Skip to main content
SpringerLink
Log in

Fractional shifted legendre tau method to solve linear and nonlinear variable-order fractional partial differential equations

  • Original Research
  • Published:
Mathematical Sciences Aims and scope Submit manuscript

Abstract

Here, we shed light on the fractional linear and nonlinear Klein–Gorden partial differential equations via Fractional Shifted Legendre Tau Method. With this objective, the operational matrices of fractional-order shifted Legendre functions (FSLFs) are derived and combined with the Tau method to convert the fractional-order differential equations to a system of solvable algebraic equations. The validity and the efficiency of the operational matrices are tested. Our findings yield an affirmative consequence, indicating applicability of the proposed method for nonlinear equations appearing in science and engineering.

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

Similar content being viewed by others

References

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

    Book  Google Scholar 

  2. Lu, D., Liang, J., Du, X., Ma, C., Gao, Z.: Fractional elastoplastic constitutive model for soils based on a novel 3d fractional plastic flow rule. Comput. Geotech. 105, 277–290 (2019)

    Article  Google Scholar 

  3. Li, C., Guo, H., Tian, X., He, T.: Generalized thermoelastic diffusion problems with fractional order strain. Eur. J. Mech. -A/Solids 78, 103827 (2019)

    Article  MathSciNet  Google Scholar 

  4. Miles, P.R., Pash, G.T., Smith, R.C., Oates, W.S.: Global sensitivity analysis of fractional-order viscoelasticity models. In: Behavior and Mechanics of Multifunctional Materials XIII, Vol. 10968, p. 1096806. International Society for Optics and Photonics (2019)

  5. Roy, R., Akbar, M.A., Wazwaz, A.M.: Exact wave solutions for the nonlinear time fractional Sharma-Tasso-olver equation and the fractional Klein-Gordon equation in mathematical physics. Opt. Quant. Electron. 50(1), 25 (2018)

    Article  Google Scholar 

  6. Hosseini, V.R., Shivanian, E., Chen, W.: Local integration of 2-d fractional telegraph equation via local radial point interpolant approximation. Eur. Phys. J. Plus 130(2), 33 (2015)

    Article  Google Scholar 

  7. Aslefallah, M., Shivanian, E.: Nonlinear fractional integro-differential reaction-diffusion equation via radial basis functions. Eur. Phys. J. Plus 130(47), 1–9 (2015)

    Google Scholar 

  8. Hosseini, V.R., Shivanian, E., Chen, W.: Local radial point interpolation (mlrpi) method for solving time fractional diffusion-wave equation with damping. J. Comput. Phys. 312, 307–332 (2016)

    Article  MathSciNet  Google Scholar 

  9. Shivanian, E.: Spectral meshless radial point interpolation (SMRPI) method to two-dimensional fractional telegraph equation. Math. Methods Appl. Sci. 39(7), 1820–1835 (2016)

    Article  MathSciNet  Google Scholar 

  10. Shivanian, E.: Analysis of the time fractional 2-d diffusion-wave equation via moving least square (mls) approximation. Int. J. Appl. Comput. Math. 3(3), 2447–2466 (2017)

    Article  MathSciNet  Google Scholar 

  11. Shivanian, E., Jafarabadi, A.: An improved spectral meshless radial point interpolation for a class of time-dependent fractional integral equations: 2d fractional evolution equation. J. Comput. Appl. Math. 325, 18–33 (2017)

    Article  MathSciNet  Google Scholar 

  12. Alqahtani, R.T.: Approximate solution of non-linear fractional Klein–Gordon equation using spectral collocation method. Appl. Math. 6, 2175–2181 (2015)

    Article  Google Scholar 

  13. Wazwaz, A.: Compacton solitons and periodic solutions for some forms of nonlinear Klein-Gordon equations. Chaos Solitons Fract. 28, 1005–1013 (2006)

    Article  MathSciNet  Google Scholar 

  14. Elgarayhi, A.: New periodic wave solutions for the shallow water equations and the generalized Klein-Gordon equation. Commun. Nonlinear Sci. Numer. Simul. 13, 877–888 (2008)

    Article  MathSciNet  Google Scholar 

  15. Golmankhaneh, A.K., Golmankhaneh, A.K., Baleanu, D.: On nonlinear fractional Klein-Gordon equation. Signal Process. 91(3), 446–451 (2011)

    Article  Google Scholar 

  16. Khader, M., Swetlam, N., Mahdy, A.: The Chebyshev collection method for solving fractional order Klein–Gordon equation. Wseas Trans. Math. 13, 31–38 (2014)

    Google Scholar 

  17. Ortiz, E., Samara, H.: An operational approach to the Tau method for the numerical solution of nonlinear differential equations. Computing 27, 15–25 (1981)

    Article  MathSciNet  Google Scholar 

  18. Ortiz, E., Samara, H.: Numerical solution of differential eigenvalue problems with an operational approach to the Tau method. Computing 31, 95–103 (1983)

    Article  MathSciNet  Google Scholar 

  19. Shaban, M., Shivanian, E., Abbasbandy, S.: Analyzing magneto-hydrodynamic squeezing flow between two parallel disks with suction or injection by a new hybrid method based on the tau method and the homotopy analysis method. Eur. Phys. J. Plus 128(11), 133 (2013)

    Article  Google Scholar 

  20. Shaban, M., Kazem, S., Shivanian, E.: Fully discrete tau solution for some types of non-local heat transport equations. Appl. Anal. 1, 1–15 (2017)

    MATH  Google Scholar 

  21. Rida, S., Yousef, A.: On the fractional order Rodrigues formula for the Legendre polynomials. Adv. Appl. Math. Sci. 10, 509–518 (2011)

    MathSciNet  MATH  Google Scholar 

  22. Klimek, M., Agrawal, O.P.: Fractional sturm-liouville problem. Comput. Math. Appl. 66(5), 795–812 (2013)

    Article  MathSciNet  Google Scholar 

  23. Kazem, S., Abbasbandy, S., Kumar, S.: Fractional-order Legendre functions for solving fractional-order differential equations. Appl. Math. Model. 37, 5498–5510 (2013)

    Article  MathSciNet  Google Scholar 

  24. Mokhtary, P.: Operational Tau method for nonlinear multi-order FDEs. Iranian J. Numer. Anal. Optim. 4, 43–55 (2014)

    MATH  Google Scholar 

  25. Kazem, S., Shaban, M., Rad, J.A.: Solution of Coupled Burger’s equation based on operational matrices of d–dimensional orthogonal functions. Z. Naturforsch 67(a), 267–274 (2007)

  26. Wang, G., Hashemi, M.: Lie symmetry analysis and soliton solutions of time-fractional k(m, n) equation. Pramana 88(1), 7 (2017)

    Article  Google Scholar 

  27. Hashemi, M., Baleanu, D.: Lie symmetry analysis and exact solutions of the time fractional gas dynamics equation 18, 3–4 (2016)

  28. Kheybari, S., Darvishi, M.T., Hashemi, M.S.: Numerical simulation for the space-fractional diffusion equations. Appl. Math. Comput. 348, 57–69 (2019)

    MathSciNet  MATH  Google Scholar 

  29. Hashemi, M.S., Inc, M., Yusuf, A.: On three-dimensional variable order time fractional chaotic system with nonsingular kernel. Chaos Solitons Fract 133, 109628 (2020)

    Article  MathSciNet  Google Scholar 

  30. Razzaghi, M., Oppenheimer, S., Ahmad, F.: Tau method approximation for radiative transfer problems in a slab medium. JQSRT 72, 439–447 (2002)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Chemistry, University of Minnesota, Minneapolis, MN, 55455, USA

    Maliheh Shaban Tameh

  2. Department of Applied Mathematics, Imam Khomeini International University, Qazvin, 34149-16818, Iran

    Elyas Shivanian

Authors
  1. Maliheh Shaban Tameh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elyas Shivanian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elyas Shivanian.

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

Tameh, M.S., Shivanian, E. Fractional shifted legendre tau method to solve linear and nonlinear variable-order fractional partial differential equations. Math Sci 15, 11–19 (2021). https://doi.org/10.1007/s40096-020-00351-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40096-020-00351-8

Keywords

Search

Navigation