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Taylor wavelets collocation technique for solving fractional nonlinear singular PDEs

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Abstract

A novel technique has been introduced to solve the Emden–Fowler equations. It has been derived from the Taylor wavelets collocation method. The proposed scheme has been successfully implemented in order to solve the singular equations. The singular problem converts to a system of algebraic equations that can be solved numerically. Moreover, the technique is very effective to remove the strong singularity point at \(x = 0\). The numerical experiments have been checked out with the exact and approximate solutions that have been achieved by others including the Adomian decomposition method (Wazwaz in Appl Math Comput 166:638–651, 2005), Modified Homotopy Perturbation Method (Singh et al. J Math Chem 54(4):918–931, 2016). Also, the error analysis of the technique has been considered.

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References

  1. Ahmadnezhad, G.H., Aghazadeh, NSh.: Haar wavelet iteration method for solving time fractional Fisher’s equation. Comput. Methods Differ. Equ. 8(3), 505–522 (2020)

    MathSciNet  Google Scholar 

  2. Bataineh, A.S., Noorani, M.S.M., Hashim, I.: Solutions of time-dependent Emden-Fowler type equations by homotopy analysis method. Phys. Lett. 371, 72–82 (2007)

    Article  Google Scholar 

  3. Celik, I.: Chebyshev Wavelet collocation method for solving generalized Burgers-Huxley equation. Math. Method Appl. Sci 39(3), 366–377 (2016)

    Article  MathSciNet  ADS  Google Scholar 

  4. Gupta, A.K., Saha Ray, S.: Numerical treatment for the solution of fractional fifth order Sawada-Kotera equation using Taylor wavelet method. Appl. Math. Model. 39(17), 5121–5130 (2015)

    Article  MathSciNet  Google Scholar 

  5. Heydari, M.H., Hooshmandasl, M.R., Ghaini, F.M.M., Cattani, C.: Wavelets method for the time fractional diffusion-wave equation. Phys. Lett. A 379, 71–76 (2015)

    Article  MathSciNet  CAS  ADS  Google Scholar 

  6. Kaur, H., Mittal, R., Mishra, V.: Haar wavelet approximate solutions for the generalized Lane-Emden equations arising in astrophysics. Comput. Phys. Commun. 184(9), 2169–2177 (2013)

    Article  MathSciNet  CAS  ADS  Google Scholar 

  7. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    Google Scholar 

  8. Luchko, Y., Gorenflo, R.: An operational method for solving fractional differential equations with the caputo derivatives. Acta Math. Vietnam 24, 207–233 (1999)

    MathSciNet  Google Scholar 

  9. Murio, D.A.: Implicit finite difference approximation for time fractional diffusion equations. Comput. Math. Appl. 56, 1138–1145 (2008)

    Article  MathSciNet  Google Scholar 

  10. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    Google Scholar 

  11. Ravi Kanth, A., Bhattacharya, V.: Cubic spline for a class of nonlinear singular boundary value problems arising in physiology. Appl. Math. Comput. 174(1), 768–774 (2006)

    MathSciNet  Google Scholar 

  12. Rahimkhani, P., Ordokhani, Y., Babolian, E.: A new operational matrix based on Bernoulli wavelets for solving fractional delay differential equations. Numer. Algorithm 74(1), 223–245 (2017)

    Article  MathSciNet  Google Scholar 

  13. Razzaghi, M., Razzaghi, M.: Taylor series direct method for variational problems. J. Frank. Inst. 325(1), 125–131 (1988)

    Article  MathSciNet  Google Scholar 

  14. Shiralashetti, S.C., Deshi, A.B.: An efficient Haar wavelet collocation method for the numerical solution of multi-term fractional differential equations. Nonliner Dyn. 83(2), 293–303 (2016)

    Article  MathSciNet  Google Scholar 

  15. Singh, R., Singh, S., Wazwaz, A.M.: A modified homotopy perturbation method for singular time dependent Emden-Fowler equations with boundary conditions. J. Math. Chem. 54, 918–931 (2016)

    Article  MathSciNet  CAS  Google Scholar 

  16. Singh, R., Garg, H., Guleria, V.: Haar wavelet collocation method for Lane-Emden equations with Dirichlet, Neumann and Neumann-Robin boundary conditions. J. Comput. Appl. Math. 346, 150–61 (2018)

    Article  MathSciNet  Google Scholar 

  17. Saadatmandi, A., Dehghan, M.: A tau approach for solution of the space fractional diffusion equation. Comput. Math. Appl. 62, 1135–1142 (2011)

    Article  MathSciNet  Google Scholar 

  18. Singh, R., Kumar, J.: An efficient numerical technique for the solution of nonlinear singular boundary value problems. Comput. Phys. Commun. 185(4), 1282–1289 (2014)

    Article  MathSciNet  CAS  ADS  Google Scholar 

  19. Singh, R., Singh, S., Wazwaz, A.M.: A modified homotopy perturbation method for singular time dependent Emden-Fowler equations with boundary conditions. J. Math. Chem. 54(4), 918–931 (2016)

    Article  MathSciNet  CAS  Google Scholar 

  20. Mohammadi, A., Aghazadeh, N., Rezapour, S.: Wavelet-Picard iterative method for solving singular fractional nonlinear partial differential equations with initial and boundary conditions. Comput. Methods Differ. Equ. 8(4), 610–638 (2020)

    MathSciNet  Google Scholar 

  21. Su, L., Wang, W., Xu, Q.: Finite difference methods for fractional dispersion equations. Appl. Math. Comput. 216, 3329–3334 (2010)

    MathSciNet  Google Scholar 

  22. Wang, L., Ma, Y., Meng, Z.: Haar wavelet method for solving fractional partial differential equations numerically. Appl. Math. Comput. 227, 66–76 (2014)

    MathSciNet  Google Scholar 

  23. Wazwaz, A.M.: Analytical solution for the time-dependent Emden-Fowler type of equations by Adomian decomposition method. Appl. Math. Comput. 166, 638–651 (2005)

    MathSciNet  Google Scholar 

  24. Wazwaz, A.M.: The variational iteration method for solving nonlinear singular boundary value problems arising in various physical models. Commun. Nonlinear Sci. Numer. Simul. 16(10), 3881–3886 (2011)

    Article  ADS  Google Scholar 

  25. Wazwaz, A.M.: Adomian decomposition method for a reliable treatment of the Emden-Fowler equation. Appl. Math. Comput. 161, 543–60 (2007)

    MathSciNet  Google Scholar 

  26. Wazwaz, A.M.: A reliable iterative method for solving the time-dependent singular Emden-Fowler equations. Central Eur. J. Eng. 3, 99–105 (2013)

    ADS  Google Scholar 

  27. Wazwaz, A.M., Rach, R.: Comparison of the Adomian decomposition method and the variational iteration method for solving the Lane-Emden equations of the first and second kinds. Kybernetes 40(10), 1305–1318 (2011)

    Article  MathSciNet  Google Scholar 

  28. Zhou, F., Xu, X.: Numerical solution of time-fractional diffusion-wave equations via chebyshev wavelets collocation method. Adv. Math. Phys. 17, 2610804 (2017)

    MathSciNet  ADS  Google Scholar 

  29. Zhou, F., Xu, X.: Numerical solution of the convection diffusion equations by the second kind chebyshev wavelets. Appl. Math. Comput. 247, 353–367 (2014)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran

    Nasser Aghazadeh & Amir Mohammadi

  2. Department of Mathematics, Izmir Institute of Technology, Izmir, Turkey

    Gamze Tanoglu

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  1. Nasser Aghazadeh
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  2. Amir Mohammadi
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  3. Gamze Tanoglu
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Correspondence to Nasser Aghazadeh.

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Aghazadeh, N., Mohammadi, A. & Tanoglu, G. Taylor wavelets collocation technique for solving fractional nonlinear singular PDEs. Math Sci 18, 41–54 (2024). https://doi.org/10.1007/s40096-022-00483-z

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