Skip to main content
SpringerLink
Log in

A New Approach for Solving a Class of Delay Fractional Partial Differential Equations

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this article, a new numerical approach has been proposed for solving a class of delay time-fractional partial differential equations. The approximate solutions of these equations are considered as linear combinations of Müntz–Legendre polynomials with unknown coefficients. Operational matrix of fractional differentiation is provided to accelerate computations of the proposed method. Using Padé approximation and two-sided Laplace transformations, the mentioned delay fractional partial differential equations will be transformed to a sequence of fractional partial differential equations without delay. The localization process is based on the space-time collocation in some appropriate points to reduce the fractional partial differential equations into the associated system of algebraic equations which can be solved by some robust iterative solvers. Some numerical examples are also given to confirm the accuracy of the presented numerical scheme. Our results approved decisive preference of the Müntz–Legendre polynomials with respect to the Legendre polynomials.

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

Similar content being viewed by others

References

  1. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

    Article  MathSciNet  Google Scholar 

  2. 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)

    Article  Google Scholar 

  3. Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection–dispersion equation. Water Resour. Res. 36, 1403–1412 (2000)

    Article  Google Scholar 

  4. Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London (2010)

    Book  Google Scholar 

  5. Perestyuk, M.O., Chernikova, O.S.: Some modern aspects of the theory of impulsive differential equations. Ukrain. Math. J. 60, 91 (2008)

    Article  MathSciNet  Google Scholar 

  6. Rezounenko, A.V., Wu, J.H.: A non-local PDE model for population dynamics with state-selective delay: local theory and global attractors. J. Comput. Appl. Math. 190, 99–113 (2006)

    Article  MathSciNet  Google Scholar 

  7. Wang, H., Hu, H.: Remarks on the perturbation methods in solving the second-order delay differential equations. Nonlinear Dyn. 33, 379–398 (2003)

    Article  Google Scholar 

  8. Khasawneh, F.A., Barton, D.A.W., Mann, B.P.: Periodic solutions of nonlinear delay differential equations using spectral element method. Nonlinear Dyn. 67, 641–658 (2012)

    Article  MathSciNet  Google Scholar 

  9. Aziz, I., Amin, R.: Numerical solution of a class of delay differential and delay partial differential equations via Haar wavelet. Appl. Math. Model. 40, 10286–10299 (2016)

    Article  MathSciNet  Google Scholar 

  10. Ghasemi, M., Fardi, M., Khoshsiar Ghaziani, R.: Numerical solution of nonlinear delay differential equations of fractional order in reproducing kernel Hilbert space. Appl. Math. Comput. 268, 815–831 (2016)

    MathSciNet  Google Scholar 

  11. Morgadoa, M.L., Fordb, N.J., Limac, P.M.: Analysis and numerical methods for fractional differential equations with delay. J. Comput. Appl. Math. 252, 159–168 (2013)

    Article  MathSciNet  Google Scholar 

  12. Tumwiine, J., Luckhaus, S., Mugisha, J.Y.T., Luboobi, L.S.: An age-structured mathematical model for the within host dynamics of malaria and the immune system. J. Math. Model. Algorithms 7, 79–97 (2008)

    Article  MathSciNet  Google Scholar 

  13. Alvarez-Vázquez, Lino J., Fernández, F.J., Muũoz-Sola, Rafael: Analysis of a multistate control problem related to food technology. J. Differ. Equ. 245, 130–153 (2008)

    Article  MathSciNet  Google Scholar 

  14. Cheng, Z., Lin, Y.Z.: The exact solution of a class of delay parabolic partial differential equation. J. Nat. Sci. Heilongjiang Univ. 25, 155–162 (2008)

    Google Scholar 

  15. Jackiewicz, Z., Zubik-Kowal, B.: Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations. Appl. Numer. Math. 56, 433–443 (2006)

    Article  MathSciNet  Google Scholar 

  16. Ouyang, Z.: Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay. Comput. Math. Appl. 61, 860–870 (2011)

    Article  MathSciNet  Google Scholar 

  17. Rihan, F.A.: Computational methods for delay parabolic and time-fractional partial differential equations. Numer. Methods Partial Differ. Equ. 26, 1556–1571 (2009)

    MathSciNet  MATH  Google Scholar 

  18. Wu, J.: A wavelet operational method for solving fractional partial differential equations numerically. Appl. Math. Comput. 214, 31–40 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Momani, S., Odibat, Z.: Comparison between the homotopy perturbation method and variational iteration method for a linear partial differential equations. Comput. Math. Appl. 54, 910–919 (2007)

    Article  MathSciNet  Google Scholar 

  20. Borwein, P., Erdélyi, T., Zhang, J.: Müntz systems and orthogonal Müntz–Legendre polynomials. Trans. Am. Math. Soc. 2, 523–542 (1994)

    MATH  Google Scholar 

  21. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204. Elsevier Science B. V, Amsterdam (2006)

    Google Scholar 

  22. El-Kady, M.: Legendre approximations for solving optimal control problems governed by ordinary differential equations. Int. J. Control Sci. Eng. 4, 54–59 (2012)

    Article  Google Scholar 

  23. Esmaeili, Sh, Shamsi, M., Luchko, Y.: Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials. Comput. Math. Appl. 62, 918–929 (2011)

    Article  MathSciNet  Google Scholar 

  24. Ejlali, N., Hosseini, S.M.: A pseudospectral method for fractional optimal control problems. J. Optim. Theory Appl. 174, 83–107 (2017)

    Article  MathSciNet  Google Scholar 

  25. Maleki, M., Hashim, I., Abbasbandy, S., Alsaedi, A.: Direct solution of a type of constrained fractional variational problems via an adaptive pseudospectral method. J. Comput. Appl. Math. 283, 41–57 (2015)

    Article  MathSciNet  Google Scholar 

  26. Turut, V., Güzel, N.: On solving partial differential equations of fractional order by using the variational iteration method and multivariate Padé approximations. Eur. J. Pure Appl. Math. 6, 147–171 (2013)

    MathSciNet  MATH  Google Scholar 

  27. Turut, V., Güzel, N.: Multivariate Padé approximation for solving nonlinear partial differential equations of fractional order. Abstr. Appl. Anal. 2013, Article ID 746401 (2013)

    Article  Google Scholar 

  28. Cuyt, A.: How well can the concept of Padé approximant be generalized to the multivariate case? J. Comput. Appl. Math. 105, 25–50 (1999)

    Article  MathSciNet  Google Scholar 

  29. Baker, G.A., Graves-Morris, P.R.: Padé Approximants, vol. 59. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  30. Matsuzuka, I., Nagasawa, K., Kitahama, A.: A proposal for two-sided Laplace transforms and its application to electronic circuits. Appl. Math. Comput. 100, 1–11 (1999)

    MathSciNet  MATH  Google Scholar 

  31. Pol, V.B., Bremmer, H.: Operational Calculus Based on the Two-sided Laplace Integral. Cambridge University Press, London (1955)

    MATH  Google Scholar 

  32. Fox, W.P.: Mathematical Modeling with Maple. Brooks Cole, Boston (2011)

    Google Scholar 

  33. Sun, Zh, Zhang, Z.: A linearized compact difference scheme for a class of nonlinear delay partial differential equations. Appl. Math. Model. 37, 742–752 (2013)

    Article  MathSciNet  Google Scholar 

  34. Lee, A.Y.: Hereditary optimal control problems: numerical method based upon a padé approximation. J. Optim. Theory Appl. 56, 157–166 (1988)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics School of Mathematical Sciences, Shahrood University of Technology, P.O. Box 3619995161-316, Shahrood, Iran

    Soleiman Hosseinpour & Alireza Nazemi

  2. Department of Mathematics, Kosar University of Bojnord, P. O. Box 9415615458, Bojnord, Iran

    Emran Tohidi

Authors
  1. Soleiman Hosseinpour
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alireza Nazemi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Emran Tohidi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Soleiman Hosseinpour.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hosseinpour, S., Nazemi, A. & Tohidi, E. A New Approach for Solving a Class of Delay Fractional Partial Differential Equations. Mediterr. J. Math. 15, 218 (2018). https://doi.org/10.1007/s00009-018-1264-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-018-1264-z

Keywords

Mathematics Subject Classification

Search

Navigation