International Journal of Nonlinear Analysis and Applications
A numerical method for solving variable order fractional optimal control problems
Document Type : Research Paper
Authors
1 Department of Mathematics , Payame Noor University (PNU), P.O. Box 19395-4697, Tehran, Iran
2 Department of Mathematics, University of Mazandaran, Babolsar, Iran
3 Department of Mathematical Sciences, University of South Africa, UNISA 0003, South Africa
Abstract
This study is devoted to introducing a computational technique based on Bernstein polynomials to solve variable order fractional optimal control problems (VO-FOCPs). This class of problems generated by dynamical systems describe with variable order fractional derivatives in the Caputo sense. In the proposed method, the Bernstein operational matrix of the fractional variable-order derivatives will be derived. Then, this matrix is used to obtain an approximate solution to mentioned problems. With the use of Gauss-Legendre quadrature rule and the mentioned operational matrix, the considered VO-FOCPs are reduced to a system of equations that are solved to get approximate solutions. The obtained results show the accuracy of the numerical technique.
Keywords
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diffusion limit in Mathematical Finance, Trends Math. (2001) 171–180.
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fractional telegraph equation using transcendental Bernstein series, Engin. Comput. 36 (2020) 867–878.
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J. Opt. Theory Appl. 163(3) (2014) 884–899.
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functions, J. Vib. Cont. 24(9) (2018) 1621–1631.
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optimal control problems, Nonlinear Dyn. 88(2) (2017) 1013–1026.
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modelling, Int. J. Diff. Equ. 2010 (2010) Article ID 846107.
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sedimenting particle, Phys. D: Nonlinear Phenom. 240(13) (2011) 1111–1118.
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Funct. 1(4) (1993) 277–300.
[33] Y. Shekari, A. Tayebi and M. H. Heydari, A meshfree approach for solving 2D variable-order fractional nonlinear
diffusion-wave equation, Comput. Meth. Appl. Mech. Engin. 350 (2019) 154–168.
[34] J. J. Shyu, S. C. Pei and C. H. Chan, An iterative method for the design of variable fractional-order FIR
differintegrators, Signal Proces. 89(3) (2009) 320–327.
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equation, Int. J. Bifurc. Chaos 22(4) (2012).
[36] H. Sun, W. Chen, H. Wei, and Y. Chen, A comparative study of constant-order and variable-order fractional
models in characterizing memory property of systems, The European Physical Journal Special Topics, 193 (2011),
185–192.
[37] H. G. Sun, Y. Zhang, W. Chen, and D. M. Reeves, Use of a variable-index fractional-derivative model to capture
transient dispersion in heterogeneous media, Journal of Contaminant Hydrology, 157 (2014), 47–58.
[38] L. F. Wang, Y. P. Ma, and Y. Q. Yang, Legendre polynomials method for solving a class of variable order
fractional differential equation, Computer Modeling in Engineering and Sciences, 101(2) (2014), 97–111.
[39] W. Yonthanthum, A. Rattana and M. Razzaghi, An approximate method for solving fractional optimal control
problems by the hybrid of block-pulse functions and Taylor polynomials, Opt. Cont. Appl. Meth. 39(2) (2018)
873–887.
[40] W. K. Zahra and M. M. Hikal, Non standard finite difference method for solving variable order fractional optimal
control problems, J. Vib. Cont. 23(6) (2017) 948–958.
[41] M. A. Zaky, E. H. Doha, T. M. Taha and D. Baleanu, New recursive approximations for variable-order fractional
operators with applications, Math. Model. Anal. 23(2) (2018) 227–239.
Nonlinearity and Chaos, World Scientific, 2012.
[2] A.H. Bhrawy, S.S. Ezz-Eldien, E.H. Doha, M.A. Abdelkawy and D. Baleanu, Solving fractional optimal control
problems within a Chebyshev-Legendre operational technique, Int. J. Cont. 90(6) (2017) 1230–1244.
[3] A. H. Bhrawy and M.A. Zaky, Numerical algorithm for the variable-order Caputo fractional functional differential
equation, Nonlinear Dyn. 85 (2016) 1815–1823.
[4] A.H. Bhrawy, and M.A. Zaky, Numerical simulation for two-dimensional variable-order fractional nonlinear cable
equation, Nonlinear Dyn. 80 (2015) 101–116.
[5] Y.M. Chen, D. Y. Liu, B. Li and Y. Sun, Numerical solution for the variable order linear cable equation with
Bernstein polynomials, Appl. Math. Comput. 238 (2014) 329–341.
[6] Y.M. Chen, Y.Q. Wei, D. Y. Liu and H. Yu, Numerical solution for a class of nonlinear variable order fractional
differential equations with Legendre wavelets, Appl. Math. Lett. 46 (2015) 83–88.
[7] C.F.M. Coimbra, Mechanics with variable-order differential operators, Ann. Phys. 12 (2003) 692–703.
[8] M.S. Dahaghin and H. Hassani, An optimization method based on the generalized polynomials for nonlinear
variable-order time fractional diffusion-wave equation, Nonlinear Dyn. 88 (2017) 1587–1598.
[9] E.H. Doha, M.A. Abdelkawy, A.Z.M. Amin, and D. Baleanu, Spectral technique for solving variable-order
fractional Volterra integro-differential equations, Numerical Meth. Partial Diff. Integral Equ. Integ. Non-integer
Order 34(5) (2018) 1659–1677.
[10] E. H. Doha, A. H. Bhrawy, D. Baleanu, S. S. Ezz-Eldien and R. M. Hafez, An efficient numerical scheme
based on the shifted orthonormal jacobi polynomials for solving fractional optimal control problems, Adv. Diff.
Equ. (2015) 1–17.
[11] R. M. Ganji and H. Jafari, A numerical approach for multi-variable orders differential equations using Jacobi
polynomials, Int. J. Appl. Comput. Math. 5 (2019).
[12] R. M. Ganji and H. Jafari, Numerical solution of variable order integro-differential equations, Adv. Math. Model.
Appl. 4(1) (2019), 64–69.
[13] R. Gorenflo, F. Mainardi, E. Scalas and M. Raberto, Fractional calculus and continuous-time finance III: the
diffusion limit in Mathematical Finance, Trends Math. (2001) 171–180.
[14] H. Hassani, Z. Avazzadeh and J. A. T. Machado, Numerical approach for solving variable-order space-time
fractional telegraph equation using transcendental Bernstein series, Engin. Comput. 36 (2020) 867–878.
[15] M.H. Heydari, A new direct method based on the Chebyshev cardinal functions for variable-order fractional
optimal control problems, J. Franklin Inst. 355(12) (2018) 4970–4995.
[16] M.H. Heydari and Z. Avazzadeh, A new wavelet method for variable-order fractional optimal controls, Asian J.
Cont. 20 (2018) 1804–1817.[17] M. H. Heydari and Z. Avazzadeh, A computational method for solving two-dimensional nonlinear variable-order
fractional optimal control problems, Asian J. Cont. 22 (2020) 1112–1126.
[18] D. Ingman, J. Suzdalnitsky and M. Zeifman, Constitutive dynamic-order model for nonlinear contact phenomena,
ASME J. Appl. Mech. 67 (2000) 383–390.
[19] H. Jafari, and H. Tajadodi, Fractional order optimal control problems via the operational matrices of Bernstein
polynomials, UPB Sci. Bull. Series A: Appl. Math. Phys. 76(3) (2014) 115–128.
[20] H. Jafari, H. Tajadodi and R.M. Ganji, A numerical approach for solving variable order differential equations
based on Bernstein polynomials, Comput. Math. Meth. 1(5) (2019) 1–10.
[21] C. F. Lorenzo and T. T. Hartley, Initialization, conceptualization and application in the generalized fractional
calculus, Crit. Rev. Biomed. Eng. 35(6) (2007) 447–553.
[22] C. F. Lorenzo and T. T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dyn. 29
(2002) 57–98.
[23] A. Lotfi, A combination of variational and penalty methods for solving a class of fractional optimal control
problems, J. Opt. Theory Appl. 174(1) (2017) 65–82.
[24] A. Lotfi, Epsilon penalty method combined with an extension of the Ritz method for solving a class of fractional
optimal control problems with mixed inequality constraints, Appl. Numerical Math. 135 (2019) 497–509.
[25] A. Lotfi, and S. A. Yousefi, Epsilon-Ritz method for solving a class of fractional constrained optimization problems,
J. Opt. Theory Appl. 163(3) (2014) 884–899.
[26] S. Mashayekhi and M. Razzaghi, An approximate method for solving fractional optimal control problems by hybrid
functions, J. Vib. Cont. 24(9) (2018) 1621–1631.
[27] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
[28] H. T. C. Pedro, M. H. Kobayashi, J. M. C. Pereira and C. F. M. Coimbra, Variable order modeling of diffusiveconvective effects on the oscillatory flow past a sphere, IFAC Proc.Vol. 39(11) (2006) 454–459.
[29] K. Rabiei, Y. Ordokhani and E. Babolian, The Boubaker polynomials and their application to solve fractional
optimal control problems, Nonlinear Dyn. 88(2) (2017) 1013–1026.
[30] L.E.S. Ramirez and C.F.M. Coimbra, On the selection and meaning of variable order operators for dynamic
modelling, Int. J. Diff. Equ. 2010 (2010) Article ID 846107.
[31] L.E.S. Ramirez and C. F. M. Coimbra, On the variable order dynamics of the nonlinear wake caused by a
sedimenting particle, Phys. D: Nonlinear Phenom. 240(13) (2011) 1111–1118.
[32] S. Samko and B. Ross, Integration and differentiation to a variable fractional order, Integral Trans. Special
Funct. 1(4) (1993) 277–300.
[33] Y. Shekari, A. Tayebi and M. H. Heydari, A meshfree approach for solving 2D variable-order fractional nonlinear
diffusion-wave equation, Comput. Meth. Appl. Mech. Engin. 350 (2019) 154–168.
[34] J. J. Shyu, S. C. Pei and C. H. Chan, An iterative method for the design of variable fractional-order FIR
differintegrators, Signal Proces. 89(3) (2009) 320–327.
[35] H. Sun, W. Chen, C. Li and Y. Chen, Finite difference schemes for variable-order time fractional diffusion
equation, Int. J. Bifurc. Chaos 22(4) (2012).
[36] H. Sun, W. Chen, H. Wei, and Y. Chen, A comparative study of constant-order and variable-order fractional
models in characterizing memory property of systems, The European Physical Journal Special Topics, 193 (2011),
185–192.
[37] H. G. Sun, Y. Zhang, W. Chen, and D. M. Reeves, Use of a variable-index fractional-derivative model to capture
transient dispersion in heterogeneous media, Journal of Contaminant Hydrology, 157 (2014), 47–58.
[38] L. F. Wang, Y. P. Ma, and Y. Q. Yang, Legendre polynomials method for solving a class of variable order
fractional differential equation, Computer Modeling in Engineering and Sciences, 101(2) (2014), 97–111.
[39] W. Yonthanthum, A. Rattana and M. Razzaghi, An approximate method for solving fractional optimal control
problems by the hybrid of block-pulse functions and Taylor polynomials, Opt. Cont. Appl. Meth. 39(2) (2018)
873–887.
[40] W. K. Zahra and M. M. Hikal, Non standard finite difference method for solving variable order fractional optimal
control problems, J. Vib. Cont. 23(6) (2017) 948–958.
[41] M. A. Zaky, E. H. Doha, T. M. Taha and D. Baleanu, New recursive approximations for variable-order fractional
operators with applications, Math. Model. Anal. 23(2) (2018) 227–239.
)
Volume 12, Special Issue
December 2021Pages 755-765
- Receive Date: 03 June 2020
- Revise Date: 11 November 2020
- Accept Date: 30 November 2020