Skip to main content
Springer Nature Link
Log in

A numerical method based on Legendre wavelet and quasilinearization technique for fractional Lane-Emden type equations

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this research, we study the numerical solution of fractional Lane-Emden type equations, which emerge mainly in astrophysics applications. We propose a numerical approach making use of Legendre wavelets and the quasilinearization technique. The nonlinear term in fractional Lane-Emden type equations is iteratively linearized using the quasilinearization technique. The linearized equations are then solved using the Legendre wavelet collocation method. The proposed method is quite effective to overcome the singularity in fractional Lane-Emden type equations. Convergence and error analysis of the proposed method are given. We solve some test problems to compare the effectiveness of the proposed method with some other numerical methods in the literature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Availability of supporting data

Not applicable

References

  1. Atanackovic, T., Pilipovic, S., Stanković, B., Zorica, D.: Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes (2014). https://doi.org/10.1002/9781118577530

    Article  Google Scholar 

  2. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific (2000) https://doi.org/10.1142/3779

  3. Machado, J.A.T., Silva, M.F., Barbosa, R.S., Jesus, I.S., Reis, C.M., Marcos, M.G., Galhano, A.F.: Some Applications of Fractional Calculus in Engineering. Mathematical Problems in Engineering 2010, 1–34 (2010). https://doi.org/10.1155/2010/639801

    Article  Google Scholar 

  4. Momani, S., Odibat, Z.: Analytical approach to linear fractional partial differential equations arising in fluid mechanics. Phys. Lett. A 355(4–5), 271–279 (2006). https://doi.org/10.1016/j.physleta.2006.02.048

    Article  Google Scholar 

  5. Debnath, L.: Recent applications of fractional calculus to science and engineering. Int. J. Math. Math. Sci. 2003(54), 3413–3442 (2003)

    Article  MathSciNet  Google Scholar 

  6. de Barros, L.C., Lopes, M.M., Pedro, F.S., Esmi, E., dos Santos, J.P.C., Sánchez, D.E.: The memory effect on fractional calculus: an application in the spread of covid-19. Comput. Appl. Math. 40(72) (2021). https://doi.org/10.1007/s40314-021-01456-z

  7. Wu, G.-C.: A fractional variational iteration method for solving fractional nonlinear differential equations. Comput. Math. Appl. 61(8), 2186–2190 (2011). https://doi.org/10.1016/j.camwa.2010.09.010

    Article  MathSciNet  Google Scholar 

  8. Jafari, H., Daftardar-Gejji, V.: Solving a system of nonlinear fractional differential equations using adomian decomposition. J. Comput. Appl. Math. 196(2), 644–651 (2006). https://doi.org/10.1016/j.cam.2005.10.017

    Article  MathSciNet  Google Scholar 

  9. Wang, Q.: Homotopy perturbation method for fractional kdv-burgers equation. Chaos, Solitons Fractals 35(5), 843–850 (2008). https://doi.org/10.1016/j.chaos.2006.05.074

    Article  MathSciNet  Google Scholar 

  10. Li, C., Zeng, F.: The finite difference methods for fractional ordinary differential equations. Numer. Funct. Anal. Optim. 34(2), 149–179 (2013). https://doi.org/10.1080/01630563.2012.706673

    Article  MathSciNet  Google Scholar 

  11. Srivastava, H.M., Saad, K.M., Khader, M.M.: An efficient spectral collocation method for the dynamic simulation of the fractional epidemiological model of the ebola virus. Chaos, Solitons Fractals 140, 110174 (2020). https://doi.org/10.1016/j.chaos.2020.110174

    Article  MathSciNet  Google Scholar 

  12. Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation methods for linear and nonlinear variable order fpdes. J. Comput. Phys. 293, 312–338 (2015). https://doi.org/10.1016/j.jcp.2014.12.001

    Article  MathSciNet  Google Scholar 

  13. Hao, Z., Zhang, Z.: Fast spectral Petrov-Galerkin method for fractional elliptic equations. Appl. Numer. Math. 162, 318–330 (2021). https://doi.org/10.1016/j.apnum.2020.12.026

    Article  MathSciNet  Google Scholar 

  14. Kim, H., Kim, K.H., Jang, B.: Shifted jacobi spectral-galerkin method for solving fractional order initial value problems. J. Comput. Appl. Math. 380, 112988 (2020). https://doi.org/10.1016/j.cam.2020.112988

    Article  MathSciNet  Google Scholar 

  15. Marco Gallegati, W.S.: Wavelet Applications in Economics and Finance. Dynamic Modeling and Econometrics in Economics and Finance. Springer (2014)

    Book  Google Scholar 

  16. Teolis, A.: Computational Signal Processing with Wavelets. Springer, Applied and Numerical Harmonic Analysis (1998)

    Book  Google Scholar 

  17. Mehra, M.: Applications of Wavelet in Inverse Problems, pp. 157–171. Springer, Singapore (2018). https://doi.org/10.1007/978-981-13-2595-3_10

  18. Faheem, M., Khan, A., Raza, A.: A high resolution hermite wavelet technique for solving space-time-fractional partial differential equations. Math. Comput. Simul. 194, 588–609 (2022). https://doi.org/10.1016/j.matcom.2021.12.012

    Article  MathSciNet  Google Scholar 

  19. Yuttanan, B., Razzaghi, M.: Legendre wavelets approach for numerical solutions of distributed order fractional differential equations. Appl. Math. Model. 70, 350–364 (2019). https://doi.org/10.1016/j.apm.2019.01.013

    Article  MathSciNet  Google Scholar 

  20. Wang, L., Ma, Y., Meng, Z.: Haar wavelet method for solving fractional partial differential equations numerically. Appl. Math. Comput. 227, 66–76 (2014). https://doi.org/10.1016/j.amc.2013.11.004

    Article  MathSciNet  Google Scholar 

  21. Wang, Y., Fan, Q.: The second kind chebyshev wavelet method for solving fractional differential equations. Appl. Math. Comput. 218(17), 8592–8601 (2012). https://doi.org/10.1016/j.amc.2012.02.022

    Article  MathSciNet  Google Scholar 

  22. Aghazadeh, N., Mohammadi, A., Tanoglu, G.: Taylor wavelets collocation technique for solving fractional nonlinear singular pdes. Math. Sci. (2022). https://doi.org/10.1007/s40096-022-00483-z

    Article  Google Scholar 

  23. Chandrasekhar, S.: An Introduction to the Study of Stellar Structure. Dover Publications (2010)

    Google Scholar 

  24. Gouari, Y., Dahmani, Z., Farooq, S.E., Ahmad, F.: Fractional singular differential systems of lane-emden type: Existence and uniqueness of solutions. Axioms 9(3) (2020). https://doi.org/10.3390/axioms9030095

  25. Mohammadi, A., Ahmadnezhad, G., Aghazadeh, N.: Chebyshev-quasilinearization method for solving fractional singular nonlinear Lane-Emden equations. Commun. Math. 30(1) (2022). https://doi.org/10.46298/cm.9895

  26. Saeed, U.: Haar Adomian Method for the Solution of Fractional Nonlinear Lane-Emden Type Equations Arising in Astrophysics. Taiwan. J. Math. 21(5), 1175–1192 (2017). https://doi.org/10.11650/tjm/7969

  27. Gümgüm, S.: Taylor wavelet solution of linear and nonlinear lane-emden equations. Applied Numerical Mathematics 158, 44–53 (2020). https://doi.org/10.1016/j.apnum.2020.07.019

    Article  MathSciNet  Google Scholar 

  28. Parand, K., Dehghan, M., Rezaei, A.R., Ghaderi, S.M.: An approximation algorithm for the solution of the nonlinear lane-emden type equations arising in astrophysics using hermite functions collocation method. Computer Physics Communications 181(6), 1096–1108 (2010). https://doi.org/10.1016/j.cpc.2010.02.018

    Article  MathSciNet  Google Scholar 

  29. Singh, R., Garg, H., Guleria, V.: Haar wavelet collocation method for lane-emden equations with dirichlet, neumann and neumann-robin boundary conditions. Journal of Computational and Applied Mathematics 346, 150–161 (2019). https://doi.org/10.1016/j.cam.2018.07.004

    Article  MathSciNet  Google Scholar 

  30. Bellman, R.E., Kalaba, R.E.: Quasilinearization and Nonlinear Boundary-value Problems. RAND Corporation, Santa Monica, CA (1965)

    Google Scholar 

  31. Mandelzweig, V.B.: Quasilinearization method and its verification on exactly solvable models in quantum mechanics. Journal of Mathematical Physics 40(12), 6266–6291 (1999). https://doi.org/10.1063/1.533092

    Article  MathSciNet  Google Scholar 

  32. Yuttanan, B., Razzaghi, M., Vo, T.N.: Legendre wavelet method for fractional delay differential equations. Applied Numerical Mathematics 168, 127–142 (2021). https://doi.org/10.1016/j.apnum.2021.05.024

    Article  MathSciNet  Google Scholar 

  33. Inc., W.R.: Mathematica, Version 13.1. Champaign, IL, 2022. https://www.wolfram.com/mathematica

  34. Shiralashetti, S.C., Deshi, A.B., Mutalik Desai, P.B.: Haar wavelet collocation method for the numerical solution of singular initial value problems. Ain Shams Engineering Journal 7(2), 663–670 (2016). https://doi.org/10.1016/j.asej.2015.06.006

    Article  Google Scholar 

  35. Wazwaz, A.-M.: Adomian decomposition method for a reliable treatment of the emden-fowler equation. Applied Mathematics and Computation 161(2), 543–560 (2005). https://doi.org/10.1016/j.amc.2003.12.048

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Izmir Institute of Technology, Izmir, Türkiye

    Fatih İdiz, Gamze Tanoğlu & Nasser Aghazadeh

Authors
  1. Fatih İdiz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gamze Tanoğlu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nasser Aghazadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the preparation of the text and presentation of the results; they also read and approved the final manuscript.

Corresponding author

Correspondence to Nasser Aghazadeh.

Ethics declarations

Ethics approval

Not applicable

Competing interests

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

İdiz, F., Tanoğlu, G. & Aghazadeh, N. A numerical method based on Legendre wavelet and quasilinearization technique for fractional Lane-Emden type equations. Numer Algor 95, 181–206 (2024). https://doi.org/10.1007/s11075-023-01568-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-023-01568-z

Keywords