![[PDF]](/pb-assets/Icons/adobe-pdf-icon-1498649896243.png){kind=icon}
Abstract
The aim of this paper is to generalize the Euler–Lagrange equation obtained by Almeida et al., where fractional variational problems for Lagrangians, depending on fractional operators and depending on indefinite integrals, were studied. The new problem that we address here is for cost functionals, where the interval of integration is not the whole domain of the admissible functions, but a proper subset of it. Furthermore, we present a numerical method, based on Jacobi polynomials for solving this problem.
References
| Agrawal, OP (2007a) Generalized Euler–Lagrange equations and transversality conditions for FVPs in terms of the Caputo derivative. Journal of Vibration and Control 13: 1217–1237. Google Scholar | SAGE Journals | ISI | |
| Agrawal, OP (2007b) Fractional variational calculus in terms of Riesz fractional derivatives. Journal of Physics A 40: 6287–6303. Google Scholar | Crossref | |
| Almeida, R, Torres, DFM (2011) Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Communication in Nonlinear Sciences and Numerical Simulations 16: 1490–1500. Google Scholar | Crossref | ISI | |
| Almeida, R, Pooseh, S, Torres, DFM (2012) Fractional variational problems depending on indefinite integrals. Nonlinear Analysis 75: 1009–1025. Google Scholar | Crossref | ISI | |
| Atanacković, TM, Konjik, S, Pilipović, S (2008) Variational problems with fractional derivatives: Euler–Lagrange equations. Journal of Physics A 41: 095201–095201. Google Scholar | Crossref | |
| Canuto, C, Quarteroni, A, Hussaini, MY, Zang, TA (2007) Spectral methods. Evolution to Complex Geometries and Applications to Fluid Dynamics, Berlin, Heidelberg: Springer-Verlag. Google Scholar | |
| Esmaeili, S, Shamsi, M (2011) A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations. Communication in Nonlinear Sciences and Numerical Simulations 16: 3646–3654. Google Scholar | Crossref | ISI | |
| Gautschi, W (2004) Orthogonal Polynomials: Computation and Approximation, New York: Oxford University Press. Google Scholar | |
| Gregory, J (2008) Generalizing variational theory to include the indefinite integral, higher derivatives, and a variety of means as cost variables. Methods and Applications of Analysis 15: 427–435. Google Scholar | Crossref | |
| Jarad, F, Abdeljawad, T, Baleanu, D (2010) Fractional variational principles with delay within Caputo derivatives. Reports on Mathematical Physics 65: 17–28. Google Scholar | Crossref | ISI | |
| Kilbas, AA, Srivastava, HM, Trujillo, JJ (2006) Theory and Applications of Fractional Differential Equations. (North-Holland Mathematics Studies, Vol. 204), Amsterdam: Elsevier. Google Scholar | |
| Malinowska, AB, Torres, DFM (2010) Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative. Computers and Mathematics with Applications 59: 3110–3116. Google Scholar | Crossref | ISI | |
| Martins, N, Torres, DFM (2011) Generalizing the variational theory on time scales to include the delta indefinite integral. Computers and Mathematics with Applications 61: 2424–2435. Google Scholar | Crossref | ISI | |
| Trefethen, LN (2000) Spectral method in matlab, Philadelphia, PA: SIAM. Google Scholar | Crossref | |
| van Brunt, B (2004) The Calculus of Variations, New York: Universitext/Springer. Google Scholar | Crossref |
