentielles du second ordre et d\u2019ordre sup rieure dont l\u2019intgrale gnrale est uniforme. Acta Math 25:185","journal-title":"Acta Math"},{"key":"2982_CR2","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1007\/BF02393211","volume":"33","author":"B Gambier","year":"1910","unstructured":"Gambier B (1910) Sur les quations diffrentielles du second ordre et du premier degr dont l\u2019intgrale gnrale est points critiques fixes. Acta Math 33:155","journal-title":"Acta Math"},{"key":"2982_CR3","first-page":"41","volume":"104","author":"E Picard","year":"1887","unstructured":"Picard E (1887) Sur une classe d\u2019\u00e9quations diffrentielles. CR Acad Sci Paris 104:41\u201343","journal-title":"CR Acad Sci Paris"},{"key":"2982_CR4","unstructured":"Ince EL (1956) Ordinary differential equations. Dover, New York"},{"key":"2982_CR5","unstructured":"Ablowitz MJ, Clarkson PA (1997) Solitons, nonlinear evolution equations and inverse scattering, vol 149"},{"key":"2982_CR6","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970883","volume-title":"Solitons and the inverse scattering trans-form","author":"MJ Ablowitz","year":"1981","unstructured":"Ablowitz MJ, Segur H (1981) Solitons and the inverse scattering trans-form. SIAM, Philadelphia"},{"key":"2982_CR7","doi-asserted-by":"crossref","unstructured":"Its AR, Novokshenov VY (1986) The isomonodromic deformation method in the theory of Painleve equations, vol 1191","DOI":"10.1007\/BFb0076661"},{"key":"2982_CR8","first-page":"395","volume":"3","author":"NA Lukashevich","year":"1967","unstructured":"Lukashevich NA (1967) Theory of the fourth Painleve equation. Diff Equat 3:395\u2013399","journal-title":"Diff Equat"},{"key":"2982_CR9","first-page":"732","volume":"4","author":"NA Lukashevich","year":"1968","unstructured":"Lukashevich NA (1968) Solution of fifth equation of Painleve. Diff Urav 4:732\u2013735","journal-title":"Diff Urav"},{"key":"2982_CR10","first-page":"264","volume":"3","author":"NA Lukashevich","year":"1967","unstructured":"Lukashevich NA, Yablonskii AI (1967) On a class of solutions of the sixth Painleve equations. Diff Urav 3:264","journal-title":"Diff Urav"},{"key":"2982_CR11","doi-asserted-by":"publisher","first-page":"2033","DOI":"10.1063\/1.525260","volume":"23","author":"AS Fokas","year":"1982","unstructured":"Fokas AS, Ablowitz MJ (1982) On a unified approach to transformations and elementary solutions of Painlev equations. J Math Phy 23:2033\u20132042","journal-title":"J Math Phy"},{"key":"2982_CR12","doi-asserted-by":"crossref","unstructured":"Gromak VI, Laine I, Shimomura S (2002) Painleve differential equations in the complex plane. Walter de Gruyter, Berlin, New York","DOI":"10.1515\/9783110198096"},{"key":"2982_CR13","doi-asserted-by":"crossref","unstructured":"Okamoto K, Conte R (1999) The Painleve property, one century later","DOI":"10.1007\/978-1-4612-1532-5"},{"key":"2982_CR14","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1007\/BF01458459","volume":"275","author":"K Okamoto","year":"1986","unstructured":"Okamoto K (1986) Studies on Painleve equation III, second and fourth Painleve equation, PII and PIV. Math Ann 275:221\u2013255","journal-title":"Math Ann"},{"key":"2982_CR15","doi-asserted-by":"publisher","first-page":"201","DOI":"10.24033\/bsmf.633","volume":"28","author":"P Painleve","year":"1900","unstructured":"Painleve P (1900) Memoire sur les equations differentielles dont 1\u2019integrale generale est uniforme. Bull Soc Math Fr 28:201\u2013 261","journal-title":"Bull Soc Math Fr"},{"key":"2982_CR16","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF02419020","volume":"25","author":"P Painleve","year":"1902","unstructured":"Painleve P (1902) Memoire Sur les equations differentielles du second ordre et d\u2019ordre superieur dont 1\u2019integrale generale est uniforme. Acta Math 25:1\u201385","journal-title":"Acta Math"},{"key":"2982_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF02393211","volume":"33","author":"B Gambier","year":"1910","unstructured":"Gambier B (1910) Sur les equations differentielles du second ordre et du premier degre dont 1\u2019integrale generale est a points critiques fixes. Acta Math 33:1\u201355","journal-title":"Acta Math"},{"key":"2982_CR18","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1007\/BF02393131","volume":"34","author":"J Chazy","year":"1911","unstructured":"Chazy J (1911) Sur les equations differentielles du troisieme ordre et d\u2019ordre suprieur dont 1\u2019intgrale generale a ses points critiques fixes. Acta Math 34:317\u2013385","journal-title":"Acta Math"},{"key":"2982_CR19","first-page":"1","volume":"48","author":"R Garnier","year":"1912","unstructured":"Garnier R (1912) Sur des equations differentielles du troisieme ordre dont l\u2019integrale generale est uniforme et sur une classe d\u2019equations nouvelles d\u2019ordre superieur. Ann Sci Ecole Normale Sup 48:1\u2013126","journal-title":"Ann Sci Ecole Normale Sup"},{"key":"2982_CR20","unstructured":"Exton H (1973) Non-linear ordinary differential equations with fixed critical points. Rend Mat (6):419\u2013462"},{"key":"2982_CR21","first-page":"764","volume":"21","author":"IP Martynov","year":"1985","unstructured":"Martynov IP (1985) Analytic properties of solutions of a third-order differential equation. Differents Uravn 21:764\u2013771","journal-title":"Differents Uravn"},{"key":"2982_CR22","first-page":"937","volume":"21","author":"IP Martynov","year":"1985","unstructured":"Martynov IP (1985) Third-order equations without moving critical singularities. Differents Uravn 21:937\u2013946","journal-title":"Differents Uravn"},{"key":"2982_CR23","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF02412437","volume":"66","author":"FJ Bureau","year":"1964","unstructured":"Bureau FJ (1964) Differential equations with fixed critical points. Ann Mat Pura Appl (IV) 66:1\u2013116","journal-title":"Ann Mat Pura Appl (IV)"},{"key":"2982_CR24","doi-asserted-by":"publisher","first-page":"1103","DOI":"10.1103\/PhysRevLett.38.1103","volume":"38","author":"MJ Ablowitz","year":"1977","unstructured":"Ablowitz MJ, Segur H (1977) Exact linearization of a Painleve transcendent. Phys Rev Lett 38:1103\u20131106","journal-title":"Phys Rev Lett"},{"key":"2982_CR25","doi-asserted-by":"publisher","first-page":"715","DOI":"10.1063\/1.524491","volume":"21","author":"MJ Ablowitz","year":"1980","unstructured":"Ablowitz MJ, Ramani A, Segur H (1980) A connection between nonlinear evolution equations and ordinary differential equations of P-type. I. J Math Phys 21:715\u2013721","journal-title":"J Math Phys"},{"key":"2982_CR26","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1002\/sapm197961131","volume":"61","author":"H Airault","year":"1979","unstructured":"Airault H (1979) Rational solutions of Painleve equations. Stud Appl Math 61:31\u201353","journal-title":"Stud Appl Math"},{"key":"2982_CR27","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1016\/S0375-9601(96)00795-5","volume":"224","author":"NA Kudryashov","year":"1997","unstructured":"Kudryashov NA (1997) The first and second Painleve equations of higher order and some relations between them. Phys Lett A 224:353\u2013360","journal-title":"Phys Lett A"},{"key":"2982_CR28","doi-asserted-by":"publisher","first-page":"5749","DOI":"10.1063\/1.533055","volume":"40","author":"PR Gordoa","year":"1999","unstructured":"Gordoa PR, Pickering A (1999) Non-isospectral scattering problems: a key to integrable hierarchies. J Math Phys 40:5749\u20135786","journal-title":"J Math Phys"},{"key":"2982_CR29","doi-asserted-by":"publisher","first-page":"557","DOI":"10.1088\/0305-4470\/33\/3\/309","volume":"33","author":"PR Gordoa","year":"2000","unstructured":"Gordoa PR, Pickering A (2000) On a new non-isospectral variant of the Boussinesq hierarchy. J Phys A 33:557\u2013567","journal-title":"J Phys A"},{"key":"2982_CR30","doi-asserted-by":"publisher","first-page":"327","DOI":"10.2977\/prims\/1145477227","volume":"37","author":"PR Gordoa","year":"2001","unstructured":"Gordoa PR, Joshi N, Pickering A (2001) On a generalized 2 + 1 dispersive water wave hierarchy. Publ Res Inst Math Sci (Kyoto) 37:327\u2013347","journal-title":"Publ Res Inst Math Sci (Kyoto)"},{"key":"2982_CR31","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1016\/S0375-9601(02)00972-6","volume":"301","author":"A Pickering","year":"2002","unstructured":"Pickering A (2002) Coalescence limits for higher order Painleve equations. Phys Lett A 301:275\u2013280","journal-title":"Phys Lett A"},{"key":"2982_CR32","first-page":"117","volume":"297","author":"T Kawai","year":"2004","unstructured":"Kawai T, Koike T, Nishikawa Y, Takei Y (2004) On the stokes geometry of higher order Painleve equations, Analyse complexe, systemes dynamiques, sommabilite des series divergentes et theories galoisiennes. II. Astrisque No 297:117\u2013166","journal-title":"Astrisque No"},{"key":"2982_CR33","doi-asserted-by":"publisher","first-page":"073504","DOI":"10.1063\/1.2217647","volume":"47","author":"PR Gordoa","year":"2006","unstructured":"Gordoa PR, Joshi N, Pickering A (2006) Second and fourth Painlev hierarchies and Jimbo-Miwa linear problems. J Math Phys 47:073504","journal-title":"J Math Phys"},{"key":"2982_CR34","unstructured":"Koike T (2007) On the Hamiltonian structures of the second and the fourth Painlev hierarchies, and the degenerate Garnier systems, algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlev hierarchies. RIMS Kkyroku Bessatsu, B2, Res Inst Math Sci (RIMS), Kyoto pp 99\u2013127"},{"key":"2982_CR35","doi-asserted-by":"publisher","first-page":"611","DOI":"10.1016\/j.physleta.2008.12.028","volume":"373","author":"AH Sakka","year":"2009","unstructured":"Sakka AH (2009) On special solutions of second and fourth Painleve hierarchies. Phys Lett A 373:611\u2013615","journal-title":"Phys Lett A"},{"key":"2982_CR36","doi-asserted-by":"publisher","first-page":"7933","DOI":"10.1088\/0305-4470\/32\/45\/309","volume":"32","author":"U Mugan","year":"1999","unstructured":"Mugan U, Jrad F (1999) Painleve test and the first Painleve hierarchy. J Phys A 32:7933\u20137952","journal-title":"J Phys A"},{"key":"2982_CR37","doi-asserted-by":"publisher","first-page":"282","DOI":"10.2991\/jnmp.2002.9.3.4","volume":"9","author":"U Mugan","year":"2002","unstructured":"Mugan U, Jrad F (2002) Painleve test and higher order differential equations. J Nonlinear Math Phys 9:282\u2013310","journal-title":"J Nonlinear Math Phys"},{"key":"2982_CR38","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1515\/zna-2004-0309","volume":"59","author":"U Mugan","year":"2004","unstructured":"Mugan U, Jrad F (2004) Non-polynomial third order equations which pass the Painleve test. Z Naturforsch A 59:163\u2013180","journal-title":"Z Naturforsch A"},{"key":"2982_CR39","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1111\/1467-9590.00130","volume":"104","author":"CM Cosgrove","year":"2000","unstructured":"Cosgrove CM (2000) Higher-order Painleve equations in the polynomial class. I. Bureau symbol P2. Stud Appl Math 104:1\u201365","journal-title":"Stud Appl Math"},{"key":"2982_CR40","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1111\/j.1467-9590.2006.00346.x","volume":"116","author":"CM Cosgrove","year":"2006","unstructured":"Cosgrove CM (2006) Higher-order Painleve equations in the polynomial class. II. Bureau symbol P1. Stud Appl Math 116:321\u2013 413","journal-title":"Stud Appl Math"},{"key":"2982_CR41","doi-asserted-by":"publisher","first-page":"393","DOI":"10.1016\/j.jmaa.2010.03.053","volume":"368","author":"D Dai","year":"2010","unstructured":"Dai D, Zhang L (2010) On tronque solutions of the first Painlev hierarchy. J Math Anal Appl 368:393\u2013399","journal-title":"J Math Anal Appl"},{"key":"2982_CR42","doi-asserted-by":"publisher","first-page":"344","DOI":"10.1016\/0022-0396(87)90175-6","volume":"68","author":"G Lebeau","year":"1987","unstructured":"Lebeau G, Lochak P (1987) On the second painleve equation: the connection formula via a Riemann-Hilbert problem and other results. Journal of Uiffekential Equations 68:344\u2013372","journal-title":"Journal of Uiffekential Equations"},{"issue":"3","key":"2982_CR43","doi-asserted-by":"publisher","first-page":"282","DOI":"10.2991\/jnmp.2003.10.3.3","volume":"10","author":"K Kajiwara","year":"2003","unstructured":"Kajiwara K (2003) On a q-difference Painleve III equation: II. rational solutions. J Nonlinear Math Phys 10 (3):282\u2013303","journal-title":"J Nonlinear Math Phys"},{"key":"2982_CR44","doi-asserted-by":"publisher","first-page":"817","DOI":"10.1016\/j.jmaa.2006.07.083","volume":"330","author":"S Yoshikatsu","year":"2007","unstructured":"Yoshikatsu S (2007) Value distribution of the fifth Painlev transcendentsin sectorial domains. J Math Anal 330:817\u2013828","journal-title":"J Math Anal"},{"key":"2982_CR45","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1016\/S0167-2789(01)00372-4","volume":"161","author":"R Contea","year":"2002","unstructured":"Contea R, Musetteb M (2002) New contiguity relation of the sixth Painlev equation from a truncation. A Physica D 161:129\u2013 141","journal-title":"A Physica D"},{"key":"2982_CR46","doi-asserted-by":"publisher","unstructured":"Ahmad I, Raja MAZ, Bilal M, Ashraf F (2016) Neural network methods to solve the Lane-Emden type equations arising in thermodynamic studies of the spherical gas cloud model. Neural Comput and Applic 1\u201316. doi:\n                    10.1007\/s00521-016-2400-y","DOI":"10.1007\/s00521-016-2400-y"},{"key":"2982_CR47","first-page":"360","volume":"261","author":"I Ahmad","year":"2015","unstructured":"Ahmad I, Mukhtar A (2015) Stochastic approach for the solution of multi-pantograph differential equation arising in cell-growth model, Appl. Math Comput 261:360","journal-title":"Math Comput"},{"key":"2982_CR48","doi-asserted-by":"publisher","unstructured":"Ahmad I, Ahmad S, Bilal M, Anwar N (2016) Stochastic numerical treatment for solving Falkner-Skan equations using feed forward neural networks. Neural Comput and Applic 1\u201314. doi:\n                    10.1007\/s00521-016-2427-0","DOI":"10.1007\/s00521-016-2427-0"},{"key":"2982_CR49","unstructured":"He JH (1999) Homotopy perturbation technique. Comput Math Appl Mech Engy pp 178\u2013257"},{"key":"2982_CR50","doi-asserted-by":"publisher","first-page":"231","DOI":"10.3233\/FI-2016-1384","volume":"146","author":"O Abu Arqub","year":"2016","unstructured":"Abu Arqub O (2016) Approximate solutions of DASs with nonclassical boundary conditions using novel reproducing kernel algorithm. Fundamenta Informaticae 146:231\u2013254","journal-title":"Fundamenta Informaticae"},{"key":"2982_CR51","doi-asserted-by":"publisher","unstructured":"Abu Arqub O, Maayah B (2016) Solutions of Bagley-Torvik and Painlev equations of fractional order using iterative reproducing kernel algorithm. Neural Comput and Applic. doi:\n                    10.1007\/s00521-016-2484-4","DOI":"10.1007\/s00521-016-2484-4"}],"container-title":["Neural Computing and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00521-017-2982-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-017-2982-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-017-2982-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,14]],"date-time":"2020-05-14T07:53:29Z","timestamp":1589442809000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00521-017-2982-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,4,22]]},"references-count":51,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["2982"],"URL":"https:\/\/doi.org\/10.1007\/s00521-017-2982-z","relation":{},"ISSN":["0941-0643","1433-3058"],"issn-type":[{"value":"0941-0643","type":"print"},{"value":"1433-3058","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,4,22]]},"assertion":[{"value":"28 October 2016","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 March 2017","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 April 2017","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Compliance with ethical standards"}},{"value":"There is no conflict of interest among all the authors.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interests"}}]}}