Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
권호 표지
DOI QR코드

DOI QR Code

ENHANCED SEMI-ANALYTIC METHOD FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

  • JANG, BONGSOO (DEPARTMENT OF MATHEMATICAL SCIENCES, ULSAN NATIONAL INSTITUTE OF SCIENCE AND TECHONOLOGY(UNIST)) ;
  • KIM, HYUNJU (DEPARTMENT OF MATHEMATICS, NORTH GREENVILLE UNIVERSITY)
  • Received : 2019.09.21
  • Accepted : 2019.12.02
  • Published : 2019.12.25
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we propose a new semi-analytic approach based on the generalized Taylor series for solving nonlinear differential equations of fractional order. Assuming the solution is expanded as the generalized Taylor series, the coefficients of the series can be computed by solving the corresponding recursive relation of the coefficients which is generated by the given problem. This method is called the generalized differential transform method(GDTM). In several literatures the standard GDTM was applied in each sub-domain to obtain an accurate approximation. As noticed in [19], however, a direct application of the GDTM in each sub-domain loses a term of memory which causes an inaccurate approximation. In this work, we derive a new recursive relation of the coefficients that reflects an effect of memory. Several illustrative examples are demonstrated to show the effectiveness of the proposed method. It is shown that the proposed method is robust and accurate for solving nonlinear differential equations of fractional order.

Keywords

References

  1. M.Caputo, Linear models of dissipation whose Q is almost frequency independent. Part II, J.Roy.Austral.Soc. 13 (1967) 529-539 https://doi.org/10.1111/j.1365-246X.1967.tb02303.x
  2. Podlubny I, Fractional Differential Equations, New York: Academic Press, 1999.
  3. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
  4. K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, 2010.
  5. Kai Diethelm, Neville J. Ford, Alan D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equation, Nonlinear Dyn. 29 (2009) 2-22.
  6. Junying Cao, Chuanju Xu, A high order schema for the numerical solution of the fractional ordinary differential equations, J. comput. phys. 238 (2013) 154-168. https://doi.org/10.1016/j.jcp.2012.12.013
  7. Jun-Sheng Duan, Temuer Chaolu, Randolph Rach, Solutions of the initial value problem for nonlinear fractional ordinary differential equations by the Rach-Adomian-Meyers modified decomposition method, Appl. Math. Comput. 218 (2012) 8370-8392. https://doi.org/10.1016/j.amc.2012.01.063
  8. Jun-Sheng Duan, Temuer Chaolu, Randolph Rach, Lei Lua, The Adomian decomposition method with convergence acceleration techniques for nonlinear fractional differential equations, Comput. Math. Appl. 66 (2013) 726-736.
  9. I. Hashim, O. Abdulaziz, S. Momani, Homotopy analysis method for fractional IVPs, Commun Nonlinear Sci Numer Sim. 14 (2009) 674-684. https://doi.org/10.1016/j.cnsns.2007.09.014
  10. A. Elsaid, Homotopy analysis method for solving a class of fractional partial differential equations, Commun Nonlinear Sci Numer Sim. 16 (2011) 3655-3664. https://doi.org/10.1016/j.cnsns.2010.12.040
  11. Xindong Zhang, Jianping Zhao, Juan Liu, Bo Tang, Homotopy perturbation method for two dimensional time-fractional wave equation, Appl. Math. Model. 38 (23) (2014) 5545-5552. https://doi.org/10.1016/j.apm.2014.04.018
  12. O. Abdulaziz, I. Hashim, S. Momani, Application of homotopy-perturbation method to fractional IVPs, J. Comput. Appl. Math. 216 (2008) 574-584. https://doi.org/10.1016/j.cam.2007.06.010
  13. Guo-Cheng Wu, A fractional variational iteration method for solving fractional nonlinear differential equations, Comput. Math. Appl. 61 (2011) 2186-2190. https://doi.org/10.1016/j.camwa.2010.09.010
  14. Yasir Khan, Naeem Faraz, Ahmet Yildirim, Qingbiao Wu, Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science, Comput. Math. Appl. 62 (2011) 2273-2278. https://doi.org/10.1016/j.camwa.2011.07.014
  15. Guo-Cheng Wu, Dumitru Baleanus, Variational iteration method for the Burgers' flow with fractional derivatives-New Lagrange multipliers, Appl. Math. Model. 37 (9) (2013) 6183-6190. https://doi.org/10.1016/j.apm.2012.12.018
  16. Z.Odibat, N. Shawagfeh, Generalized Talyor's formula, Appl. Math. Comput. 186 (2007) 286-29 https://doi.org/10.1016/j.amc.2006.07.102
  17. Zaid Odibat, Shaher Momani, Vedat Suat Erturk, Generalized differential transform method: Application to differential equations of fractional order, Appl. Math. Comput. 197 (2008) 467-477. https://doi.org/10.1016/j.amc.2007.07.068
  18. A. Elsaid Fractional differential transform method combined with the Adomian polynomials, Appl. Math. Comput. 218 (2012) 6899-6911. https://doi.org/10.1016/j.amc.2011.12.066
  19. Kyunghoon Kim, Bongsoo Jang, A semi-analytic method with an effect of memory for solving fractional differential equations, Advances in difference equations 371 (2013)
  20. Bongsoo Jang, Efficient analytic method for solving nonlinear fractional differential equations, Appl. Math. Model. 38 (5-6) (2014) 1775-1787. https://doi.org/10.1016/j.apm.2013.09.018
  21. A. Di Matteo, A. Pirrotta, Generalized differential transform method for nonlinear boundary value problem of fractional order, Commun. Nonlinear Sci. Numer. Sim. 29 (2015) 88-101. https://doi.org/10.1016/j.cnsns.2015.04.017
  22. Z. Odibat, C. Bertelle, M.A. Aziz-Alaoui, G. Duchamp, A multi-step differential transform method and application to non-chaotic or chaotic systems, Comput. Math. Appl. 59 (2010) 1462-1472. https://doi.org/10.1016/j.camwa.2009.11.005
  23. V. Erturk, Z. Odibat, S. Momani, An approximate solution of a fractional order differential equation model of human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells, Comput. Math. Appl. 62 (2011) 992-1002.
  24. Eman Abuteena, Shaher Momanib, Ahmad Alawneh, Solving the fractional nonlinear Bloch system using the multi-step generalized differential transform method, Comput. Math. Appl. 68 (2014) 2124-2132. https://doi.org/10.1016/j.camwa.2013.05.013