A numerical scheme based on non-discretization of data for boundary value problems of fractional order differential equations

  • Kamal ShahEmail author
  • JinRong Wang
Original Paper


In this article, we develop a powerful method for the numerical solution of boundary value problems (BVPs) of fractional order differential equations (FDEs). Omitting the discretization of data by using Bernstein polynomials, we construct the required scheme. With the help of this scheme we convert the concerned FDEs to algebraic equations whose solutions led us to the numerical solution of the considered problem. Numerical examples are provided to illustrate our main results. Also comparison of the results with the exact solutions and other method like (Haar wavelets) is provided to justify the efficiency of the proposed scheme.


Fractional differential equations Boundary value problem Numerical solutions Bernstein polynomials Discretization of data 

Mathematics Subject Classifications

34L05 65L05 65T99 34G10 



We are really thankful to the reviewers for their useful suggestions which improved this paper very well.

Compliance with ethical standards

Competing interests

We declare that no competing interest exist regarding this paper.

Authors contribution

All authors equally contributed this paper.


  1. 1.
    Arikoglu, A., Ozkol, I.: Solution of fractional differential equations by using differential transform method. Chaos Solitons Fractals. 34, 1473–1481 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arikoglu, A., Ozkol, I.: Solution of fractional integro-differential equations by using fractional differential transform method. Chaos Solitons Fractals. 40, 521–529 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bai, Z., Dong, X., Yin, C.: Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions. Bound. Value Probl. 2016(63), 11 (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bai, Z., Zhang, S., Sun, S., Yin, C.: Monotone iterative method for fractional differential equations. Electron. J. Diff. Equ. 2016(6), 1–8 (2016)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Benchohra, M., Graef, J.R., Hamani, S.: Existence results for boundary value problems with nonlinear fractional differential equations. J. Appl. Anal. 87, 851–863 (2008)CrossRefGoogle Scholar
  6. 6.
    Bhattia, M.I., Bracken, P.: Solutions of differential equations in a Bernstein polynomial basis. J. Comput. Appl. Math. 205, 272–280 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Blank, L.: Numerical treatment of differential equations of fractional order, Numerical Analysis Report 287, Manchester Centre for Computational Mathematics (1996)Google Scholar
  8. 8.
    Baillie, R.T.: Long memory processes and fractional integration in econometrics. J. Econom. 73, 5–59 (1996)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Darania, P., Ebadian, A.: A method for the numerical solution of the integro-differential equations. Appl. Math. Comput. 188, 657–668 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Diethelm, K., Walz, G.: Numerical solution of fractional order differential equations by extropolation. Numer. Algorithms. 16, 231–253 (1997)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Das, S.: Analytical solution of a fractional diffusion equation by variational iteration method. Comput. Math. Appl. 57, 483–487 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    El-Wakil, S.A., Elhanbaly, A., Abdou, M.A.: Adomian decomposition method for solving fractional nonlinear differential equations. Appl. Math. Comput. 182, 313–324 (2006)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Erturk, V.S., Momani, S.: Solving systems of fractional differential equations using differential transform method. J. Comput. Appl. Math. 215, 142–151 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Erturk, V.S., Momani, S., Odibat, Z.: Application of generalized differential transform method to multi-order fractional differential equations. Comm. Nonlinear Sci. Numer. Simul. 13, 1642–1654 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Herrmann, R.: Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore (2014)CrossRefGoogle Scholar
  16. 16.
    Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)CrossRefGoogle Scholar
  17. 17.
    He, J.H.: Some applications of nonlinear fractional differential equations and their approximations. Bull. Sci. Technol. 15, 86–90 (1999)Google Scholar
  18. 18.
    Gzyl, H., Palacios, J.L.: On the approximation properties of Bernstein polynomials via probabilistic tools. Bol. Asoc. Mat. Venez. 10, 5–13 (2003)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Gmez-Aguilar, J.F., Rosales-Garca, J.J., Gua-Caldern, M.: RLC electrical circuit of non-integer order. Cent. Eur. J. Phys. 11(10), 1361–5 (2013)Google Scholar
  20. 20.
    Hashim, I., Abdulaziz, O., Momani, S.: Homotopy analysis method for fractional IVPs. Commun. Nonlinear Sci. Numer. Simul. 14, 674–684 (2009)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ichise, M., Nagayanagi, Y., Kojima, T.: An analog simulation of non-integer order transfer functions for analysis of electrode processes. J. Electroanal. Chem. Interfacial Electrochem. 33, 253–265 (1971)CrossRefGoogle Scholar
  22. 22.
    Inca, M., Kiliç, B.: Classification of traveling wave solutions for time-fractional fifth-order KdV-like equation. Waves Random Complex Media 2014, 393–404 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Netw. 32, 245–256 (2012)CrossRefGoogle Scholar
  24. 24.
    Kilbas, A.A., Marichev, O.I., Samko, S.G.: Fractional Integrals and Derivatives (Theory and Applications). Gordon and Breach, Basel (1993)zbMATHGoogle Scholar
  25. 25.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, North-Holland athematics Studies, vol. 204. Elsevier, Amsterdam (2006)Google Scholar
  26. 26.
    Khalil, H., Shah, K., Khan, R.A.: Approximate solution of boundary value problems using shifted Legendre polynomials. Appl. Comput. Math. 16, 1–15 (2017)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Lakshmikantham, V., Leela, S., Vasundhara, J.: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge (2009)zbMATHGoogle Scholar
  28. 28.
    Liua, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker-Planck equation. J. Comput. Appl. Math. 146, 209–219 (2004)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Li, Y., Shah, K.: Numerical solutions of coupled systems of fractional order partial differential equations. Adv. Math. Phys. 2017, 14 (2017)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)zbMATHGoogle Scholar
  31. 31.
    Magin, R.L.: Fractional calculus in bioengineering-part 2. J. Crit. Rev. Biomed. Eng. 32, 105–193 (2004)CrossRefGoogle Scholar
  32. 32.
    Magin, R.L.: Fractional calculus in bioengineering. J. Crit. Rev. Biomed. Eng. 32, 1–104 (2004)CrossRefGoogle Scholar
  33. 33.
    Lundstrom, B.N., Higgs, M.H., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nat. Neurosci. 11(11), 1335–1342 (2008)CrossRefGoogle Scholar
  34. 34.
    Mandal, B.N., Bhattacharya, S.: Numerical solution of some classes of integral equations using Bernstein polynomials. Appl. Math. Comput. 190, 1707–1716 (2007)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Momani, S., Odibat, Z.: Numerical approach to differential equations of fractional order. J. Comput. Appl. Math. 207, 96–110 (2007)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Momani, S., Odibat, Z.: Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys. Lett. A 365, 345–350 (2007)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Mahsud, Y., Shah, N.A., Vieru, D.: Influence of time-fractional derivatives on the boundary layer flow of Maxwell fluids. Chin. J. Phys. 55, 1340–1351 (2017)CrossRefGoogle Scholar
  38. 38.
    Moshrefi-Torbati, M., Hammond, J.K.: Physical and geometrical interpretation of fractional operators. J. Frankl. Inst. 335(6), 1077–1086 (1998)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Nigmatullin, R.R.: Fractional integral and its physical interpretation. J. Theor. Math. Phys. 90(3), 242–251 (1992)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Ordokhani, Y., Davaei, S.: Approximate solutions of differential equations by using the Bernstein polynomials. ISRN Appl. Math. 2011, 1–15 (2011). (Art. ID 787694)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Odibat, Z., Momani, S.: Numerical methods for nonlinear partial differential equations of fractional order. Appl. Math. Model. 32, 28–39 (2008)CrossRefGoogle Scholar
  42. 42.
    Odibat, Z., Shawagfeh, N.: Generalized Taylor’s formula. Appl. Math. Comput. 186, 286–293 (2007)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Podlubny, I.: Fractional Differential Equations, Mathematics in Science and Engineering. Academic Press, New York (1999)zbMATHGoogle Scholar
  44. 44.
    Rehman, M., Khan, R.: A note on boundary value problems for a coupled system of fractional differential equations. Comput. Math. Appl. 61, 2630–2637 (2011)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Rossikhin, Y.A., Shitikova, M.V.: Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. J. Appl. Mech. Rev. 50, 15–67 (1997)CrossRefGoogle Scholar
  46. 46.
    Rehman, M., Khan, R.A.: The legender wavelet method for solving fractional differential equation. Commun. Nonlinear Sci. Numer. Simul. 16, 4163–4173 (2011)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Rahman, M.: Boundary Value Problems for Fractional Differential Equations: Existence Theory and Numerical Solutions; PhD dissertation, Nust University, Pakistan (2011)Google Scholar
  48. 48.
    Rosales Garcia, J.J., Calderon, M.G., Martinez Ortiz, J., Baleanu, D.: Motion of a particle in a resisting medium using fractional calculus approach. Proc. Roman. Acad. Series A. 14(1), 42–7 (2013)MathSciNetGoogle Scholar
  49. 49.
    Rosales, J., Guia, M., Gomez, F., Aguilar, F., Martinez, J.: Two dimensional fractional projectile motion in a resisting medium. Cent. Eur. J. Phys. 12(7), 517–20 (2014)Google Scholar
  50. 50.
    Su, X.: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 22, 64–69 (2009)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Saadatmandi, A., Deghan, M.: A new operational matrix for solving fractional-order differential equation. Comput. Math. Appl. 59, 1326–1336 (2010)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Shah, K., Khalil, H., Khan, R.A.: A generalized scheme based on shifted Jacobi polynomials for numerical simulation of coupled systems of multi-term fractional order partial differential equations. Lond. Math. Soc. J. Comput. Math. 20, 11–29 (2017)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Shah, K.: Multi Point Boundary Value Problems for Systems Of Fractional Differential Equations: Existence Theory and Numerical Simulations, Phd Disertation, University of Malakand, Pakistan (2016)Google Scholar
  54. 54.
    Sweilam, N.H., Khader, M.M., Al-Bar, R.F.: Numerical studies for a multi-order fractional differential equation. Phys. Lett. A. 371, 26–33 (2007)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Wang, Y., Liu, L., Wu, Y.: Positive solutions for a nonlocal fractional differential equation. Nonlinear Anal. 74, 3599–3605 (2011)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Yousefi, S.A., Behroozifar, M.: Operational matrices of Bernstein polynomials and their applications. Int. J. Syst. Sci. 41, 709–716 (2010)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Yang, X.J.: Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems. Thermal Sci. 21(3), 1161–1171 (2017)CrossRefGoogle Scholar
  58. 58.
    Yang, X.J., Machado, J.T.: A new fractional operator of variable order: application in the description of anomalous diffusion. Phys. A 481, 276–283 (2017)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Yang, X.J., Machado, J.T., Baleanu, D., Cattani, C.: On exact traveling-wave solutions for local fractional Korteweg-de Vries equation, Chaos: an interdisciplinary. J. Nonlinear Sci. 26(8), 084312–084320 (2016)zbMATHGoogle Scholar
  60. 60.
    Shah, K., Wang, J., Khalil, H., Khan, R.A.: Existence and numerical solutions of a coupled system of integral BVP for fractional differential equations. Adv. Differ. Equ. 2018(149), 1–21 (2018)MathSciNetGoogle Scholar
  61. 61.
    Peng, S., Wang, J., Yu, X.: Stable manifolds for some fractional differential equations. Nonlinear Anal. Model. Control 23, 642–663 (2018)CrossRefGoogle Scholar
  62. 62.
    Wang, J., Ibrahim, A.G., O’Regan, D.: Topological structure of the solution set for fractional non-instantaneous impulsive evolution inclusions. J. Fixed Point Theory Appl. 20, 1–25 (2018). (Art.59)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Wang, Y., Liu, L., Zhang, X., Wu, Y.: Positive solutions of a fractional semipositone differential system arising from the study of HIV infection models. Appl. Math. Comput. 258, 312–324 (2015)MathSciNetzbMATHGoogle Scholar
  64. 64.
    Zhang, X., Liu, L., Wu, Y.: Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives. Appl. Math. Comput. 219, 1420–1433 (2012)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Zhang, X., Liu, L., Wu, Y.: Variational structure and multiple solutions for a fractional advection-dispersion equation. Comput. Math. Appl. 68, 1794–1805 (2014)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Zhang, X., Mao, C., Liu, L., Wu, Y.: Exact iterative solution for an abstract fractional dynamic system model for bioprocess. Qual. Theory Dyn. Syst. 16, 205–222 (2017)MathSciNetCrossRefGoogle Scholar
  67. 67.
    Zhang, X., Liu, L., Wu, Y., Wiwatanapataphee, B.: Nontrivial solutions for a fractional advection dispersion equation in anomalous diffusion. Appl. Math. Lett. 66, 1–8 (2017)MathSciNetCrossRefGoogle Scholar
  68. 68.
    Zhu, B., Liu, L., Wu, Y.: Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equation with delay. Appl. Math. Lett. 61, 73–79 (2016)MathSciNetCrossRefGoogle Scholar
  69. 69.
    Zhang, J., Wang, J.: Numerical analysis for a class of Navier–Stokes equations with time fractional derivatives. Appl. Math. Comput. 336, 481–489 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Royal Academy of Sciences, Madrid 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MalakandKhyber PakhtunkhwaPakistan
  2. 2.Department of MathematicsGuizhou UniversityGuiyangChina
  3. 3.School of Mathematical SciencesQufu Normal UniversityQufuChina

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