zbMATH — the first resource for mathematics

Direct solution of a type of constrained fractional variational problems via an adaptive pseudospectral method. (English) Zbl 1311.65087
Summary: This paper presents an adaptive Legendre-Gauss pseudospectral method for solving a type of constrained fractional variational problems (FVPs). The fractional derivative is defined in the Caputo sense. In the presented method, by dividing the domain of the problem into a uniform mesh the given FVP reduces to a nonlinear mathematical programming problem, and there is no need to solve the complicated fractional Euler-Lagrange equations. The method developed in this paper adjusts both the mesh spacing and the number of collocation points on each subinterval in order to improve the accuracy. The method is easy to implement and yields very accurate results. Some error estimates and convergence properties of the method are discussed. Numerical examples are included to confirm the efficiency and convergence of the proposed method.

65K10 Numerical optimization and variational techniques
49J15 Existence theories for optimal control problems involving ordinary differential equations
49M37 Numerical methods based on nonlinear programming
Full Text: DOI
[1] Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, (1999), Academic Press New York · Zbl 0924.34008
[2] Das, S., Functional fractional calculus for system identification and controls, (2008), Springer New York · Zbl 1154.26007
[3] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., (Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, (2006), Elsevier Science B.V. Amsterdam)
[4] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach Yverdon · Zbl 0818.26003
[5] Riewe, F., Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E (3), 53, 2, 1890-1899, (1996)
[6] Malinowska, A. B.; Torres, D. F.M., Introduction to the fractional calculus of variations, (2012), Imp. Coll. Press London · Zbl 1218.49026
[7] Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J. J., (Fractional Calculus, Series on Complexity, Nonlinearity and Chaos, vol. 3, (2012), World Scientific Publishing Co. Pvt. Ltd. Hackensack, NJ)
[8] Tikhomirov, V. M., Store about maxima and minima, (1990), American Mathematics Society Providence, RI · Zbl 0746.49001
[9] Elsgolts, L., Differential equations and calculus of variations, (1977), Mir Moscow, (translated from the Russian by G. Yankovsky)
[10] Agrawal, O. P.; Muslih, S. I.; Baleanu, D., Generalized variational calculus in terms of multi-parameters fractional derivatives, Commun. Nonlinear Sci. Numer. Simul., 16, 12, 4756-4767, (2011) · Zbl 1236.49030
[11] Almeida, R.; Pooseh, S.; Torres, D. F.M., Fractional variational problems depending on indefinite integrals, Nonlinear Anal., 75, 3, 1009-1025, (2012) · Zbl 1236.49042
[12] Atanacković, T. M.; Konjik, S.; Pilipović, S., Variational problems with fractional derivatives: Euler-Lagrange equations, J. Phys. A: Math. Theor., 41, 9, 095201, (2008) · Zbl 1175.49020
[13] Frederico, G. S.F.; Torres, D. F.M., Fractional noether’s theorem in the Riesz-Caputo sense, Appl. Math. Comput., 217, 3, 1023-1033, (2010) · Zbl 1200.49019
[14] Jelicic, Z. D.; Petrovacki, N., Optimality conditions and a solution scheme for fractional optimal control problems, Struct. Multidiscip. Optim., 38, 6, 571-581, (2009) · Zbl 1274.49035
[15] Klimek, M., Fractional sequential mechanics models with symmetric fractional derivative, Czech. J. Phys., 51, 12, 1348-1354, (2001) · Zbl 1064.70507
[16] Odzijewicz, T.; Malinowska, A. B.; Torres, D. F.M., Fractional calculus of variations in terms of a generalized fractional integral with applications to physics, Abstr. Appl. Anal., 2012, (2012), Art. ID 871912, 24 pages, http://dx.doi.org/10.1155/2012/871912 · Zbl 1242.49019
[17] Agrawal, O. M.P., Formulation of Euler-Lagrange equations for fractional variational problems, J. Math. Anal. Appl., 272, 368-379, (2002) · Zbl 1070.49013
[18] Almeida, R.; Torres, D. F.M., Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives, Commun. Nonlinear Sci. Numer. Simul., 16, 1490-1500, (2011) · Zbl 1221.49038
[19] Almeida, R.; Torres, D. F.M., Calculus of variations with fractional derivatives and fractional integrals, Appl. Math. Lett., 22, 1816-1820, (2009) · Zbl 1183.26005
[20] Agrawal, O. M.P., Generalized variational problems and Euler-Lagrange equations, Comput. Math. Appl., 59, 1852-1864, (2010) · Zbl 1189.49029
[21] Agrawal, O. M.P., Generalized Euler-Lagrange equations and transversality conditions for FVPs in terms of Caputo derivative, J. Vib. Control, 13, 1217-1237, (2007) · Zbl 1158.49006
[22] Yousefi, S. A.; Dehghan, M.; Lotfi, A., Generalized Euler-Lagrange equations for fractional variational problems with free boundary conditions, Comput. Math. Appl., 62, 987-995, (2011) · Zbl 1228.49016
[23] Malinowska, A. B.; Torres, D. F.M., Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative, Comput. Math. Appl., 59, 3110-3116, (2010) · Zbl 1193.49023
[24] Wang, D.; Xiao, A., Fractional variational integrators for fractional variational problems, Commun. Nonlinear Sci. Numer. Simul., 17, 602-610, (2012) · Zbl 1239.49028
[25] Wang, D.; Xiao, A., Fractional variational integrators for fractional Euler-Lagrange equations with holonomic constraints, Commun. Nonlinear Sci. Numer. Simul., 18, 905-914, (2013) · Zbl 1261.35149
[26] Bourdin, L.; Cresson, J.; Greff, I.; Inizan, P., Variational integrator for fractional Euler-Lagrange equations, Appl. Numer. Math., 71, 14-23, (2013) · Zbl 1284.65183
[27] Agrawal, O. M.P.; Mehedi Hasan, M.; Tangpong, X. W., A numerical scheme for a class of parametric problem of fractional variational calculus, J. Comput. Nonlinear Dyn., 7, 2, 021005, (2012), 6 pages, http://dx.doi.org/10.1115/1.4005464
[28] Almeida, R.; Torres, D. F.M., Leitmann’s direct method for fractional optimization problems, Appl. Math. Comput., 217, 3, 956-962, (2010) · Zbl 1200.65049
[29] Lotfi, A.; Yousefi, S. A., A numerical technique for solving a class of fractional variational problems, J. Comput. Appl. Math., 237, 633-643, (2013) · Zbl 1253.65105
[30] Agrawal, O. M.P., A general finite element formulation for fractional variational problems, J. Math. Anal. Appl., 337, 1-12, (2008) · Zbl 1123.65059
[31] Pooseh, S.; Almeida, R.; Torres, D. F.M., Discrete direct methods in the fractional calculus of variations, Comput. Math. Appl., 66, 5, 668-676, (2013) · Zbl 1350.49033
[32] Pooseh, S.; Almeida, R.; Torres, D. F.M., Approximation of fractional integrals by means of derivatives, Comput. Math. Appl., 64, 10, 3090-3100, (2012) · Zbl 1268.41024
[33] Pooseh, S.; Almeida, R.; Torres, D. F.M., Numerical approximations of fractional derivatives with applications, Asian J. Control, 15, 3, 698-712, (2013) · Zbl 1327.93165
[34] Pooseh, S.; Almeida, R.; Torres, D. F.M., Fractional order optimal control problems with free terminal time, J. Ind. Manag. Optim., 10, 2, 363-381, (2014) · Zbl 1278.26013
[35] Pooseh, S.; Almeida, R.; Torres, D. F.M., A discrete time method to the first variation of fractional order variational functionals, Cent. Eur. J. Phys., 11, 10, 1262-1267, (2013)
[36] Maleki, M.; Mashali-Firouzi, M., A numerical solution of problems in calculus of variation using direct method and nonclassical parameterization, J. Comput. Appl. Math., 234, 1364-1373, (2010) · Zbl 1189.65132
[37] Maleki, M.; Hashim, I.; Abbasbandy, S., Pseudospectral methods based on nonclassical orthogonal polynomials for solving nonlinear variational problems, Int. J. Comput. Math., 91, 7, 1552-1573, (2014) · Zbl 1304.49064
[38] Garg, D.; Ptterson, M.; Hager, W. W.; Rao, A. V.; Benson, D. A., A unified framework for the numerical solution of optimal control problems using pseudospectral methods, Automatica, 46, 1843-1851, (2010) · Zbl 1219.49028
[39] Elnagar, G. N.; Kazemi, M. A., Pseudospectral Legendre-based optimal computation of nonlinear constrained variational problems, J. Comput. Appl. Math., 88, 363-375, (1997) · Zbl 0898.65032
[40] Fornberg, B., A practical guide to pseudospectral methods, (1998), Cambridge University Press · Zbl 0912.65091
[41] Canuto, C.; Quarteroni, A.; Hussaini, M. Y.; Zang, T. A., Spectral methods: fundamentals in single domains, (2006), Springer · Zbl 1093.76002
[42] Trefethen, L. N., Spectral methods in MATLAB, (2000), SIAM Philadelphia · Zbl 0953.68643
[43] Darby, C. L.; Hager, W. W.; Rao, A. V., An \(h p\)-adaptive pseudospectral method for solving optimal control problems, Optim. Control Appl. Methods, 32, 476-502, (2011) · Zbl 1266.49066
[44] Maleki, M.; Hashim, I., Adaptive pseudospectral methods for solving constrained linear and nonlinear time-delay optimal control problems, J. Franklin Inst., 351, 811-839, (2014) · Zbl 1293.49086
[45] Maleki, M.; Hashim, I.; Tavassoli Kajani, M.; Abbasbandy, S., An adaptive pseudospectral method for fractional order boundary value problems, Abstr. Appl. Anal., 2012, (2012), Art. ID 381708, 19 pages, http://dx.doi.org/10.1155/2012/381708 · Zbl 1261.34009
[46] Boggs, P. T.; Kearsley, A. J.; Tolle, J. W., A global convergence analysis of an algorithm for large-scale nonlinear optimization problems, SIAM J. Optim., 9, 4, 833-862, (1999) · Zbl 0986.90078
[47] Askey, R., Mean convergence of orthogonal series and lagnange interpolation, Acta Math. Acad. Sci. Hungar., 23, 71-85, (1972) · Zbl 0253.41003
[48] Kantorovich, L. V.; Ahilov, G. P., Functional analysis, (1982), Pergamon Press Oxford
[49] Guo, B. Y.; Wang, L., Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces, J. Approx. Theory, 128, 1-41, (2004) · Zbl 1057.41003
[50] Guo, B. Y.; Wang, Z. Q., Legendre-Gauss collocation methods for ordinary differential equations, Adv. Comput. Math., 30, 249-280, (2009) · Zbl 1162.65375
[51] Marzban, H. R.; Razzaghi, M., Rationalized Haar approach for nonlinear constrained optimal control problems, Appl. Math. Model., 34, 174-183, (2010) · Zbl 1185.49032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.