Skip to main content
SpringerLink
Log in

A Modified Numerical Method Based on Bernstein Wavelets for Numerical Assessment of Fractional Variational and Optimal Control Problems

  • Research Paper
  • Published:
Iranian Journal of Science and Technology, Transactions of Electrical Engineering Aims and scope Submit manuscript

Abstract

The main idea of this work is to present a novel scheme based on Bernstein wavelets for finding numerical solution of two classes of fractional optimal control problems (FOCPs) and one class of fractional variational problems (FVPs). First, we present an approximation for fractional derivative using the Laplace transform. Then, the obtained integer-order problems are converted into equivalent variational problems. By using the Bernstein wavelets, activation functions and the Gauss–Legendre integration scheme, problems are transformed to algebraic systems of equations. Finally, these systems are solved employing Newton’s iterative scheme. Error bound for the best approximation is given. Also, we propose a scheme to determine the number of basis functions necessary to get a certain precision. In order to verify that the mentioned scheme is applicable and powerful for solving FOCPs and FVP, some numerical experiments have been provided.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Agrawal OP (2008) A general finite element formulation for fractional variational problems. J Math Anal Appl 337:1–12

    Article  MathSciNet  MATH  Google Scholar 

  • Agrawal OP (2004) A general formulation and solution scheme for fractional optimal control problems. Nonl Dyn 38:323–337

    Article  MathSciNet  MATH  Google Scholar 

  • Alipour M, Rostamy D, Baleanu D (2013) Solving multidimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices. J Vib Control 19:2523–2540

    Article  MathSciNet  MATH  Google Scholar 

  • Alizadeh A, Effati S (2016) An iterative approach for solving fractional optimal control problems. J Vib Control 1:1–19

    MATH  Google Scholar 

  • Alizadeh A, Effati S, Heydari A (2017) Numerical schemes for fractional optimal control problems. J Dyn Syst Meas Control 139(8):081002

    Article  Google Scholar 

  • Babolian E, Vahidi AR, Shoja A (2014) An efficient method for nonlinear fractional differential equations: combination of the Adomian decomposition method and spectral method. Indian J Pure Appl Math 45:1017–1028

    Article  MathSciNet  MATH  Google Scholar 

  • Bell WW (2004) Special functions for scientists and engineers. Dover Publications Inc, Mineola, NY

    MATH  Google Scholar 

  • Bhatti MI, Bracken P (2007) Solutions of differential equations in a Bernstein polynomial basis. J Comput Appl Math 205:272–280

    Article  MathSciNet  MATH  Google Scholar 

  • Bohannan GW (2008) Analog fractional order controller in temperature and motor control applications. J Vib Control 14(9–10):1487–1498

    Article  MathSciNet  Google Scholar 

  • Chen S, Liu F, Turner I, Anh V (2013) An implicit numerical method for the two-dimensional fractional percolation equation. Appl Math Comput 219:4322–4331

    MathSciNet  MATH  Google Scholar 

  • Chui CK (1997) Wavelets. A mathematical tool for signal analysis. SIAM monographs on Mathematical Modeling and Computation. Philadelphia, SIAM. 13

  • Dehestani H, Ordokhani Y, Razzaghi M (2020) Modified wavelet method for solving fractional variational problems. J Vib Control 27(5–6):582–596

    MathSciNet  MATH  Google Scholar 

  • Dehestani H, Ordokhani Y, Razzaghi M (2019) On the applicability of Genocchi wavelets method for different kinds of fractional order differential equations with delay. Numer Linear Algebra Appl 26(5):e2259

    Article  MathSciNet  MATH  Google Scholar 

  • Dehestani H, Ordokhani Y, Razzaghi M (2022) Fractional-Lucas optimization method for evaluating the approximate solution of the multi-dimensional fractional differential equations. Eng Comput 38:481–495

    Article  Google Scholar 

  • Ding Y, Wang Z, Ye H (2012) Optimal control of a fractional-order HIV-immune system with memory. IEEE Trans Contr Syst Tech 30:763–769

    Article  Google Scholar 

  • Ezz-Eldien SS, Bhrawy AH, ElKalaawy AA (2018) Direct numerical method for isoperimetric fractional variational problems based on operational matrix. J Vib Control 24(14):3063–3076

    Article  MathSciNet  MATH  Google Scholar 

  • Freed AD, Diethelm K (2006) Fractional calculus in biomechanics: A 3d viscoelastic model using regularized fractional derivative kernels with application to the human calcaneal fat pad. Biomech Model Mechanobiol 5:203–215

    Article  Google Scholar 

  • Heydari MH, Hooshmandasl MR, Maalek Ghaini FM, Cattani C (2016) Wavelets method for solving fractional optimal control problems. Appl Math Comput 286:139–154

    MathSciNet  MATH  Google Scholar 

  • Jajarmi A, Baleanu D (2018) Suboptimal control of fractional-order dynamic systems with delay argument. J Vib Control 24(12):1

    Article  MathSciNet  MATH  Google Scholar 

  • Jesus IS, Machado JAT (2008) Fractional control of heat diffusion systems. Nonlinear Dyn 54(3):263–282

    Article  MathSciNet  MATH  Google Scholar 

  • Khader MM (2015) An efficient approximate method for solving fractional variational problems. Appl Math Model 39:1643–1649

    Article  MathSciNet  MATH  Google Scholar 

  • Kheiri H, Jafari M (2019) Fractional optimal control of an HIV/AIDS epidemic model with random testing and contact tracing. J Appl Math Comput 60:387–411

    Article  MathSciNet  MATH  Google Scholar 

  • Kiryakova VS (1994) Generalized fractional falculus and applications. Longman Sci. Techn., Harlow, John Wiley and Sons, New York

  • Kreyszig E (1978) Introductory Functional Analysis with Applications. Wiley, New York

    MATH  Google Scholar 

  • Li Y, Sun N, Zheng B, Wang Q, Zhang Y (2014) Wavelet operational matrix method for solving the Riccati differential equation. Commun Nonlinear Sci Numer Simul 19:483–493

    Article  MathSciNet  MATH  Google Scholar 

  • Lotfi A, Yousefi SA (2013) A numerical technique for solving a class of fractional variational problems. J Comput Appl Math 237:633–643

    Article  MathSciNet  MATH  Google Scholar 

  • Lotfi A, Yousefi SA, Dehghan M (2013) Numerical solution of a class of fractional optimal control problems via the Legendre orthonormal basis combined with the operational matrix and the Gauss quadrature rule. J Comput Appl Math 250:143–160

    Article  MathSciNet  MATH  Google Scholar 

  • Mashayekhi S, Razzaghi M (2018) An approximate method for solving fractional optimal control problems by hybrid functions. J Vib Control 24(9):1621–1631

    Article  MathSciNet  MATH  Google Scholar 

  • Morgado ML, Rebelo M, Ferras LL, Ford NJ (2017) Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method. Appl Numer Math 114:108–123

    Article  MathSciNet  MATH  Google Scholar 

  • Nemati A (2017) Numerical solution of 2D fractional optimal control problems by the spectral method combined with Bernstein operational matrix. Int J Control 91(12):2632–2645

    Article  MATH  Google Scholar 

  • Ordokhani Y, Rahimkhani P (2018) A numerical technique for solving fractional variational problems by Müntz-Legendre polynomials. J Appl Math Comput 58:75–94

    Article  MathSciNet  MATH  Google Scholar 

  • Popovic JK, Spasic DT, Tosic J et al (2015) Fractional model for pharmacokinetics of high dose methotrexate in children with acutelymphoblastic leukaemia. Commun Nonlinear Sci Numer Simul 22:451–471

    Article  MathSciNet  Google Scholar 

  • Rabiei K, Ordokhani Y, Babolian E (2018) Fractional-order Boubaker functions and their applications in solving delay fractional optimal control problems. J Vib Control 24(15):3370–3383

    Article  MathSciNet  MATH  Google Scholar 

  • Rabiei K, Ordokhani Y, Babolian E (2018) Numerical solution of 1D and 2D fractional optimal control of system via Bernoulli polynomials. Int J Appl Comput Math 7

  • Rabiei K, Parand K (2020) Collocation method to solve inequality constrained optimal control problems of arbitrary order. Eng Comput 36:115–125

    Article  Google Scholar 

  • Rahimkhani P, Ordokhani Y (2020) Approximate solution of nonlinear fractional integro-differential equations using fractional alternative Legendre functions. J Comput Appl Math 365:112365

    Article  MathSciNet  MATH  Google Scholar 

  • Rahimkhani P, Ordokhani Y (2021) Orthonormal Bernoulli wavelets neural network method and its application in astrophysics. Comput Appl Math 40(78):1–24

    MathSciNet  MATH  Google Scholar 

  • Rahimkhani P, Ordokhani Y (2020) The bivariate Müntz wavelets composite collocation method for solving space-time fractional partial differential equations. Comput Appl Math 39:115

    Article  MATH  Google Scholar 

  • Rahimkhani P, Ordokhani Y, Babolian E (2016) An efficient approximate method for solving delay fractional optimal control problems. Nonl Dyn 86:1649–1661

    Article  MathSciNet  MATH  Google Scholar 

  • Rahimkhani P, Ordokhani Y, Babolian E (2018) Müntz-Legendre wavelet operational matrix of fractional-order integration and its applications for solving the fractional pantograph differential equations. Numer Algorithms 77(4):1283–1305

    Article  MathSciNet  MATH  Google Scholar 

  • Rahimkhani P, Ordokhani Y, Babolian E (2017) Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet. J Comput Appl Math 309:493–510

    Article  MathSciNet  MATH  Google Scholar 

  • Rahimkhani P, Ordokhani Y, Lima PM (2019) An improved composite collocation method for distributed-order fractional differential equations based on fractional Chelyshkov wavelets. Appl Numer Math 145:1–27

    Article  MathSciNet  MATH  Google Scholar 

  • Razzaghi M, Yousefi S (2000) Legendre wavelets direct method for variational problems. Math Comput Simul 53:185–192

    Article  MathSciNet  Google Scholar 

  • Ren J, Sun Z, Dai W (2016) New approximations for solving the Caputo-type fractional partial differential equations. Appl Math Model 40(4):2625–2636

    Article  MathSciNet  MATH  Google Scholar 

  • Sabermahani S, Ordokhani Y, Yousefi SA (2019) Fractional order Lagrange polynomials: an application for solving delay fractional optimal control problems. Trans Inst Meas Control 41(11):2997–3009

    Article  Google Scholar 

  • Sahu PK, Saha Ray S (2018) Comparison on wavelets techniques for solving fractional optimal control problems. J Vib Control 24(6):1185–1201

    Article  MathSciNet  MATH  Google Scholar 

  • Samko S, Kilbas AA, Marichev O (1993) Fractional integrals and derivatives: theory and applications. Gordon and Breach, Yverdon

    MATH  Google Scholar 

  • Soltanpour Moghadam A, Arabameri M, Baleanu D, Barfeie M (2020) Numerical solution of variable fractional order advection-dispersion equation using Bernoulli wavelet method and new operational matrix of fractional order derivative. Math Methods Appl Sci 43:3936–3953

    MathSciNet  MATH  Google Scholar 

  • Suarez IJ, Vinagre BM, Chen YQ (2008) A fractional adaptation scheme for lateral control of an AGV. J Vib Control 14(9–10):1499–1511

    Article  MathSciNet  MATH  Google Scholar 

  • Tripathy MC, Mondal D, Biswas K, Sen S (2015) Design and performance study of phase-locked loop using fractional-order loop filter. Int J Circuit Theory Appl 43(6):776–792

    Article  Google Scholar 

  • Wang JR, Zhou Y (2011) A class of fractional evolution equations and optimal controls. Nonlinear Anal RealWorld Appl 12:262–272

    Article  MathSciNet  MATH  Google Scholar 

  • Yang S, Xiao A, Su H (2010) Convergence of the variational iteration method for solving multi- order fractional differential equations. Comput Math Appl 60:2871–2879

    Article  MathSciNet  MATH  Google Scholar 

  • Zaky MA, Doha EH, Tenreiro Machado JA (2018) A spectral framework for fractional variational problems based on fractional Jacobi functions. Appl Numer Math 132:51–72

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The second author is supported by the Alzahra university within project 99/1/159. Also, we express our sincere thanks to the anonymous referees for valuable suggestions that improved the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran

    Parisa Rahimkhani & Yadollah Ordokhani

Authors
  1. Parisa Rahimkhani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yadollah Ordokhani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yadollah Ordokhani.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rahimkhani, P., Ordokhani, Y. A Modified Numerical Method Based on Bernstein Wavelets for Numerical Assessment of Fractional Variational and Optimal Control Problems. Iran J Sci Technol Trans Electr Eng 46, 1041–1056 (2022). https://doi.org/10.1007/s40998-022-00522-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40998-022-00522-4

Keywords

Mathematics Subject Classification

Search

Navigation