Skip to main content
SpringerLink
Log in

Lucas Wavelet Scheme for Fractional Bagley–Torvik Equations: Gauss–Jacobi Approach

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

A novel technique called as Lucas wavelet scheme (LWS) is prepared for the treatment of Bagley–Torvik equations (BTEs). Lucas wavelets for the approximation of unknown functions appearing in BTEs are introduced. Fractional derivatives are evaluated by employing Gauss–Jacobi quadrature formula. Further, well-known least square method (LSM) is adopted to compute the residual function, and the system of algebraic equation is obtained. Convergence criterion is derived and error bounds are obtained for the established technique. The scheme is investigated by choosing some reliable test problems through tables and figures, which ensures the convenience, validity and applicability of LWS.

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
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Data Availability Statement

No data were used to support this work.

References

  1. Bhatti, M.I., Bracken, P.: Solutions of differential equations in a Bernstein polynomial basis. J. Comput. Appl. Math. 205(1), 272–280 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cheon, G.S., Kim, H., Shapiro, L.W.: A generalization of Lucas polynomial sequence. Discret. Appl. Math. 157(5), 920–927 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Khalil, H., Khan, R.A.: A new method based on Legendre polynomials for solutions of the fractional two-dimensional heat conduction equation. Comput. Math. Appl. 67(10), 1938–1953 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Graps, A.: An introduction to wavelets. IEEE Comput. Sci. Eng. 2(2), 50–61 (1995)

    Article  Google Scholar 

  5. Misiti, M., Misiti, Y., Oppenheim, G., Poggi, J.M.: Wavelets and their Applications. Wiley, Hoboken (2013)

    MATH  Google Scholar 

  6. Farge, M.: Wavelet transforms and their applications to turbulence. Annu. Rev. Fluid Mech. 24(1), 395–458 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. Szu, H.H., Yang, X.Y., Telfer, B.A., Sheng, Y.: Neural network and wavelet transform for scale-invariant data classification. Phys. Rev. E 48(2), 1497–1501 (1993)

    Article  Google Scholar 

  8. Vidakovic, B.: A note on random densities via wavelets. Stat. Probab. Lett. 26(4), 315–321 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Peyrin, F., Zaim, M., Goutte, R.: Construction of wavelet decompositions for tomographic images. J. Math. Imaging Vis. 3(1), 105–122 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  12. Tuan, N.H., Ganji, R.M., Jafari, H.: A numerical study of fractional rheological models and fractional Newell–Whitehead–Segel equation with non-local and non-singular kernel. Chin. J. Phys. 68, 308–320 (2020)

    Article  MathSciNet  Google Scholar 

  13. Jafari, H., Ganji, R.M., Sayevand, K., Baleanu, D.: A numerical approach for solving fractional optimal control problems with Mittag–Leffler kernel. J. Vib. Control 10775463211016968 (2021)

  14. Jafari, H., Ganji, R.M., Nkomo, N.S., Lv, Y.P.: A numerical study of fractional order population dynamics model. Results Phys. 27, 104456 (2021)

    Article  Google Scholar 

  15. Singh, H.: Numerical simulation for fractional delay differential equations. Int. J. Dyn. Control 9(2), 463–474 (2021)

    Article  MathSciNet  Google Scholar 

  16. Saeed, U., Rehman, M., Iqbal, M.A.: Modified Chebyshev wavelet methods for fractional delay-type equations. Appl. Math. Comput. 264, 431–442 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Singh, H., Srivastava, H.M., Kumar, D.: A reliable numerical algorithm for the fractional vibration equation. Chaos Solitons Fractals 103, 131–138 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Singh, H., Wazwaz, A.M.: Computational method for reaction diffusion-model arising in a spherical catalyst. Int. J. Appl. Comput. Math. 7(3), 1–11 (2021)

    Article  MathSciNet  Google Scholar 

  19. Yousefi, S., Razzaghi, M.: Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations. Math. Comput. Simul. 70(1), 1–8 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Saeed, U.: Hermite wavelet method for fractional delay differential equations. J. Differ. Equ. Article ID 359093, 8 pages (2014)

  21. Alshbool, M.H.T., Bataineh, A.S., Hashim, I., Isik, O.R.: Solution of fractional-order differential equations based on the operational matrices of new fractional Bernstein functions. J. King Saud Univ. Sci. 29(1), 1–18 (2017)

    Article  Google Scholar 

  22. Horadam, A.F., Mahon, J.M.: Pell and Pell–Lucas polynomials. Fibonacci Quart. 23(1), 7–20 (1985)

    MathSciNet  MATH  Google Scholar 

  23. Abd-Elhameed, W.M., Youssri, Y.H.: Generalized Lucas polynomial sequence approach for fractional differential equations. Nonlinear Dyn. 89(2), 1341–1355 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Oruc, Ö.: A new algorithm based on Lucas polynomials for approximate solution of 1D and 2D nonlinear generalized Benjamin–Bona–Mahony–Burgers equation. Comput. Math. Appl. 74(12), 3042–3057 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Bagley, R.L., Torvik, P.J.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27(3), 201–210 (1983)

    Article  MATH  Google Scholar 

  26. Bota, C., Caruntu, B., Pasca, M.S., Ţucu, D., Lapadat, M.: Least squares differential quadrature method for the generalized Bagley–Torvik fractional differential equation. Math. Probl. Eng. (2020)

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

    Article  MATH  Google Scholar 

  28. Podlubny, I., Skovranek, T., Jara, B.M.V.: Matrix approach to discretization of fractional derivatives and to solution of fractional differential equations and their systems. In: IEEE Conference on Emerging Technologies & Factory Automation, pp. 1–6 (2009)

  29. Bagley, R.L., Torvik, P.J.: Fractional calculus-a different approach to the analysis of viscoelastically damped structures. Am. Inst. Aeronaut. Astronaut. 21(5), 741–748 (1983)

    Article  MATH  Google Scholar 

  30. Ray, S.S., Bera, R.K.: Analytical solution of the Bagley–Torvik equation by adomian decomposition method. Appl. Math. Comput. 168(1), 398–410 (2005)

    MathSciNet  MATH  Google Scholar 

  31. Ghorbani, A., Alavi, A.: Application of He’s variational iteration method to solve semidifferential equations of nth order. Math. Probl. Eng. 1–9 (2008)

  32. Bansal, M.K., Jain, R.: Analytical solution of Bagley–Torvik equation by generalize differential transform. Int. J. Pure and Appl. Math. 110(2), 265–273 (2016)

    Article  Google Scholar 

  33. Cenesiz, Y., Keskin, Y., Kurnaz, A.: The solution of the Bagley–Torvik equation with the generalized Taylor collocation method. J. Franklin Inst. 347(2), 452–466 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Karaaslan, M.F., Celiker, F., Kurulay, M.: Approximate solution of the Bagley–Torvik equation by hybridizable discontinuous Galerkin methods. Appl. Math. Comput. 285, 51–58 (2016)

    MathSciNet  MATH  Google Scholar 

  35. Singh, H., Pandey, R.K., Srivastava, H.M.: Solving non-linear fractional variational problems using Jacobi polynomials. Mathematics 7(3), 224 (2019)

    Article  Google Scholar 

  36. Singh, H., Srivastava, H.M.: Jacobi collocation method for the approximate solution of some fractional-order Riccati differential equations with variable coefficients. Physica A 523, 1130–1149 (2019)

    Article  MathSciNet  Google Scholar 

  37. Singh, H., Srivastava, H.M.: Numerical simulation for fractional-order Bloch equation arising in nuclear magnetic resonance by using the Jacobi polynomials. Appl. Sci. 10(8), 2850 (2020)

    Article  Google Scholar 

  38. Singh, H.: Jacobi collocation method for the fractional advection-dispersion equation arising in porous media. Numer. Methods Partial Differ. Equ. (2020)

  39. Ray, S.S.: On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley Torvik equation. Appl. Math. Comput. 218(9), 5239–5248 (2012)

    MathSciNet  MATH  Google Scholar 

  40. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications. Elsevier, Amsterdam (1998)

    MATH  Google Scholar 

  41. Heydari, M.H., Hooshmandasl, M.R., Ghaini, F.M., Mohammadi, F.: Wavelet collocation method for solving multiorder fractional differential equations. J. Appl. Math. Article ID 542401, 19 pages (2012)

  42. Balaji, S., Hariharan, G.: An efficient operational matrix method for the numerical solutions of the fractional Bagley–Torvik equation using wavelets. J. Math. Chem. 57(8), 1885–1901 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  43. Srivastava, H.M., Shah, F.A., Abass, R.: An application of the Gegenbauer wavelet method for the numerical solution of the fractional Bagley–Torvik equation. Russ. J. Math. Phys. 26(1), 77–93 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  44. Cetin, M., Sezer, M., Güler, C.: Lucas polynomial approach for system of high-order linear differential equations and residual error estimation. Math. Probl. Eng. Article ID 625984 (2015)

  45. Abd-Elhameed, W.M., Youssri, Y.H.: Spectral solutions for fractional differential equations via a novel Lucas operational matrix of fractional derivatives. Rom. J. Phys. 61(5–6), 795–813 (2016)

    Google Scholar 

  46. Ali, I., Haq, S., Nisar, K.S., Baleanu, D.: An efficient numerical scheme based on Lucas polynomials for the study of multidimensional Burgers-type equation. Adv. Differ. Equ. 1, 1–24 (2021)

    MathSciNet  Google Scholar 

  47. Golub, G.H., Welsch, J.H.: Calculation of Gauss quadrature rule. Math. Comput. 23(106), 221–230 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  48. Pang, G., Chena, W., Sze, K.Y.: Gauss–Jacobi-type quadrature rules for fractional directional integrals. Comput. Math. Appl. 66(5), 597–607 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  49. Kumar, R., Koundal, R., Srivastava, K., Baleanu, D.: Normalized Lucas wavelets: an application to Lane–Emden and pantograph differential equations. Eur. Phys. J. Plus 135(11), 1–24 (2020)

    Article  Google Scholar 

  50. Dehestani, H., Ordokhani, Y., Razzaghi, M.: Combination of Lucas wavelets with Legendre-Gauss quadrature for fractional Fredholm-Volterra integro-differential equations. J. Comput. Appl. Math. 382, 113070 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  51. Koundal, R., Kumar, R., Kumar, R., Srivastava, K., Baleanu, D.: A novel collocated-shifted Lucas polynomial approach for fractional integro-differential equations. Int. J. Appl. Comput. Math. 7(4), 1–19 (2021)

    Article  MathSciNet  Google Scholar 

  52. El-Gamel, M., El-Hady, M.A.: Numerical solution of the Bagley–Torvik equation by Legendre-collocation method. SeMA J. 74(4), 371–383 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  53. Mekkaoui, T., Hammouch, Z.: Approximate analytical solutions to the Bagley–Torvik equation by the fractional iteration method. Ann. Univ. Craiova Math. Comput. Sci. Ser. 39(2), 251–256 (2012)

    MathSciNet  MATH  Google Scholar 

  54. Fazli, H., Nieto, J.J.: An investigation of fractional Bagley–Torvik equation. Open Math. 17(1), 499–512 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  55. Sakar, M.G., Saldir, O., Akgül, A.: A novel technique for fractional Bagley–Torvik equation. Proc. Natl. Acad. Sci. India Sect. A Phys. Sci. 89(3), 539–545 (2019)

    Article  MathSciNet  Google Scholar 

  56. Xu, Y., Liu, Q., Liu, J., Chen, Y.: A novel method for solving the Bagley–Torvik equation as ordinary differential equation. J. Comput. Nonlinear Dyn. 14(8), 081005 (2019)

    Article  Google Scholar 

  57. Zahra, W.K., Daele, M.V.: Discrete spline methods for solving two point fractional Bagley–Torvik equation. Appl. Math. Comput. 296, 42–56 (2017)

    MathSciNet  MATH  Google Scholar 

  58. Ramadan, M.A., Moatimid, G.M., Taha, M.H.: One-step new iterative method for solving Bagley–Torvik fractional differential equation. Iran. J. Sci. Technol. Trans. A Sci. 43(5), 2493–2500 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The listed authors would like to thank the reviewers for their constructive and useful suggestions. The first author (Reena Koundal) also thanks Central University of Himachal Pradesh for providing the resources to conduct the present research work.

Funding

Dr. K. Srivastava’s (third author) work is supported by DST, Ministry of Science and Technology, India through WOS-A vide their File No.SR/WOS-A/PM-20/2018.

Author information

Authors and Affiliations

  1. School of Mathematics, Computers & Information Sciences, Central University of Himachal Pradesh, Dharamshala, India

    Reena Koundal, Rakesh Kumar & K. Srivastava

  2. Department of Mathematics, Cankaya University, Eskisehir Yolu29.km, 06810, Ankara, Turkey

    D. Baleanu

  3. Institute of Space Sciences, Magurele, Bucharest, Romania

    D. Baleanu

  4. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

    D. Baleanu

Authors
  1. Reena Koundal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rakesh Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K. Srivastava
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. D. Baleanu
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Reena Koundal: Formal analysis, Methodology, Software, Writing-original draft. Rakesh Kumar: Project administration, Supervision, Visualization, Writing-review and editing. K. Srivastava: Investigation. D. Baleanu: Investigation.

Ethics declarations

Conflict of interest

Authors have no conflict of interest.

Ethical approval

We have declared that the information about all the listed authors are correct. During the submission of the manuscript, we have followed all the rules of ethical standards.

Informed consent

The listed authors have confirmed their authorship.

Additional information

Publisher's Note

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

Supported by DST, Ministry of Science and Technology, India through WOS-A vide their File No.SR/WOS-A/PM-20/2018 (K. Srivastava).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Koundal, R., Kumar, R., Srivastava, K. et al. Lucas Wavelet Scheme for Fractional Bagley–Torvik Equations: Gauss–Jacobi Approach. Int. J. Appl. Comput. Math 8, 3 (2022). https://doi.org/10.1007/s40819-021-01206-z

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40819-021-01206-z

Keywords

Search

Navigation