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

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.

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References

  1. 1.

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

    Google Scholar 

  2. 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. 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)

    MathSciNet  Article  Google Scholar 

  4. 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. 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. 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. 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. 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)

    MathSciNet  Article  Google Scholar 

  9. 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)

    MathSciNet  Article  Google Scholar 

  10. 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)

    MathSciNet  Article  Google Scholar 

  11. 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)

    MathSciNet  Article  Google Scholar 

  12. 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. 13.

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

    MathSciNet  Article  Google Scholar 

  14. 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)

    MathSciNet  Article  Google Scholar 

  15. 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. 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. 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)

    MathSciNet  Article  Google Scholar 

  18. 18.

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

    MathSciNet  Article  Google Scholar 

  19. 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. 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. 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. 22.

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

    MathSciNet  Article  Google Scholar 

  23. 23.

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

    MathSciNet  Article  Google Scholar 

  24. 24.

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

    MATH  Google Scholar 

  25. 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. 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. 27.

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

  28. 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. 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)

    MathSciNet  Article  Google Scholar 

  30. 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 

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Correspondence to Elyas Shivanian.

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Tameh, M.S., Shivanian, E. Fractional shifted legendre tau method to solve linear and nonlinear variable-order fractional partial differential equations. Math Sci (2020). https://doi.org/10.1007/s40096-020-00351-8

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Keywords

  • Klein–Gorden partial differential equations
  • Fractional-order Legendre functions
  • Operational matrices
  • Caputo fractional derivatives