Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Numerical Solutions of Fractional Differential Equations with Variable Coefficients by Taylor Basis Functions

  • Kammanee, Athassawat (Applied Analysis Research Unit, Division of Computational Science, Faculty of Science,Prince of Songkla University, Centre of Excellence in Mathematics, CHE)
  • Received : 2020.01.31
  • Accepted : 2020.09.08
  • Published : 2021.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, numerical techniques are presented for solving initial value problems of fractional differential equations with variable coefficients. The method is derived by applying a Taylor vector approximation. Moreover, the operational matrix of fractional integration of a Taylor vector is provided in order to transform the continuous equations into a system of algebraic equations. Furthermore, numerical examples demonstrate that this method is applicable and accurate.

Keywords

Acknowledgement

The authors greatly acknowledge the valuable comment of Prof. Mohsen Razzaghi (Mississippi State University) and Assoc. Prof. Varayu Boonpongkrong (Prince of Songkla University) to obtain better result. Moreover, the author also gratefully acknowledge the copy-editing service of the Research and Development Office (RDO/PSU), and the helpful edits and comments by Dr. Seppo Karrila.

References

  1. A. Arikoglu and I. Ozkol, Solution of fractional differential equations by using differential transform method, Chaos Solitons Fract, 34(2007), 1473-1481. https://doi.org/10.1016/j.chaos.2006.09.004
  2. S. El-Wakil, A. Elhanbaly and M. Abdou, Adomian decomposition method for solving fractional nonlinear differential equations, Appl. Math. Comput., 182(2006), 313-324. https://doi.org/10.1016/j.amc.2006.02.055
  3. M. Fouda, A. Elwakil, A. Radwan and A. Allagui, Power and energy analysis of fractional-order electrical energy storage devices, Energy, 111(2016), 785-792. https://doi.org/10.1016/j.energy.2016.05.104
  4. G. Groza and M. Razzaghi, A Taylor series method for the solution of the linear initial-boundary-value problems for partial differential equations, Comput. Math. Appl., 66(2013), 1329-1343. https://doi.org/10.1016/j.camwa.2013.08.004
  5. M. K. Ishteva, Properties and applications of the Caputo fractional operator, Master's thesis, University Karlsruhe, Germany, 2005.
  6. S. Kazema, S. Abbasbandya and S. Kumar, Fractional-order Legendre functions for solving fractional-order differential equations, Appl. Math. Model., 37(2013), 5498- 5510. https://doi.org/10.1016/j.apm.2012.10.026
  7. V. S. Krishnasamy and M. Razzaghi, The numerical solution of the Bagley-Torvik equation with fractional Taylor method, J. Comput. Nonlinear Dynam., 11(2016), 051010, 6 pp. https://doi.org/10.1115/1.4032390
  8. P. Kumar, S. Kumar and B. Raman, A fractional order variational model for the robust estimation of optical flow from image sequences, Optik - International Journal for Light and Electron Optics, 127(2016), 8710-8727. https://doi.org/10.1016/j.ijleo.2016.05.118
  9. Y. Li and W. Zhao, Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Comput., 216(2010), 2276-2285. https://doi.org/10.1016/j.amc.2010.03.063
  10. H. R. Marzban and M. Razzaghi, Analysis of time-delay systems via hybrid of blockpulse functions and Taylor series, J. Vib. Control, 11(2005), 1455-1468. https://doi.org/10.1177/1077546305058662
  11. J. H. Mathews, Numerical methods for mathematics, Science, and Engineering, Prentice Hall, 1992.
  12. M. Meerschaert and C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., 56(2006), 80-90. https://doi.org/10.1016/j.apnum.2005.02.008
  13. K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley & Sons., 1993.
  14. Z. M. Odibat, A study on the convergence of variational iteration method, Math. Comput. Modelling, 51(2010), 1181-1192. https://doi.org/10.1016/j.mcm.2009.12.034
  15. Z. M. Odibat and S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul., 7(2006), 27-34.
  16. D. Rostamy and E. Mottaghi, Stability analysis of a fractional-order epidemics model with multiple equilibriums, Adv. Difference Equ., (2016), Paper No. 170, 11 pp.
  17. M. Yi and J. Huang, Wavelet operational matrix method for solving fractional differential equations with variable coefficients, Appl. Math. Comput., 230(2014), 383-394. https://doi.org/10.1016/j.amc.2013.06.102