299
Views
6
CrossRef citations to date
0
Altmetric
Regular Articles

Application of the extended Chebyshev cardinal wavelets in solving fractional optimal control problems with ABC fractional derivative

, &
Pages 2694-2708 | Received 03 Oct 2021, Accepted 22 Mar 2022, Published online: 06 Apr 2022

References

  • Abdeljawad, T., & Baleanu, D. (2017). Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel. The Journal of Nonlinear Sciences and Applications, 10, 1098–1107. https://.org/10.22436/jnsa.010.03.20
  • Agrawal, O. P. (2004). A general formulation and solution scheme for fractional optimal control problem. Nonlinear Dynamics, 38(1–4), 323–337. https://doi.org/10.1007/s11071-004-3764-6
  • Algahtani, O. J. J. (2016). Comparing the Atangana–Baleanu and Caputo-Fabrizio derivative with fractional order: Allen Cahn model. Chaos, Solitons and Fractals, 89(5), 552–559. https://doi.org/10.1016/j.chaos.2016.03.026
  • Alipour, M., Rostamy, D., & Baleanu, D. (2013). Solving multi-dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices. Journal of Vibration and Control, 19(16), 2523–2540. https://doi.org/10.1177/1077546312458308
  • Atangana, A., & Baleanu, D. (2016). New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Thermal Science, 20(2), 763–769. https://doi.org/10.2298/TSCI160111018A
  • Baghani, O. (2021). Second Chebyshev wavelets (SCWS) method for solving finite-time fractional linear quadratic optimal control problems. Mathematics and Computers in Simulation, 190, 343–361. https://doi.org/10.1016/j.matcom.2021.05.017
  • Canuto, C., Hussaini, M., Quarteroni, A., & Zang, T. (1988). Spectral methods in fluid dynamics. Springer-Verlag.
  • Cao, S. L W., & Wang, Y. (2022). On spectral Petrov–Galerkin method for solving optimal control problem governed by a two-sided fractional diffusion equation. Computers & Mathematics with Applications, 107, 104–116. https://doi.org/10.1016/j.camwa.2021.12.020
  • Caputo, M., & Fabrizio, M. (2015). A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications, 1(2), 73–85. http://dx.doi.org/10.12785/pfda/010201
  • Chen, Y., Yi, M., & Yu, C. (2012). Error analysis for numerical solution of fractional differential equation by Haar wavelets method. Journal of Computational Science, 3(5), 367–373. https://doi.org/10.1016/j.jocs.2012.04.008
  • Dehestania, H., Ordokhani, Y., & Razzaghi, M. (2020). Fractional-order Bessel wavelet functions for solving variable order fractional optimal control problems with estimation error. International Journal of System Science, 51(6), 1032–1052. https://doi.org/10.1080/00207721.2020.1746980
  • Doungmo Gouf, E. F., & Khan, Y. (2021). A new auto-replication in systems of attractors with two and three merged basins of attraction via control. Communications in Nonlinear Science and Numerical Simulation, 96(3), 105709. https://doi.org/10.1016/j.cnsns.2021.105709
  • Doungmo Goufo, E. F. (2018). Mathematical analysis of peculiar behavior by chaotic, fractional and strange multiwing attractors. International Journal of Bifurcation and Chaos, 28(10), 1850125. https://doi.org/10.1142/S0218127418501250
  • Heydari, M. H. (2020a). Chebyshev cardinal functions for a new class of nonlinear optimal control problems generated by Atangana–Baleanu-Caputo variable-order fractional derivative. Chaos, Solitons and Fractals, 130(4), 109401. https://doi.org/10.1016/j.chaos.2019.109401
  • Heydari, M. H. (2020b). Numerical solution of nonlinear 2D optimal control problems generated by Atangana–Riemann–Liouville fractal–fractional derivative. Applied Numerical Mathematics, 150(4), 507–518. https://doi.org/10.1016/j.apnum.2019.10.020
  • Heydari, M. H., & Razzaghi, M. (2022). Extended Chebyshev cardinal wavelets for nonlinear fractional delay optimal control problems. International Journal of Systems Science, 53(5), 1048–1067. https://doi.org/10.1080/00207721.2021.1987579
  • Hosseininia, M., & Heydari, M. H. (2019). Legendre wavelets for the numerical solution of nonlinear variable-order time fractional 2D reaction-diffusion equation involving Mittag-Leffler non-singular kernel. Chaos, Solitons and Fractals, 127(1), 400–407. https://doi.org/10.1016/j.chaos.2019.07.017
  • Hosseinpour, S., Nazemi, A., & Tohidi, E. (2019). Müntz-Legendre spectral collocation method for solving delay fractional optimal control problems. Journal of Computational and Applied Mathematics, 351(14), 344–363. https://doi.org/10.1016/j.cam.2018.10.058
  • Jafari, H., Ganji, R. M., Sayevand, K., & Baleanu, D. (2021). A numerical approach for solving fractional optimal control problems with Mittag-Leffler kernel. Journal of Vibration and Control. https://doi.org/10.1177/10775463211016967.
  • Kumar, N., & Mehra, M. (2021a). Legendre wavelet collocation method for fractional optimal control problems with fractional Bolza cost. Numerical Methods for Partial Differential Equations, 37(2), 1693–1724. https://doi.org/10.1002/num.v37.2
  • Kumar, N., & Mehra, M. (2021b). Collocation method for solving nonlinear fractional optimal control problems by using Hermite scaling function with error estimates. Optimal Control Applications and Methods, 42(2), 417–444. https://doi.org/10.1002/oca.v42.2
  • Maleki, M., & Dadkhah Tirani, M. (2011). Chebyshev finite difference method for solving constrained quadratic optimal control problems. Journal of Mathematical Extension, 52(1), 1–21.
  • Marzban, H. R., & Razzaghi, M. (2010). Rationalized Haar approach for nonlinear constrained optimal control problems. Applied Mathematical Modelling, 34(1), 174–183. https://doi.org/10.1016/j.apm.2009.03.036
  • Mehandiratta, V., Mehra, M., & Leugering, G. (2021). Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control and Related Fields, 11(1), 189–209. https://doi.org/10.3934/mcrf.2020033
  • Mohammadi, F., Moradi, L., Baleanu, D., & Jajarmi, A. (2018). A hybrid functions numerical scheme for fractional optimal control problems: application to nonanalytic dynamic systems. Journal of Vibration and Control, 24(21), 5030–5043. https://doi.org/10.1177/1077546317741769
  • Moradi, L., Mohammadi, F., & Baleanu, D. (2019). A direct numerical solution of time-delay fractional optimal control problems by using Chelyshkov wavelets. Journal of Vibration and Control, 25(2), 310–324. https://doi.org/10.1177/1077546318777338
  • Ordokhani, Y., & Razzaghi, M. (2005). Linear quadratic optimal control problems with inequality constraints via rationalized Haar functions. Dynamics of Continuous, Discrete and Impulsive Systems Series B, 12(5), 761–773.
  • Podlubny, I. (1998). Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier.
  • Postavaru, O., & Toma, A. (2022). A numerical approach based on fractional-order hybrid functions of block-pulse and Bernoulli polynomials for numerical solutions of fractional optimal control problems. Mathematics and Computers in Simulation, 194(1), 269–284. https://doi.org/10.1016/j.matcom.2021.12.001
  • Rabiei, K., & Ordokhani, Y. (2019). Boubaker hybrid functions and their application to solve fractional optimal control and fractional variational problems. Applications of Mathematics, 63(5), 541–567. https://doi.org/10.21136/AM
  • Rabiei, K., Ordokhani, Y., & Babolian, E. (2018). Fractional-order Boubaker functions and their applications in solving delay fractional optimal control problems. Journal of Vibration and Control, 24(15), 3370–3383. https://doi.org/10.1177/1077546317705041
  • Rakhshan, S. A., Effati, S., & Kamyad, A. V. (2018). Solving a class of fractional optimal control problems by the Hamilton-Jacobi-Bellman equation. Journal of Vibration and Control, 24(9), 1741–1756. https://doi.org/10.1177/1077546316668467
  • Singh, A. K., & Mehra, M. (2021). Wavelet collocation method based on Legendre polynomials and its application in solving the stochastic fractional integro-differential equations. Journal of Computational Science, 51(2), 101342. https://doi.org/10.1016/j.jocs.2021.101342
  • Sweilam, N. H., & Al-Ajami, T. M. (2015). Legendre spectral collocation method for solving some types of fractional optimal control problems. Journal of Advanced Research, 6(3), 393–403. https://doi.org/10.1016/j.jare.2014.05.004
  • Tajadodi, H., Khan, A., Gómez-Aguilar, J. F., & Khan, H. (2021). Optimal control problems with Atangana–Baleanu fractional derivative. Optimal Control Applications and Methods, 42(1), 96–109. https://doi.org/10.1002/oca.v42.1
  • Valian, F., Ordokhani, Y., & Vali, M. A. (2020). Numerical solution of fractional optimal control problems with inequality constraint using the fractional-order Bernoulli wavelet functions. Iranian Journal of Science and Technology, Transactions of Electrical Engineering, 44(4), 1513–1528. https://doi.org/10.1007/s40998-020-00327-3 .
  • Wang, F., Zhang, Z., & Zhou, Z. (2021). A spectral Galerkin approximation of optimal control problem governed by fractional advection-diffusion-reaction equations. Journal of Computational and Applied Mathematics, 386(6), 113233. https://doi.org/10.1016/j.cam.2020.113233
  • Xu, X., Xiong, L., & Zhou, F. (2021). Solving fractional optimal control problems with inequality constraints by a new kind of Chebyshev wavelets method. Journal of Computational Science, 54(1), 101412. https://doi.org/10.1016/j.jocs.2021.101412

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.