This paper presents an efficient approximation schemes for the numerical solution of a fractional variational problem (FVP) and fractional optimal control problem (FOCP). As basis function for the trial solution, we employ the shifted Jacobi orthonormal polynomial. We state and derive a new operational matrix of right-sided Caputo fractional derivative of such polynomial. The new methodology of the present schemes is based on the derived operational matrix with the help of the Gauss–Lobatto quadrature formula and the Lagrange multiplier technique. Accordingly, the aforementioned problems are reduced into systems of algebraic equations. The error bound for the operational matrix of right-sided Caputo derivative is analyzed. In addition, the convergence of the proposed approaches is also included. The results obtained through numerical procedures and comparing our method with other methods demonstrate the high accuracy and powerful of the present approach.
Issue Section:
Research Papers
References
1.
Oldham
, K. B.
, and Spanier
, J.
, 1974
, Fractional Calculus: Theory and Applications, Differentiation and Integration to Arbitrary Order
, Academic Press
, London
.2.
Podlubny
, I.
, 1999
, Fractional Differential Equations
, Academic Press
, San Diego, CA
.3.
Hilfer
, R.
, 2000
, Applications of Fractional Calculus in Physics
, World Scientific
, Hackensack, NJ
.4.
Abel
, N. H.
, 1965
, Oeuvres completes de Niels Henrik Abel, Christiana: Imprimerie de Grondahl and Son
, Johnson Reprint Corporation, New York, Vol. VIII
, p. 621
.5.
El-Nabulsi
, R. A.
, and Torres
, D. F. M.
, 2007
, “Necessary Optimality Conditions for Fractional Action-Like Integrals of Variational Calculus With Riemann-Liouville Derivatives of Order (α, β)
,” Math. Methods Appl. Sci.
, 30
(15
), pp. 1931
–1939
.6.
Frederico
, G. S. F.
, and Torres
, D. F. M.
, 2008
, “Fractional Conservation Laws in Optimal Control Theory
,” Nonlinear Dyn.
, 53
(3
), pp. 215
–222
.7.
Baleanu
, D.
, Muslih
, S. I.
, Rabei
, E. M.
, Alireza Golmankhaneh
, K.
, and Ali Golmankhaneh
, K.
, 2011
, “On Fractional Hamiltonian Systems Possessing First-Class Constraints Within Caputo Derivatives
,” Rom. Rep. Phys.
, 63
(1
), pp. 3
–8
.8.
Jumarie
, G.
, 2007
, “Fractional Hamilton-Jacobi Equation for the Optimal Control of Nonrandom Fractional Dynamics With Fractional Cost Function
,” J. Appl. Math. Comput.
, 23
(1–2), pp. 215
–228
.9.
Jarad
, F.
, Abdeljawad
, T.
, and Dumitru
, B.
, 2010
, “Fractional Variational Principles With Delay Within Caputo Derivatives
,” Rep. Math. Phys.
, 65
(1
), pp. 17
–28
.10.
Bauer
, P. S.
, 1931
, “Dissipative Dynamical Systems—I
,” Proc. Natl. Acad. Sci. U. S. A.
, 17
(5
), pp. 311
–314
.11.
Riewe
, F.
, 1996
, “Nonconservative Lagrangian and Hamiltonian Mechanics
,” Phys. Rev. E
, 53
(2
), pp. 1890
–1899
.12.
Jumarie
, G.
, 2008
, “Stock Exchange Fractional Dynamics Defined as Fractional Exponential Growth Driven by (Usual) Gaussian White Noise. Application to Fractional Black-Scholes Equations
,” Insur.: Math. Econ.
, 42
(1
), pp. 271
–287
.13.
Hu
, F.
, Zhu
, W. Q.
, and Chen
, L. C.
, 2012
, “Stochastic Fractional Optimal Control of Quasi-Integrable Hamiltonian System With Fractional Derivative Damping
,” Nonlinear Dyn.
, 70
(2), pp. 1459
–1472
.14.
Muslih
, S. I.
, and Baleanu
, D.
, 2005
, “Hamiltonian Formulation of Systems With Linear Velocities Within Riemann-Liouville Fractional Derivatives
,” J. Math. Anal. Appl.
, 304
(2
), pp. 599
–606
.15.
Frederico
, G. S. F.
, and Torres
, D. F. M.
, 2010
, “Fractional Noether’s Theorem in the Riesz-Caputo Sense
,” Appl. Math. Comput.
, 217
(3
), pp. 1023
–1033
.16.
Saadatmandi
, A.
, and Dehghan
, M.
, 2010
, “A New Operational Matrix for Solving Fractional-Order Differential Equations
,” Comput. Math. Appl.
, 59
(3
), pp. 1326
–1336
.17.
Bhrawy
, A. H.
, Zaky
, M. A.
, and Machado
, J. A. T.
, 2017
, “Numerical Solution of the Two-Sided Space-Time Fractional Telegraph Equation Via Chebyshev Tau Approximation
,” J. Optim. Theory Appl.
, 174
(1
), pp. 321
–341
.18.
Ezz-Eldien
, S. S.
, 2016
, “New Quadrature Approach Based on Operational Matrix for Solving a Class of Fractional Variational Problems
,” J. Comput. Phys.
, 317
, pp. 362
–381
.19.
Ezz-Eldien
, S. S.
, Bhrawy
, A. H.
, and El-Kalaawy
, A. A.
, 2017
, “Direct Numerical Method for Isoperimetric Fractional Variational Problems Based on Operational Matrix
,” J. Vib. Control
, epub.20.
Ezz-Eldien
, S. S.
, Hafez
, R. M.
, Bhrawy
, A. H.
, Baleanu
, D.
, and El-Kalaawy
, A. A.
, 2017
, “New Numerical Approach for Fractional Variational Problems Using Shifted Legendre Orthonormal Polynomials
,” J. Optim. Theory Appl.
, 174
(1
), pp. 295
–320
.21.
Doha
, E. H.
, Bhrawy
, A. H.
, Baleanu
, D.
, Ezz-Eldien
, S. S.
, and Hafez
, R. M.
, 2015
, “An Efficient Numerical Scheme Based on the Shifted Orthonormal Jacobi Polynomials for Solving Fractional Optimal Control Problems
,” Adv. Differ. Equations
, 2015
(15
), p. 17
.22.
Bhrawy
, A. H.
, and Ezz-Eldien
, S. S.
, 2016
, “A New Legendre Operational Technique for Delay Fractional Optimal Control Problems
,” Calcolo
, 53
(4), pp. 521
–543
.23.
Kreyszig
, E.
, 1978
, Introductory Functional Analysis With Applications
, Wiley
, New York
.24.
Rivlin
, T. J.
, 1981
, An Introduction to the Approximation of Functions
, Dover Publications
, Mineola, NY
.25.
Canuto
, C.
, Hussaini
, M. Y.
, Quarteroni
, A.
, and Zang
, T. A.
, 2006
, Spectral Methods Fundamentals in Single Domains
, Springer
, Berlin
.26.
Dehghan
, M.
, Hamedi
, E. A.
, and Arab
, H. K.
, 2016
, “A Numerical Scheme for the Solution of a Class of Fractional Variational and Optimal Control Problems Using the Modified Jacobi Polynomials
,” J. Vib. Control
, 22
(6
), pp. 1547
–1559
.Copyright © 2018 by ASME
You do not currently have access to this content.
