Skip to main content
SpringerLink
Log in

New Numerical Approach for Fractional Variational Problems Using Shifted Legendre Orthonormal Polynomials

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This paper reports a new numerical approach for numerically solving types of fractional variational problems. In our approach, we use the fractional integrals operational matrix, described in the sense of Riemann–Liouville, with the help of the Lagrange multiplier technique for converting the fractional variational problem into an easier problem that consisting of solving an algebraic equations system in the unknown coefficients. Several numerical examples are introduced, combined with their approximate solutions and comparisons with other numerical approaches, for confirming the accuracy and applicability of the proposed approach.

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
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Kirchner, J.W., Feng, X., Neal, C.: Fractal stream chemistry and its implications for containant transport in catchments. Nature 403, 524–526 (2000)

    Article  Google Scholar 

  2. Baillie, R.T.: Long memory processes and fractional integration in econometrics. J. Econom. 73, 5–59 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Couceiro, M.S., Ferreira, N.M.F., Tenreiro Machado, J.A.: Application of fractional algorithms in the control of a robotic bird. Commun. Nonlinear Sci. Numer. Simul. 15, 895–910 (2010)

    Article  Google Scholar 

  4. Ahmad, W.M., El-Khazali, R.: Fractional-order dynamical models of love. Chaos Solitons Fractals 33, 1367–1375 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Sierociuk, D., Dzielinski, D., Sarwas, G., Petras, I., Podlubny, I., Skovranek, T.: Modelling heat transfer in heterogeneous media using fractional calculus. Philos. Trans. R. Soc. A 371, 20130146 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Abdelkawy, M.A., Ezz-Eldien, S.S., Amin, A.Z.M.: Jacobi spectral collocation scheme for solving Abels integral equations. Prog. Fract. Differ. Appl. 1, 187–200 (2015)

    Google Scholar 

  7. Tarasov, V.E.: Fractional vector calculus and fractional Maxwell’s equations. Ann. Phys. 323, 2756–2778 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jiang, Y., Wang, X., Wang, Y.: On a stochastic heat equation with first order fractional noises and applications to finance. J. Math. Anal. Appl. 396, 656–669 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jiang, Y., Ma, J.: Higher order finite element methods for time fractional partial differential equations. J. Comput. Appl. Math. 235, 3285–3290 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Song, L., Wang, W.: A new improved Adomian decomposition method and its application to fractional differential equations. Appl. Math. Model. 37, 1590–1598 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhou, H., Tian, W., Deng, W.: Quasi-compact finite difference schemes for space fractional diffusion equations. J. Sci. Comput. 56, 45–66 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Heydari, M.H., Hooshmandasl, M.R., Mohammadi, F.: Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions. Appl. Math. Comput. 234, 267–276 (2014)

    MathSciNet  MATH  Google Scholar 

  13. Neamaty, A., Agheli, B., Darzi, R.: Solving fractional partial differential equation by using wavelet operational method. J. Math. Comput. Sci. 7, 230–240 (2013)

    Google Scholar 

  14. Bhrawy, A.H., Taha, T.M., Machado, J.A.T.: A review of operational matrices and spectral techniques for fractional calculus. Nonlinear Dyn. 81, 1023–1052 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bhrawy, A.H., Abdelkawy, M.A.: A fully spectral collocation approximation for multi-dimensional fractional Schrodinger equations. J. Comput. Phys. 294, 462–483 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang, L., Ma, Y., Meng, Z.: Haar wavelet method for solving fractional partial differential equations numerically. Appl. Math. Comput. 227, 66–76 (2014)

    MathSciNet  MATH  Google Scholar 

  17. Ma, J., Liu, J., Zhou, Z.: Convergence analysis of moving finite element methods for space fractional differential equations. J. Comput. Appl. Math. 255, 661–670 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bhrawy, A.H., Doha, E.H., Ezz-Eldien, S.S., Abdelkawy, M.A.: A numerical technique based on the shifted Legendre polynomials for solving the time-fractional coupled KdV equation. Calcolo (2015). doi:10.1007/s10092-014-0132-x

  19. Bhrawy, A.H.: A highly accurate collocation algorithm for 1+1 and 2+1 fractional percolation equations. J. Vib. Control (2015). doi:10.1177/1077546315597815

  20. Bhrawy, A.H., Zaky, M.A.: Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation. Nonlinear Dyn. 80, 101–116 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: An efficient Legendre spectral tau matrix formulation for solving fractional sub-diffusion and reaction sub-diffusion equations. J. Comput. Nonlinear Dyn. 10, 021019 (2015)

    Article  Google Scholar 

  22. Bhrawy, A.H., Doha, E.H., Baleanu, D., Ezz-Eldien, S.S.: A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations. J. Comput. Phys. 293, 142–156 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Bhrawy, A.H., Zaky, M.A.: A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations. J. Comput. Phys. 281, 876–895 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Bhrawy, A.H., Doha, E.H., Machado, J.A.T., Ezz-Eldien, S.S.: An efficient numerical scheme for solving multi-dimensional fractional optimal control problems with a quadratic performance index. Asian J. Control (2015). doi:10.1002/asjc.1109

  25. Bhrawy, A.H., Doha, E.H., Baleanu, D., Ezz-Eldien, S.S., Abdelkawy, M.A.: An accurate numerical technique for solving fractional optimal control problems. Proc. Romanian Acad. A 16, 47–54 (2015)

    MathSciNet  Google Scholar 

  26. Bhrawy, A.H., Ezz-Eldien, S.S.: A new Legendre operational technique for delay fractional optimal control problems. Calcolo (2015). doi:10.1007/s10092-015-0160-1

  27. Ezz-Eldien, S.S., Doha, E.H., Baleanu, D., Bhrawy, A.H.: A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems. J. Vib. Control (2015). doi:10.1177/1077546315573916

  28. Gelfand, I.M., Fomin, S.V.: Calculus of Variations. Prentice Hall, New Jersey (1963)

    MATH  Google Scholar 

  29. Dym, C.L., Shames, I.H.: Solid Mechanics: A Variational Approach. Mc Graw-Hill, New York (1973)

    MATH  Google Scholar 

  30. Elsgolts, L.: Differential Equations and the Calculus of Variations. Mir, Moscow (1977)

    Google Scholar 

  31. Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53, 1890–1899 (1996)

    Article  MathSciNet  Google Scholar 

  32. Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55, 3581–3592 (1997)

    Article  MathSciNet  Google Scholar 

  33. Malinowska, A.B., Torres, D.F.M.: Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative. Comput. Math. Appl. 59, 3110–3116 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Malinowska, A.B., Torres, D.F.M.: Fractional calculus of variations for a combined Caputo derivative. Fract. Calc. Appl. Anal. 14, 523–537 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Agrawal, O.P.: Fractional variational calculus and the transversality conditions. J. Phys. A 39, 10375–10384 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Agrawal, O.P.: Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A 40, 6287–6303 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Agrawal, O.P.: A general finite element formulation for fractional variational problems. J. Math. Anal. Appl. 337, 1–12 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. Almeida, R., Torres, D.F.M.: Calculus of variations with fractional derivatives and fractional integrals. Appl. Math. Lett. 22, 1816–1820 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Bastos, N., Ferreira, R., Torres, D.F.M.: Discrete-time fractional variational problems. Signal Process. 91, 513–524 (2011)

    Article  MATH  Google Scholar 

  40. Pooseh, S., Almeida, R., Torres, D.F.M.: A discrete time method to the first variation of fractional order variational functionals. Cent. Eur. J. Phys. 11, 1262–1267 (2013)

    Google Scholar 

  41. Pooseh, S., Almeida, R., Torres, D.F.M.: Discrete direct methods in the fractional calculus of variations. Comput. Math. Appl. 66, 668–676 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. Wang, D., Xiao, A.: Fractional variational integrators for fractional variational problems. Commun. Nonlinear Sci. Numer. Simul. 17, 602–610 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  43. Dehghan, M., Hamedi, E.A., Khosravian-Arab, H.: A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials. J. Vib. Control (2014). doi:10.1177/1077546314543727

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Assiut University, New Valley Branch, El-Kharja, 72511, Egypt

    Samer S. Ezz-Eldien

  2. Department of Basic Science, Institute of Information Technology, Modern Academy, Cairo, Egypt

    Ramy M. Hafez

  3. Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

    Ali H. Bhrawy & Ahmed A. El-Kalaawy

  4. Department of Mathematics and Computer Sciences, Faculty of Art and Sciences, Çankaya University, 0630, Balgat, Ankara, Turkey

    Dumitru Baleanu

  5. Institute of Space Sciences, Magurele-Bucharest, Romania

    Dumitru Baleanu

Authors
  1. Samer S. Ezz-Eldien
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ramy M. Hafez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ali H. Bhrawy
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Dumitru Baleanu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Ahmed A. El-Kalaawy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dumitru Baleanu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ezz-Eldien, S.S., Hafez, R.M., Bhrawy, A.H. et al. New Numerical Approach for Fractional Variational Problems Using Shifted Legendre Orthonormal Polynomials. J Optim Theory Appl 174, 295–320 (2017). https://doi.org/10.1007/s10957-016-0886-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-016-0886-1

Keywords

Mathematics Subject Classification

Search

Navigation