International Journal of Nonlinear Analysis and Applications

Efficient analytical method for the solution of some fractional -order nonlinear differential equations

Document Type : Research Paper

Authors

Department of Mathematics, College of Education For Pure Sciences, Ibn Al-Haitham, University of Baghdad, Iraq

Abstract

A novel technique called the Variational Adomian decomposition method (VIADM) is used to approximate an analytical solution for several types of non-linear fractional differential equations. Some examples are presented to back up our findings. The solution procedure and results indicated that the proposed method is very effective, reliable, and straightforward. The results show how effective and precise the present technology is at resolving various nonlinear problems in applied science. The MATLAB software carried out all the computations and graphics. Fractional derivatives are mentioned in Caputo Sense. Moreover, a graphical representation was made for the solution of some examples. For integer and fractional order problems, solution graphs are shown.

Keywords

[1] Y.O. Hasan and L.M. Zhu, Modified Adomian decomposition method for singular initial value problems in the second-order ordinary differential equations, Surv. Math. Appl. 3 (2008), 183–193.
[2] J.H. He, Variational iteration method-a kind of non-linear analytical technique: some examples, Int. J. Non-Linear Mech. 34 (1999), no. 4, 699–708.
[3] J.H. He, Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput. 114 (2000), no. 2–3, 115–123.
[4] J.H. He, Variational principles for some nonlinear partial differential equations with variable coefficients, Chaos, Solitons Fractals 19 (2004), no. 4, 847–851.
[5] R. Hilfer, Applications of fractional calculus in physics, World Scientific, Singapore, 2000.
[6] H. Jafari, H.K. Jassim, F. Tchier and D. Baleanu, On the approximate solutions of local fractional differential equations with local fractional operator, Entropy 18 (2016), 1–12.
[7] H.K. Jassim and W.A. Shahab, Fractional variational iteration method to solve one-dimensional second-order hyperbolic telegraph equations, J. Phys.: Conf. Ser. 1032 (2018), no. 1, 012015.
[8] C. Jin and M. Liu, A new modification of Adomian decomposition method for solving a kind of evolution equation, Appl. Math. Comput. 169 (2005), 953–962.
[9] S. Liao, Homotopy analysis method: a new analytical technique for nonlinear problems, Commun. Nonlinear Sci. Numer. Simul. 2 (1997), no. 2, 95–100.
[10] K.B. Oldham and J. Spanier, The fractional calculus, Academic Press, New York, 1974.
[11] I. Podlubny, Fractional differential equations, Academic Press, New York, 1999.
[12] S. Sharma and A.J. Obaid, Mathematical modelling, analysis and design of fuzzy logic controller for the control of ventilation systems using MATLAB fuzzy logic toolbox, J. Interdiscip. Math. 23 (2020), no. 4, 843–849.
[13] K.J. Wang and G.D. Wang, Variational principle and approximate solution for the fractal generalized BenjaminBona-Mahony-Burgers equation in fluid mechanics, Fractals 29 (2020), no. 3, 2150075.
[14] A. Yildirim, Application of he’s homotopy perturbation method for solving the Cauchy reaction-diffusion problem, Comp. Math. Appl. 57 (2009), 612–618.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 2
July 2022
Pages 401-408
  • Receive Date: 04 January 2022
  • Revise Date: 18 March 2022
  • Accept Date: 07 April 2022