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Numerical solution of the nonlinear conformable space–time fractional partial differential equations

Indian Journal of Pure and Applied Mathematics volume 52, pages 407–419 (2021)Cite this article

Abstract

In this paper, a numerical approach for solving the nonlinear space-time fractional partial differential equations with variable coefficients is proposed. The fractional derivatives are described in the conformable sense. The numerical approach is based on shifted Chebyshev polynomials of the second kind and finite difference method. The proposed scheme reduces the main problem to a system of nonlinear algebraic equations. The validity and the applicability of the proposed technique are shown by numerical examples.

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References

  1. R.L. Bagley, P.J. Torvik. On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 51 (1984) 294-298.

    MATH  Google Scholar 

  2. I. Podlubny. Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calculus. App. Anal. 5 (2002) 367 - 386.

    MathSciNet  MATH  Google Scholar 

  3. I. Podlubny. Fractional differential equations, Academic press, New York, 1999.

    MATH  Google Scholar 

  4. F. Mainardi. Fractional calculus: Some basic problems in continuum and statistical mechanics, In: A. Carpinteri, F. Mainardi, Editors, Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, New York, 1997.

    MATH  Google Scholar 

  5. M.D. Thomas and P.B. Bamforth, Modelling chloride diffusion in concrete: Effect of fly ash and slag. Cem. Concr. Res.29 (4) (1999) 487-495.

    Google Scholar 

  6. A. Khitab, S. Lorente, and J.P. Ollivier, Predictive model for chloride penetration through concrete. Mag. Concr. Res. 57 (9) (2005) 511-520.

    Google Scholar 

  7. M.A. Firoozjaee, S.A. Yousefi. A numerical approach for fractional partial differential equations by using Ritz approximation. Appl. Math. Comput. 338 (2018) 711-721.

    MathSciNet  MATH  Google Scholar 

  8. H. Dehestani, Y. Ordokhani, M. Razzaghi. Fractional-order Legendre-Laguerre functions and their applications in fractional partial differential equations. Appl. Math. Comput. 336 (2018) 433-453.

    MathSciNet  MATH  Google Scholar 

  9. Shaher Momani, Zaid Odibat. A novel method for nonlinear fractional partial differential equations: Combination of DTM and generalized Taylor’s formula. J. Comput. Appl. Math. 220 (2008) 85-95.

    MathSciNet  MATH  Google Scholar 

  10. U. Saeed, M. Rehman. Haar wavelet Picard method for fractional nonlinear partial differential equations. Appl.Math.Comput. 264 (2015) 310-322.

    MathSciNet  MATH  Google Scholar 

  11. T. A. Biala, A. Q. M. Khaliq. Parallel algorithms for nonlinear time-space fractional parabolic PDEs. J. Comput. Phys. 375 (2018) 135-154.

    MathSciNet  MATH  Google Scholar 

  12. M.M. Khader, K. M. Saad. A numerical approach for solving the fractional Fisher equation using Chebyshev spectral collocation method. Chaos Soliton Fract. 110 (2018) 169-177.

    MathSciNet  MATH  Google Scholar 

  13. Z.Odibat, S. Momani. Numerical methods for nonlinear partial differential equations of fractional order. Appl. Math. Model. 32 (2008) 28-39.

    MATH  Google Scholar 

  14. Z. Mohamed, T. M. Elzaki. Applications of new integral transform for linear and nonlinear fractional partial differential equations. J. King Saud Univ. Sci. 2018

    Google Scholar 

  15. S. Zhang, S. Hong. Variable separation method for a nonlinear time fractional partial differential equation with forcing term. J. Comput. Appl. Math. 339 (2018) 297-305.

    MathSciNet  MATH  Google Scholar 

  16. A. Demir, M. A. Bayrak, E. Ozbilge. An Approximate Solution of the Time-Fractional Fisher Equation with Small Delay by Residual Power Series Method. Math. Probl. Eng. 2018 (2018) 1-8.

    MathSciNet  MATH  Google Scholar 

  17. A. M. Nagy. Numerical solution of time fractional nonlinear Klein-Gordon equation using Sinc-Chebyshev collocation method. Appl.Math.Comput. 310 (2017) 139-148.

    MathSciNet  MATH  Google Scholar 

  18. A. Yusuf, M. Inc, A. I. Aliyu, D. Baleanu. Efficiency of the new fractional derivative with nonsingular Mittag-Leffler kernel to some nonlinear partial differential equations. Chaos Soliton Fract. 116 (2018) 220-226.

    MathSciNet  MATH  Google Scholar 

  19. P. Prakash, S. Harikrishnan, M. Benchohra. Oscillation of certain nonlinear fractional partial differential equation with damping term. Appl. Math. Lett. 43 (2015) 72-79.

    MathSciNet  MATH  Google Scholar 

  20. R. Khalil, M. A. Horani, A. Yousef, M. Sababheh, A new defnition of fractional derivative. J. Comput. Appl. Math. 264 (2014) 65-70.

    MathSciNet  MATH  Google Scholar 

  21. T. Abdeljawad. On conformable fractional calculus. J. Comput. Appl. Math. 279 (2015) 57-66.

    MathSciNet  MATH  Google Scholar 

  22. Q. Feng. A new approach for seeking coefficient function solutions of conformable fractional partial differential equations based on the Jacobi elliptic equation. Chinese J. Phys. 56 (2018) 2817-2828.

    Google Scholar 

  23. C. Chen, Y. L. Jiang. Simplest equation method for some time-fractional partial differential equations with conformable derivative. Comput. Math. Appl. 75 (2018) 2978-2988.

    MathSciNet  MATH  Google Scholar 

  24. K. Hosseini, R. Ansari. New exact solutions of nonlinear conformable time- fractional Boussinesq equations using the modified Kudryashov method. Wave Random Complex. 27 (2017) 628-636.

    MathSciNet  Google Scholar 

  25. Abdelsalam UM. Exact travelling solutions of two coupled (2 + 1)-Dimensional Equations. J. Egyptian Math. Soc. 25 (2017) 125-128.

    MathSciNet  MATH  Google Scholar 

  26. H. Thabet, S. Kendre. Analytical solutions for conformable space-time fractional partial differential equations via fractional differential transform. Chaos Soliton Fract. 109 (2018) 238-245.

    MathSciNet  MATH  Google Scholar 

  27. Y. Yang. Solving a Nonlinear Multi-Order Fractional Differential Equation Using Legendre Pseudo-Spectral Method. AM 4 (2013) 113-118.

    Google Scholar 

  28. F. Mohammadi and C. Cattani. A generalized fractional-order Legendre wavelet Tau method for solving fractional differential equations. J. Comput. Appl. Math. 339 (2018) 306-316.

    MathSciNet  MATH  Google Scholar 

  29. E.H. Doha, A.H. Bhrawy, and S.S. Ezz-Eldien. Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Appl. Math. Model. 35 (2011) 5662-5672.

    MathSciNet  MATH  Google Scholar 

  30. N.H. Sweilam, A.M. Nagy, A. A. El-Sayed. Second kind shifted Chebyshev polynomials for solving space fractional order diffusion equation. Chaos Soliton Fract. 73 (2015) 141-147.

    MathSciNet  MATH  Google Scholar 

  31. N.H. Sweilam, A.M. Nagy, A. A. El-Sayed. On the numerical solution of space fractional order diffusion equation via shifted Chebyshev polynomials of the third kind. J. King Saud Univ. Sci. 28 (2016) 41-47.

    Google Scholar 

  32. N.H. Sweilam, A.M. Nagy, A. A. El-Sayed. Numerical approach for solving space fractional order diffusion equations using shifted Chebyshev polynomials of the fourth kind. Turk J. Math. 40 (2016) 1283-1297.

    MathSciNet  MATH  Google Scholar 

  33. M.M. Khader. On the numerical solutions for the fractional diffusion equation. Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 2535-42.

    MathSciNet  MATH  Google Scholar 

  34. H. Azizi, G.B. Loghmani. Numerical approximation for space fractional diffusion equations via Chebyshev finite difference method. J. Fract. Ca.l Appl. 4 (2013) 303-311.

  35. H. Azizi, G.B. Loghmani. A numerical method for space fractional diffusion equations using a semi-disrete scheme and Chebyshev collocation method. J. Math. Comput. Sci. 8 (2014) 226-235.

    Google Scholar 

  36. A. Saadatmandia, M. Dehghan. A tau approach for solution of the space fractional diffusion equation. Comput. Math. Appl. 62 (2011) 1135-1142.

    MathSciNet  MATH  Google Scholar 

  37. X.Li, C. Xu. A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 247 (2009) 2108-2131.

    MathSciNet  MATH  Google Scholar 

  38. C. Piret, E. Hanert. A radial basis functions method for fractional diffusion equations. J. Comput. Phys. 14 (2012) 71-81.

    MathSciNet  MATH  Google Scholar 

  39. M.M. Khader. An efficient approximate method for solving linear fractional Klein-Gordon equation based on the generalized Laguerre polynomials. Int. J. Comput. Math. 90 (2013) 1853-1864.

    MathSciNet  MATH  Google Scholar 

  40. H. C. Yaslan. Numerical solution of the conformable space-time fractional wave equation. Chinese J. Phys. 56 (2018) 2916-2925.

    MathSciNet  Google Scholar 

  41. J. C. Mason, D. C. Handscomb, Chebyshev Polynomials, Chapman and Hall, CRC, New York, NY, Boca Raton, 2003.

    MATH  Google Scholar 

  42. Wazwaz AM, Gorguis A. An analytic study of Fishers equation by using Adomian decomposition method. Appl. Math. Comput. 154 (2004) 609-20.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Pamukkale University, 20070, Denizli, Turkey

    H. Çerdik Yaslan

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  1. H. Çerdik Yaslan
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Correspondence to H. Çerdik Yaslan.

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Communicated by G.D. Veerappa Gowda.

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Yaslan, H.Ç. Numerical solution of the nonlinear conformable space–time fractional partial differential equations. Indian J Pure Appl Math 52, 407–419 (2021). https://doi.org/10.1007/s13226-021-00057-0

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  • DOI: https://doi.org/10.1007/s13226-021-00057-0

Keywords

  • Nonlinear space–time fractional partial differential equation
  • Conformable fractional derivative
  • Finite difference method
  • Newton method
  • Shifted Chebyshev polynomials of the second kind

Mathematics Subject Classification

  • 35G31
  • 35R11
  • 65M70