Discrete Chebyshev Polynomials for Solving Fractional Variational Problems
Abstract
In the current study, a general formulation of the discrete Chebyshev polynomials is given. The operational matrix of fractional integration for these discrete polynomials is also derived. Then, a numerical scheme based on the discrete Chebyshev polynomials and their operational matrix has been developed to solve fractional variational problems. In this method, the need for using Lagrange multiplier during the solution procedure is eliminated. The performance of the proposed scheme is validated through some illustrative examples. Moreover, the obtained numerical results were compared to the previously acquired results by the classical Chebyshev polynomials. Finally, a comparison for the required CPU time is presented, which indicates more efficiency and less complexity of the proposed method.References
P. K. Sahu, and S. S. Ray, Comparison for accurate solutions of nonlinear Hammerstein fuzzy integral equations, Mathematical Communications, vol. 21, no. 2, pp. 283–299, 2016.
F. Mohammadi, S. T. Mohyud-Din, A fractional-order Legendre collocation method for solving the Bagley-Torvik equations. Adv. Differ. Equ. vol. 269, pp. 269, 2016.
H. Khosravian-Arab, R. Almeida, Numerical solution for fractional variational problems using the Jacobi polynomials, Appl Math Model. vol. 39, no. 21, pp. 6461–6470, 2015.
S. S. Ezz-Eldien, R. M. Hafez, A. H. Bhrawy, D. Baleanu, A. A. El-Kalaawy, New numerical approach for fractional variational problems using shifted Legendre orthonormal polynomials, J. Optimiz. Theory App., vol. 174, no. 1, pp. 295–320, 2017.
F. Mohammadi, C. Cattani, A generalized fractional-order Legendre wavelet Tau method for solving fractional differential equations, J. Comput. Appl. Math., 2017.
M. A. Zaky, E. H. Doha, T. M. Taha, and D. Baleanu, New recursive approximations for variable-order fractional operators with applications, Mathematical Modelling and Analysis, vol. 23, no. 2, pp. 227–239, 2018.
M. A. Zaky, A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations, Comp. Appl. Math. vol. 37, pp. 3525–3538, 2018.
D. Xiu, G. E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM journal on scientific computing, vol. 24, no. 2, pp. 619–644, 2002.
R. Goertz, P. Öffner, On Hahn polynomial expansion of a continuous function of bounded variation, ArXiv e-prints: arXiv:1610.06748, 2016.
B. H. S. Asli, J. Flusser, New discrete orthogonal moments for signal analysis, Signal Processing, vol. 141, pp. 57–73, 2017.
R. Almeida, D. F .M. Torres, Calculus of variations with fractional derivatives and fractional integrals, Appl. Math. Lett. vol. 22, no. 12, pp.1816–1820, 2009.
A. Malinowska, D. Torres, Fractional calculus of variations for a combined Caputo derivative, Fract. Calc. Appl. Anal. vol. 14, no. 4, pp. 523–537, 2011.
B. van Brunt, The Calculus of Variations, Springer-Verlag, New York, 2004.
M.A.Zaky,andJ.T.Machado,On the formulation and numerical simulation of distributed-order fractional optimal control problems,Commun. Nonlinear. Sci. Numer. Simul., vol. 52, pp. 177–189, 2017.
D. Baleanu, S.I. Muslih, Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives, Phys. Scripta, vol. 72, no. 3, pp.119–121, 2005.
S.S. Ezz-Eldien, E.H. Doha, A.H. Bhrawy, A.A. El-Kalaawy, J.A.T. Machado, A new operational approach for solving fractional variational problems depending on indefinite integrals, Commun Nonlinear Sci Numer Simul. vol. 57, pp. 246–263, 2018.
A. B. Malinowska, T. Odzijewicz, D. F. Torres, Advanced methods in the fractional calculus of variations, Springer, 2015.
F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E., vol. 53, no. 2, pp. 1890–1899, 1996.
F. Riewe, Mechanics with fractional derivatives, Phys. Rev. E., vol. 55, no. 3, pp. 3581–3592, 1997.
I. Malmir, Optimal control of linear time-varying systems with state and input delays by Chebyshev wavelets, Statistics, Optimization & Information Computing, vol. 5, no. 4, pp. 302-324 2017. http://dx.doi.org/10.19139/soic.v5i4.341
M. Barrios, G. Reyero, An Euler-Lagrange Equation only Depending on Derivatives of Caputo for Fractional Variational Problems with Classical Derivatives, Statistics, Optimization & Information Computing, vol. 8, no. 2, pp. 590C601, 2020. https://doi.org/10.19139/soic-2310-5070-865
S. G. Samko, A.A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Langhorne, 1993.
S. S. Ray, A. Atangana, S. C. Noutchie, M. Kurulay, N. Bildik and A. Kilicman, Fractional calculus and its applications in applied mathematics and other sciences, Mathematical Problems in Engineering, 2014.
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
N. Gogin and M. Hirvensalo, On the generating function of Discrete Chebyshev Polynomials, Journal of Mathematical Sciences, vol. 224, no. 2, 2017.
A. F. Nikiforov, S. K. Suslov and V. B. Uvarov, Classical orthogonal polynomials of a discrete variable, Springer Berlin Heidelberg, 1991.
H.M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer, Boston, 2001.
J. Riordan, An Introduction to Combinatorial Analysis, New York: Wiley, 1980.
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