{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,19]],"date-time":"2025-03-19T16:31:09Z","timestamp":1742401869124},"reference-count":44,"publisher":"Elsevier BV","issue":"10","license":[{"start":{"date-parts":[[2012,11,1]],"date-time":"2012-11-01T00:00:00Z","timestamp":1351728000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2016,11,14]],"date-time":"2016-11-14T00:00:00Z","timestamp":1479081600000},"content-version":"vor","delay-in-days":1474,"URL":"https:\/\/www.elsevier.com\/open-access\/userlicense\/1.0\/"}],"funder":[{"name":"COMPETE\u2014Operational Programme Factors of Competitiveness (\u201cPrograma Operacional Factores de Competitividade\u201d)"},{"name":"Center for Research and Development in Mathematics and Applications (University of Aveiro)"},{"name":"Portuguese Foundation for Science and Technology (\u201cFCT\u2014Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia\u201d)"},{"name":"FCT"},{"name":"Bia\u0142ystok University of Technology","award":["S\/WI\/02\/2011"]},{"name":"European Union Human Capital Programme Podniesienie potencja\u0142u uczelni wyzszych jako czynnik rozwoju gospodarki opartej na wiedzy"},{"name":"FCT"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computers & Mathematics with Applications"],"published-print":{"date-parts":[[2012,11]]},"DOI":"10.1016\/j.camwa.2012.01.073","type":"journal-article","created":{"date-parts":[[2012,2,18]],"date-time":"2012-02-18T16:16:06Z","timestamp":1329581766000},"page":"3351-3366","source":"Crossref","is-referenced-by-count":49,"title":["Generalized fractional calculus with applications to the calculus of variations"],"prefix":"10.1016","volume":"64","author":[{"given":"Tatiana","family":"Odzijewicz","sequence":"first","affiliation":[]},{"given":"Agnieszka B.","family":"Malinowska","sequence":"additional","affiliation":[]},{"given":"Delfim F.M.","family":"Torres","sequence":"additional","affiliation":[]}],"member":"78","reference":[{"year":"2006","series-title":"Theory and Applications of Fractional Differential Equations","author":"Kilbas","key":"10.1016\/j.camwa.2012.01.073_br000005"},{"issue":"3","key":"10.1016\/j.camwa.2012.01.073_br000010","doi-asserted-by":"crossref","first-page":"1140","DOI":"10.1016\/j.cnsns.2010.05.027","article-title":"Recent history of fractional calculus","volume":"16","author":"Machado","year":"2011","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"issue":"2","key":"10.1016\/j.camwa.2012.01.073_br000015","doi-asserted-by":"crossref","first-page":"1890","DOI":"10.1103\/PhysRevE.53.1890","article-title":"Nonconservative Lagrangian and Hamiltonian mechanics","volume":"53","author":"Riewe","year":"1996","journal-title":"Phys. Rev. E (3)"},{"issue":"3","key":"10.1016\/j.camwa.2012.01.073_br000020","doi-asserted-by":"crossref","first-page":"3581","DOI":"10.1103\/PhysRevE.55.3581","article-title":"Mechanics with fractional derivatives","volume":"55","author":"Riewe","year":"1997","journal-title":"Phys. Rev. E (3)"},{"issue":"2","key":"10.1016\/j.camwa.2012.01.073_br000025","doi-asserted-by":"crossref","first-page":"834","DOI":"10.1016\/j.jmaa.2007.01.013","article-title":"A formulation of Noether\u2019s theorem for fractional problems of the calculus of variations","volume":"334","author":"Frederico","year":"2007","journal-title":"J. Math. Anal. Appl."},{"issue":"3","key":"10.1016\/j.camwa.2012.01.073_br000030","first-page":"956","article-title":"Leitmann\u2019s direct method for fractional optimization problems","volume":"217","author":"Almeida","year":"2010","journal-title":"Appl. Math. Comput."},{"issue":"3","key":"10.1016\/j.camwa.2012.01.073_br000035","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/s11071-007-9309-z","article-title":"Fractional conservation laws in optimal control theory","volume":"53","author":"Frederico","year":"2008","journal-title":"Nonlinear Dynam."},{"issue":"3","key":"10.1016\/j.camwa.2012.01.073_br000040","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1016\/j.sigpro.2010.07.016","article-title":"Modified optimal energy and initial memory of fractional continuous-time linear systems","volume":"91","author":"Mozyrska","year":"2011","journal-title":"Signal Process."},{"issue":"9\u201312","key":"10.1016\/j.camwa.2012.01.073_br000045","first-page":"479","article-title":"Fractional optimal control in the sense of Caputo and the fractional Noether\u2019s theorem","volume":"3","author":"Frederico","year":"2008","journal-title":"Int. Math. Forum"},{"issue":"9","key":"10.1016\/j.camwa.2012.01.073_br000050","doi-asserted-by":"crossref","first-page":"3110","DOI":"10.1016\/j.camwa.2010.02.032","article-title":"Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative","volume":"59","author":"Malinowska","year":"2010","journal-title":"Comput. Math. Appl."},{"issue":"2","key":"10.1016\/j.camwa.2012.01.073_br000055","first-page":"210","article-title":"Minimal modified energy control for fractional linear control systems with the Caputo derivative","volume":"26","author":"Mozyrska","year":"2010","journal-title":"Carpathian J. Math."},{"issue":"24","key":"10.1016\/j.camwa.2012.01.073_br000060","doi-asserted-by":"crossref","first-page":"6287","DOI":"10.1088\/1751-8113\/40\/24\/003","article-title":"Fractional variational calculus in terms of Riesz fractional derivatives","volume":"40","author":"Agrawal","year":"2007","journal-title":"J. Phys. A"},{"issue":"3","key":"10.1016\/j.camwa.2012.01.073_br000065","first-page":"1023","article-title":"Fractional Noether\u2019s theorem in the Riesz\u2013Caputo sense","volume":"217","author":"Frederico","year":"2010","journal-title":"Appl. Math. Comput."},{"issue":"5","key":"10.1016\/j.camwa.2012.01.073_br000070","doi-asserted-by":"crossref","first-page":"1569","DOI":"10.1007\/s10773-011-0668-3","article-title":"Hamilton\u2013Jacobi formulation for systems in terms of Riesz\u2019s fractional derivatives","volume":"50","author":"Rabei","year":"2011","journal-title":"Internat. J. Theoret. Phys."},{"issue":"31","key":"10.1016\/j.camwa.2012.01.073_br000075","doi-asserted-by":"crossref","first-page":"6675","DOI":"10.1088\/0305-4470\/35\/31\/311","article-title":"Stationarity-conservation laws for fractional differential equations with variable coefficients","volume":"35","author":"Klimek","year":"2002","journal-title":"J. Phys. A"},{"issue":"11","key":"10.1016\/j.camwa.2012.01.073_br000080","doi-asserted-by":"crossref","first-page":"1447","DOI":"10.1007\/s10582-006-0024-7","article-title":"Lagrangian fractional mechanics\u2014a noncommutative approach","volume":"55","author":"Klimek","year":"2005","journal-title":"Czech. J. Phys."},{"issue":"4","key":"10.1016\/j.camwa.2012.01.073_br000085","doi-asserted-by":"crossref","first-page":"523","DOI":"10.2478\/s13540-011-0032-6","article-title":"Fractional calculus of variations for a combined Caputo derivative","volume":"14","author":"Malinowska","year":"2011","journal-title":"Fract. Calc. Appl. Anal."},{"issue":"3","key":"10.1016\/j.camwa.2012.01.073_br000090","doi-asserted-by":"crossref","first-page":"033503","DOI":"10.1063\/1.3319559","article-title":"A fractional calculus of variations for multiple integrals with application to vibrating string","volume":"51","author":"Almeida","year":"2010","journal-title":"J. Math. Phys."},{"issue":"3","key":"10.1016\/j.camwa.2012.01.073_br000095","doi-asserted-by":"crossref","first-page":"033504","DOI":"10.1063\/1.2483292","article-title":"Fractional embedding of differential operators and Lagrangian systems","volume":"48","author":"Cresson","year":"2007","journal-title":"J. Math. Phys."},{"issue":"1","key":"10.1016\/j.camwa.2012.01.073_br000100","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1016\/j.nonrwa.2009.01.003","article-title":"An approach via fractional analysis to non-linearity induced by coarse-graining in space","volume":"11","author":"Jumarie","year":"2010","journal-title":"Nonlinear Anal. RWA"},{"key":"10.1016\/j.camwa.2012.01.073_br000105","first-page":"32","article-title":"Composition functionals in fractional calculus of variations","volume":"1","author":"Malinowska","year":"2010","journal-title":"Commun. Frac. Calc."},{"issue":"3","key":"10.1016\/j.camwa.2012.01.073_br000110","doi-asserted-by":"crossref","first-page":"1507","DOI":"10.1016\/j.na.2011.01.010","article-title":"Fractional variational calculus with classical and combined Caputo derivatives","volume":"75","author":"Odzijewicz","year":"2012","journal-title":"Nonlinear Anal."},{"issue":"5","key":"10.1016\/j.camwa.2012.01.073_br000115","doi-asserted-by":"crossref","first-page":"1852","DOI":"10.1016\/j.camwa.2009.08.029","article-title":"Generalized variational problems and Euler\u2013Lagrange equations","volume":"59","author":"Agrawal","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"10.1016\/j.camwa.2012.01.073_br000120","series-title":"Generalized Fractional Calculus and Applications","volume":"vol. 301","author":"Kiryakova","year":"1994"},{"year":"1999","series-title":"Fractional Differential Equations","author":"Podlubny","key":"10.1016\/j.camwa.2012.01.073_br000125"},{"key":"10.1016\/j.camwa.2012.01.073_br000130","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1007\/BF01171116","article-title":"Some properties of fractional integrals I","volume":"28","author":"Hardy","year":"1928","journal-title":"Math. Z."},{"issue":"1","key":"10.1016\/j.camwa.2012.01.073_br000135","doi-asserted-by":"crossref","first-page":"140","DOI":"10.2307\/44152475","article-title":"Functions that have no first order derivative might have fractional derivatives of all orders less than one","volume":"20","author":"Ross","year":"1994","journal-title":"Real Anal. Exchange"},{"issue":"3","key":"10.1016\/j.camwa.2012.01.073_br000140","doi-asserted-by":"crossref","first-page":"1009","DOI":"10.1016\/j.na.2011.02.028","article-title":"Fractional variational problems depending on indefinite integrals","volume":"75","author":"Almeida","year":"2012","journal-title":"Nonlinear Anal."},{"issue":"3","key":"10.1016\/j.camwa.2012.01.073_br000145","doi-asserted-by":"crossref","first-page":"1490","DOI":"10.1016\/j.cnsns.2010.07.016","article-title":"Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives","volume":"16","author":"Almeida","year":"2011","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"10.1016\/j.camwa.2012.01.073_br000150","series-title":"Variational Analysis and Generalized Differentiation. I","volume":"vol. 330","author":"Mordukhovich","year":"2006"},{"key":"10.1016\/j.camwa.2012.01.073_br000155","series-title":"Variational Analysis and Generalized Differentiation. II","volume":"vol. 331","author":"Mordukhovich","year":"2006"},{"issue":"6","key":"10.1016\/j.camwa.2012.01.073_br000160","doi-asserted-by":"crossref","first-page":"526","DOI":"10.2307\/30037526","article-title":"The isoperimetric problem","volume":"112","author":"Bl\u00e5sj\u00f6","year":"2005","journal-title":"Amer. Math. Monthly"},{"issue":"4","key":"10.1016\/j.camwa.2012.01.073_br000165","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1023\/B:OPTE.0000042033.33845.4c","article-title":"Complementary extremum principles for isoperimetric optimization problems","volume":"5","author":"Curtis","year":"2004","journal-title":"Optim. Eng."},{"issue":"2","key":"10.1016\/j.camwa.2012.01.073_br000170","doi-asserted-by":"crossref","first-page":"674","DOI":"10.1016\/j.jmaa.2009.06.029","article-title":"H\u00f6lderian variational problems subject to integral constraints","volume":"359","author":"Almeida","year":"2009","journal-title":"J. Math. Anal. Appl."},{"issue":"6","key":"10.1016\/j.camwa.2012.01.073_br000175","doi-asserted-by":"crossref","first-page":"951","DOI":"10.1177\/1077546309103268","article-title":"Isoperimetric problems on time scales with nabla derivatives","volume":"15","author":"Almeida","year":"2009","journal-title":"J. Vib. Control"},{"key":"10.1016\/j.camwa.2012.01.073_br000180","series-title":"Nonlinear Analysis and Optimization II","first-page":"123","article-title":"Isoperimetric problems of the calculus of variations on time scales","volume":"vol. 514","author":"Ferreira","year":"2010"},{"issue":"7","key":"10.1016\/j.camwa.2012.01.073_br000185","doi-asserted-by":"crossref","first-page":"072703","DOI":"10.1063\/1.2736519","article-title":"Stochastic embedding of dynamical systems","volume":"48","author":"Cresson","year":"2007","journal-title":"J. Math. Phys."},{"issue":"2","key":"10.1016\/j.camwa.2012.01.073_br000190","doi-asserted-by":"crossref","first-page":"217","DOI":"10.12775\/TMNA.2009.015","article-title":"Constants of motion for non-differentiable quantum variational problems","volume":"33","author":"Cresson","year":"2009","journal-title":"Topol. Methods Nonlinear Anal."},{"key":"10.1016\/j.camwa.2012.01.073_br000195","unstructured":"L. Bourdin, J. Cresson, I. Greff, P. Inizan, Variational integrators on fractional Lagrangian systems in the framework of discrete embeddings, 2011. arXiv:1103.0465\u00a0[math.DS]."},{"key":"10.1016\/j.camwa.2012.01.073_br000200","unstructured":"P. Inizan, Dynamique fractionnaire pour le chaos Hamiltonien, Th\u00e8se de Doctorat, Ph.D. Thesis, Institut de M\u00e9canique C\u00e9leste et de Calcul des \u00c9ph\u00e9m\u00e9ride, Paris, 2010."},{"key":"10.1016\/j.camwa.2012.01.073_br000205","unstructured":"M. Klimek, On solutions of linear fractional differential equations of a variational type, The Publishing Office of Czestochowa University of Technology, Czestochowa, 2009."},{"issue":"6","key":"10.1016\/j.camwa.2012.01.073_br000210","doi-asserted-by":"crossref","first-page":"063507","DOI":"10.1063\/1.3152608","article-title":"Solution of the fractional Langevin equation and the Mittag-Leffler functions","volume":"50","author":"Camargo","year":"2009","journal-title":"J. Math. Phys."},{"issue":"12","key":"10.1016\/j.camwa.2012.01.073_br000215","doi-asserted-by":"crossref","first-page":"1816","DOI":"10.1016\/j.aml.2009.07.002","article-title":"Calculus of variations with fractional derivatives and fractional integrals","volume":"22","author":"Almeida","year":"2009","journal-title":"Appl. Math. Lett."},{"year":"1998","series-title":"Handbook of Integral Equations","author":"Polyanin","key":"10.1016\/j.camwa.2012.01.073_br000220"}],"container-title":["Computers & Mathematics with Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0898122112001058?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0898122112001058?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2020,7,2]],"date-time":"2020-07-02T17:46:46Z","timestamp":1593712006000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0898122112001058"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,11]]},"references-count":44,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2012,11]]}},"alternative-id":["S0898122112001058"],"URL":"https:\/\/doi.org\/10.1016\/j.camwa.2012.01.073","relation":{},"ISSN":["0898-1221"],"issn-type":[{"type":"print","value":"0898-1221"}],"subject":[],"published":{"date-parts":[[2012,11]]}}}