An approximate method based on Bernstein polynomials for solving fractional PDEs with proportional delays

Document Type : Research Article


Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.


We apply a new method to solve fractional partial differential equations (FPDEs) with proportional delays. The method is based on expanding the unknown solution of FPDEs with proportional delays by the basis of Bernstein polynomials with unknown control points and uses operational matrices with the least-squares method to convert the FPDEs with proportional de lays to an algebraic system in terms of Bernstein coefficients (control points) approximating the solution of FPDEs. We use the Caputo derivatives of de gree 0 < α ≤ 1 as the fractional derivatives in our work. The main advantage of using this technique is that the method can easily be employed to a variety of FPDEs with or without proportional delays, and also the method offers a very simple and flexible framework for direct approximating of the solution of FPDEs with proportional delays. The convergence analysis of the present method is discussed. We show the effectiveness and superiority of the method by comparing the results obtained by our method with the results of some available methods in two numerical examples.


1. Aghdam, Y.E., Mesgrani, H., Javidi, M. and Nikan, O. A computational approach for the space-time fractional advection-diffusion equation arising in contaminant transport through porous media, Eng. Comput. (2020), 1–13.
2. Alipour, M., Rostamy, D. and Baleanu, D. Solving multi-dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices, J. Vib. Control, 19 (2012), 2523–2540.
3. Chakrabarti, A. and Martha, S.C. Approximate solutions of Fredholm integral equations of the second kind, Appl. Math. Comput. 211 (2009), 459–466.
4. Elagan, S.K., Sayed, M. and Higazy, M. An analytical study on fractional partial differential equations by Laplace transform operator method, Int. J. Appl. Eng. Res. 13(1) (2018), 545–549.
5. Ghomanjani, F. and Khorram, E. Approximate solution for quadratic Riccati differential equation, J, Taibah Univ. Sci. 11(2) (2017), 246–250.
6. Hassani, H., Tenreiro Machado, J.A., Avazzadeh, Z. and Naraghirad, E. Generalized shifted Chebyshev polynomials: solving a general class of nonlinear variable order fractional PDE, Commun. nonlinear Sci. 85 (2020), 1–16.
7. Javadi, S., Jani, M. and Babolian, E. A numerical scheme for space-time fractionaladvection-dispersion equation, Int. J. Nonlinear Anal. Appl. 7(2) (2016), 331–343.
8. Karimi, K. Bahadorimehr, A. and Mansoorzadeh, S. Numerical solution of nonlocal parabolic partial differential equation via Bernstein polynomial method, J. Math. 48 (2016), 47–53.
9. Kazem, S. An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations, Appl. Math. Model. 37(3) (2013), 1126–1136.
10. Ketabdari, A., Farahi, M.H. and Effati, S. A novel approximate method for solving 2D fractional optimal control problems using generalized fractional order of Bernstein functions, Manuscript in press.
11. Khan, H., Alipour, M., Jafari, H. and Khan, R.A. Approximate analytical solution of a coupled system of fractional partial differential equations by Bernstein polynomials, Int. J. Appl. Comput. Math. 2 (2016), 85–96.
12. Kreyszig, E. Introduction to functional analysis with applications, New York: Wiley, 1978.
13. Li, M., Li, G., Li, Z. and Jia, X. Determination of Time-Dependent Coefficients in Time-Fractional Diffusion Equations by Variational Iteration Method, J. Math. Res. 12(1) (2020), 1–74.
14. Li, W., Bai, L., Chen, Y., Dos Santos, S. and Li, B. Solution of linear fractional partial differential equations based on the operator matrix of fractional Bernstein polynomials and error correction, Int. J. Innov. Comput. I. 14 (2018), 211–226.
15. Momani, S. and Odibat, Z. Homotopy perturbation method for nonlinear pertial differential equations of fractional order, Phys. Lett. A, 365 (2007), 345–350.
16. Nemati, A. and Yousefi, S.A. A numerical scheme for solving two dimensional fractional optimal control problems by the Ritz method com bined with fractional operational matrix, IMA J. Math. Control Inform. 34(4) (2017), 1079–1097.
17. Rostamy, D., Karimi, K. and Mohamadi, E. Solving fractional partial differential equations by an efficient new basis, Int. J. Appl. Math. Comput. 5 (2013), 6–21.
18. Safaie, E. and Farahi, M.H. An approximation method for numerical solution of multi-dimensional feedback delay fractional optimal control problems by Bernstein polynomials, Iran J. Numerical Analysis and Optimization, 4(1) (2014), 77–94.
19. Safdari, H., Mesgarani, H., Javidi, M. and Esmaeelzade Aghdam, Y. Convergence analysis of the space fractional-order diffusion equation based on the compact finite difference scheme, J. Comput. Appl. Math. 39(2)
(2020), 1–15.
20. Sakar, M.G., Uludag, F. and Erdogan, F. Numerical solution of time fractional nonlinear PDEs with proportional delays by homotopy perturbation method, Appl. Math. Model. 40 (2016), 6639–6649.
21. Shah, R., Khan, H., Kumam, P., Arif, M. and Baleanu, D. Natural trans form decomposition method for solving fractional-order partial differential equations with proportional delay, Mathematics, 7 (2019), 1–14.
22. Singh, B.K. and Kumar, P. Fractional variational iteration method for solving fractional partial differential equations with proportional delay, Int. J. Differ. Equ. 4 (2017), 1–11.
23. Sripacharasakullert, P., Sawangtong, W. and Sawangtong, P. An approximate analytical solution of the fractional multi-dimensional Burgers equation by the homotopy perturbation method, Adv. Differ. Equ. 252 (2019), 1–12.
24. Stone, M. The generalized weierstrass approximation theorem, Math. Mag. 21 (1948), 237–254.
25. Thabet, H. and Kendre, S. New modification of Adomian decomposition method for solving a system of nonlinear fractional partial differential equations, Int. J. Adv. Appl. Math. Mech. 6(3) (2019), 1–13.
26. Turut, V. and G¨uzel, N. On solving partial differential equations of fractional order by using the variational iteration method and multivariate pad approximations, Eur. j. pure appl. math. 6(2) (2013), 147–171.
27. Wu, J. Theory and applications of partial functional differential equations, New York: Springer, 1996.
28. Wu, Q. and Zeng, X. Jacobi collocation methods for solving generalized space-fractional Burgers equations, Commun. Appl. Math. Comput. 2 (2020), 305–318.
29. Yaslan, H.. Legendre collocation method for the nonlinear space-time fractional partial differential equations, Iran J. Sci. Technol. Trans. Sci. 44 (2020), 239–249.
30. Zhang, Y. A finite difference method for fractional partial differential equation, Appl. Math. Comput. 215 (2009), 524–529.
31. Zheng, J., Sederberg, T.W. and Johnson, R.W. Least squares methods for solving differential equations using Bezier control points, Appl. Numer. Math. 48 (2004), 237–252.