Skip to main content
SpringerLink
Log in

An effective numerical method for solving nonlinear variable-order fractional functional boundary value problems through optimization technique

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

An optimization method standing on a new basis formed by the transcendental Bernstein series (TBS) is proposed. The TBS includes the unknown free coefficients and control parameters for solving nonlinear variable-order fractional functional boundary value problems (NV-FFBVP). The corresponding operational matrices for variable-order fractional derivatives are derived for expanding the solution by means of TBS. The TBS is a generalization of the Bernstein polynomials (BP) and represent a superset of BP. In the particular cases where all the control parameters are zero, the TBS method is equivalent to the BP method. The proposed technique reduces the NV-FFBVP to a system of algebraic equations and, subsequently, to the problem of finding the free coefficients and control parameters using an optimization technique. Several numerical results reveal the computational performance and reliability of the method.

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

Similar content being viewed by others

References

  1. Tavares, D., Almeida, R., Torres, D.F.M.: Caputo derivatives of fractional variable order: numerical approximations. Commun. Nonlinear Sci. Numer. Simul. 35, 69–87 (2016)

    Article  MathSciNet  Google Scholar 

  2. Zhao, X., Sun, Z.Z., Karniadakis, G.E.: Second-order approximations for variable order fractional derivatives: algorithms and applications. J. Comput. Phys. 293, 184–200 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Samko, S.G., Ross, B.: Integration and differentiation to a variable fractional order. Integr. Trans. Spec. Func. 1(4), 277–300 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Lorenzo, C.F., Hartley, T.T.: Initialized fractional calculus. Int. J. Appl. Math. 3(3), 249–265 (2000)

    MathSciNet  MATH  Google Scholar 

  5. Momani, S., Odibat, Z.: Generalized differential transform method for solving a space and time-fractional diffusion-wave equation. Phys. Lett. A. 370, 379–87 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Coimbra, C.F.M.: Mechanics with variable-order differential operators. Ann. Phys. 12(11–12), 692–703 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Samko, S.G.: Fractional integration and differentiation of variable order. Anal. Math. 21, 213–36 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Pedro, H.T.C., Kobayashi, M.H., Pereira, J.M.C., Coimbra, C.F.M.: Variable order modeling of diffusive-convective effects on the oscillatory flow past a sphere. J. Vib. Control. 14, 1569–1672 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ramirez, L.E.S., Coimbra, C.: On the variable order dynamics of the nonlinear wake caused by a sedimenting particle. Physica D 240, 1111–1118 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Sun, H., Chen, W., Wei, H., Chen, Y.: A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems. Eur. Phys. J. Spec. Top. 193, 185–192 (2011)

    Article  Google Scholar 

  11. Shyu, J.J., Pei, S.C., Chan, C.H.: An iterative method for the design of variable fractional-order FIR differintegrators. Signal Process. 89(3), 320–327 (2009)

    Article  MATH  Google Scholar 

  12. Coimbra, C.: Mechanics with variable-order differential operators. Ann. Phys. 12(11–12), 692–703 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lorenzo, C., Hartley, T.: Variable order and distributed order fractional operators. Nonlinear Dyn. 29, 57–98 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zhang, H., Liu, F., Zhuang, P., Turner, I., Anh, V.: Numerical analysis of a new space-time variable fractional order advection-dispersion equation. Appl. Math. Comput. 242, 541–550 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Zhang, S.: Existence result of solutions to differential equations of variable-order with nonlinear boundary value conditions. Commun. Nonlinear Sci. Numer. Simul. 18, 3289–3297 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Razminia, A., Dizaji, A.F., Majd, V.J.: Solution existence for non-autonomous variable-order fractional differential equations. Math. Comput. Model. 55, 1106–1117 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhang, H., Liu, F., Phanikumar, M.S., Meerschaert, M.M.: A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model. Comput. Math. Appl. 66, 693–701 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Yang, J., Yao, H., Wu, B.: An efficient numerical method for variable order fractional functional differential equation. Appl. Math. Lett. 76, 221–226 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zaky, M.A., Ezz-Eldien, S.S., Doha, E.H., Machado, J.A.T., Bhrawy, A.H.: An efficient operational matrix technique for multidimensional variable-order time fractional diffusion equations. J. Comput. Nonlinear Dyn. 11(6), 061002–8 (2016)

    Article  Google Scholar 

  20. Moghaddam, B.P., Machado, J.A.T.: Extended algorithms for approximating variable order fractional derivatives with applications. J. Sci. Comput. 71(3), 1351–1374 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Tayebi, A., Shekari, Y., Heydari, M.H.: A meshless method for solving two-dimensional variable-order time fractional advection-diffusion equation. J. Comput. Phys. 340, 655–669 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Dahaghin, MSh, Hassani, H.: An optimization method based on the generalized polynomials for nonlinear variable-order time fractional diffusion-wave equation. Nonlinear Dyn. 88(3), 1587–1598 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, X., Wu, B.: A numerical technique for variable fractional functional boundary value problems. Appl. Math. Lett. 43, 108–113 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, X., Wu, B.: A new reproducing kernel method for variable order fractional boundary value problems for functional differential equations. Appl. Math. Lett. 311, 387–393 (2017)

    MathSciNet  MATH  Google Scholar 

  25. Jia, Y.T., Xu, M.Q., Lin, Y.Z.: A numerical solution for variable order fractional functional differential equation. Appl. Math. Lett. 64, 125–130 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu, J., Li, X., Hu, X.: A RBF-based differential quadrature method for solving two-dimensional variable-order time fractional advection-diffusion equation. J. Comput. Phys. 384, 222–238 (2019)

    Article  MathSciNet  Google Scholar 

  27. Hassani, H., Avazzadeh, Z., Tenreiro Machado, J.A.: Numerical approach for solving variable-order space-time fractional telegraph equation using transcendental Bernstein series. Eng Comput. (2019). https://doi.org/10.1007/s00366-019-00736-x

    Article  Google Scholar 

  28. Zhao, T., Mao, Z., Karniadakis, G.E.: Multi-domain spectral collocation method for variable-order nonlinear fractional differential equations. Comput. Methods Appl. Mech. Eng. 348, 377–395 (2019)

    Article  MathSciNet  Google Scholar 

  29. Bhrawy, A.H., Zaky, M.A.: Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation. Nonlinear Dyn. 80(1), 101–116 (2016)

    MathSciNet  MATH  Google Scholar 

  30. Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation methods for linear and nonlinear variable order FPDEs. J. Comput. Phys. 80(1), 312–338 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Bhrawy, A.H., Zaky, M.A.: Numerical algorithm for the variable-order Caputo fractional functional differential equation. Nonlinear Dyn. 85(3), 1815–1823 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mokhtary, P., Ghoreishi, F., Srivastava, H.M.: The Müntz–Legendre Tau method for fractional differential equations. Appl. Math. Model. 40(2), 671–684 (2016)

    Article  MathSciNet  Google Scholar 

  33. Ateş, I., Zegeling, P.A.: A homotopy perturbation method for fractional-order advection-diffusion-reaction boundary-value problems. Appl. Math. Model. 47, 425–441 (2017)

    Article  MathSciNet  Google Scholar 

  34. Reutskiy, S.Y.: A new semi-analytical collocation method for solving multi-term fractional partial differential equations with time variable coefficients. Appl. Math. Model. 45, 238–254 (2017)

    Article  MathSciNet  Google Scholar 

  35. Liu, Y., Fang, Z., Li, H., He, S.: A mixed finite element method for a time-fractional fourth-order partial differential equation. Appl. Math. Comput. 243, 703–717 (2014)

    MathSciNet  MATH  Google Scholar 

  36. Shen, S., Liu, F., Chen, J., urner, I.T., Anh, V.: Numerical techniques for the variable order time fractional diffusion equation. Appl. Math. Comput. 218, 10861–10870 (2012)

    MathSciNet  MATH  Google Scholar 

  37. Garg, M., Manohar, P.: Matrix method for numerical solution of space-time fractional diffusion-wave equations with three space variables. Afr. Mat. 25(1), 161–181 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Sweilam, N.H., Khader, M.M., Almarwm, H.M.: Numerical studies for the variable-order nonlinear fractional wave equation. Fractals Calc. Appl. Anal. 15(4), 669–683 (2012)

    MathSciNet  MATH  Google Scholar 

  39. Sun, H., Chen, W., Li, C., Chen, Y.: Finite difference schemes for variable-order time fractional diffusion equation. Int. J. Bifurc. Chaos. 22(4), 1250085 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhung, P., Liu, F., Anh, V., Turner, I.: Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 47(3), 1760–1781 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. Chen, Y.M., Wei, Y.Q., Liu, D.Y., Yu, H.: Numerical solution for a class of nonlinear variable order fractional differential equations with Legendre wavelets. Appl. Math. Lett. 46, 83–88 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  42. Keshi, F.K., Moghaddam, B.P., Aghili, A.: A numerical approach for solving a class of variable-order fractional functional integral equations. Compt. Appl. Math. 37(4), 4821–4834 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  43. Li, Z., Wang, H., Xiao, R., Yang, S.: A variable-order fractional differential equation model of shape memory polymers. Chaos Solitons Fractals 102, 473–485 (2017)

    Article  MathSciNet  Google Scholar 

  44. Almeida, R., Bastos, N.R.O.: A numerical method to solve higher-order fractional differential equations. Mediterr. J. Math. 13(3), 1339–1352 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  45. Chen, Y., Liu, L., Li, B., Sun, Y.: Numerical solution for the variable order linear cable equation with Bernstein polynomials. Appl. Math. Comput. 238, 329–341 (2014)

    MathSciNet  MATH  Google Scholar 

  46. Asgari, M., Ezzati, R.: Using operational matrix of two-dimensional Bernstein polynomials for solving two-dimensional integral equations of fractional order. Appl. Math. Comput. 307, 290–298 (2017)

    MathSciNet  MATH  Google Scholar 

  47. Javadi, Sh, Babolian, E., Taheri, Z.: Solving generalized pantograph equations by shifted orthonormal Bernstein polynomials. J. Comput. Appl. Math. 303, 1–14 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  48. Hesameddini, E., Shahbazi, M.: Solving multipoint problems with linear Volterra–Fredholm integro-differential equations of the neutral type using Bernstein polynomials method. Appl. Numer. Math. 136, 122–138 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  49. Miclăuş, D., Pişcoran, L.I.: A new method for the approximation of integrals using the generalized Bernstein quadrature formula. Appl. Math. Comput. 340, 146–155 (2019)

    MathSciNet  Google Scholar 

  50. Rostamy, D., Alipour, M., Jafari, H., Baleanu, D.: Solving multi-term orders fractional differential equations by operational matrices of BPs with convergence analysis. Roman. Rep. Phys. 65(2), 334–349 (2013)

    Google Scholar 

  51. Chen, Y., Yi, M., Chen, C., Yu, C.: Bernstein polynomials method for fractional convection-diffusion equation with variable coefficients. CMES Comput. Model. Eng. Sci. 83, 639–654 (2012)

    MathSciNet  MATH  Google Scholar 

  52. Behiry, S.H.: Solution of nonlinear Fredholm integro-differential equations using a hybrid of block pulse functions and normalized Bernstein polynomials. J. Comput. Appl. Math. 260, 258–265 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  53. Wang, H., Zheng, X.: Wellposedness and regularity of the variable-order time-fractional diffusion equations. J. Math. Anal. Appl. 475(2), 1778–1802 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  54. Liu, J., Li, X., Hu, X.: A RBF-based differential quadrature method for solving two-dimensional variable-order time fractional advection-diffusion equation. J. Comput. Phys. 384, 222–238 (2019)

    Article  MathSciNet  Google Scholar 

  55. Shekari, Y., Tayebi, A., Heydari, M.H.: A meshfree approach for solving 2D variable-order fractional nonlinear diffusion-wave equation. Comput. Method. Appl. Mech. Eng. 350, 154–168 (2019)

    Article  MathSciNet  Google Scholar 

  56. Patrício, M.F.S., Ramos, H., Patrício, M.: Solving initial and boundary value problems of fractional ordinary differential equations by using collocation and fractional powers. J. Comput. Appl. Math. 345, 348–359 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  57. Malesz, W., Macias, M., Sierociuk, D.: Analytical solution of fractional variable order differential equations. J. Comput. Appl. Math. 348, 214–236 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  58. Zhou, Y., Ionescu, C., Tenreiro Machado, J.A.: Fractional dynamics and its applications. Nonlinear Dyn. 80(4), 1661–1664 (2015)

    Article  MathSciNet  Google Scholar 

  59. Chang, S.H.: Numerical solution of Troesch’s problem by simple shooting method. Appl. Math. Comput. 216, 3303–3306 (2010)

    MathSciNet  MATH  Google Scholar 

  60. Singh, R., Garg, H., Guleria, V.: Haar wavelet collocation method for Lane-Emden equations with Dirichlet, Neumann and Neumann–Robin boundary conditions. J. Comput. Appl. Math. 346, 150–161 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  61. Zahra, W.K., Van Daele, M.: Discrete spline methods for solving two point fractional Bagley–Torvik equation. Appl. Math. Comput. 296, 42–56 (2017)

    MathSciNet  MATH  Google Scholar 

  62. Yang, L., Chen, H.: Unique positive solutions for fractional differential equation boundary value problems. Appl. Math. Lett. 23, 1095–1098 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  63. Kreyszig, E.: Introductory functional analysis with applications. Wiley, London (1978)

    MATH  Google Scholar 

  64. Feng, X., Mei, L., He, G.: An efficient algorithm for solving Troesch’s problem. Appl. Math. Comput. 189, 500–507 (2007)

    MathSciNet  MATH  Google Scholar 

  65. Deeba, E., Khuri, S.A., Xie, S.: An algorithm for solving boundary value problems. J. Comput. Phys. 159(2), 125–138 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahrekord University, Shahrekord, Iran

    H. Hassani

  2. Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, R. Dr. António Bernardino de Almeida, 431, 4249-015, Porto, Portugal

    J. A. Tenreiro Machado

  3. School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    Z. Avazzadeh

Authors
  1. H. Hassani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. A. Tenreiro Machado
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Z. Avazzadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Hassani.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hassani, H., Machado, J.A.T. & Avazzadeh, Z. An effective numerical method for solving nonlinear variable-order fractional functional boundary value problems through optimization technique. Nonlinear Dyn 97, 2041–2054 (2019). https://doi.org/10.1007/s11071-019-05095-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-019-05095-2

Keywords

Search

Navigation