Numerical Comparison of FNVIM and FNHPM for Solving a Certain Type of Nonlinear Caputo Time-Fractional Partial Differential Equations

Ali Khalouta 1  and Abdelouahab Kadem 1
  • 1 Laboratory of Fundamental Mathematics and Numerical, Department of Mathematics, Faculty of Sciences, Ferhat Abbas Sétif University 1, 19000, Sétif, Algeria


This work presents a numerical comparison between two efficient methods namely the fractional natural variational iteration method (FNVIM) and the fractional natural homotopy perturbation method (FNHPM) to solve a certain type of nonlinear Caputo time-fractional partial differential equations in particular, nonlinear Caputo time-fractional wave-like equations with variable coefficients. These two methods provided an accurate and efficient tool for solving this type of equations. To show the efficiency and capability of the proposed methods we have solved some numerical examples. The results show that there is an excellent agreement between the series solutions obtained by these two methods. However, the FNVIM has an advantage over FNHPM because it takes less time to solve this type of nonlinear problems without using He’s polynomials. In addition, the FNVIM enables us to overcome the diffi-culties arising in identifying the general Lagrange multiplier and it may be considered as an added advantage of this technique compared to the FNHPM.

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  • [1] F.B.M. Belgacem and R. Silambarasan, Theory of natural transform, Mathematics in Engineering, Science and Aerospace 3 (2012), no. 1, 105–135.

  • [2] M.H. Cherif, K. Belghaba, and Dj. Ziane, Homotopy perturbation method for solving the fractional Fisher’s equation, International Journal of Analysis and Applications 10 (2016), no. 1, 9–16.

  • [3] A.M. Elsheikh and T.M. Elzaki, Variation iteration method for solving porous medium equation, International Journal of Development Research 5 (2015), no. 6, 4677–4680.

  • [4] P. Guo, The Adomian decomposition method for a type of fractional differential equations, Journal of Applied Mathematics and Physics 7 (2019), 2459–2466.

  • [5] S. Javeed, D. Baleanu, A. Waheed, M. Shaukat Khan, and H. Affan, Analysis of homotopy perturbation method for solving fractional order differential equations, Mathematics 7 (2019), no. 1, Art. 40, 14 pp.

  • [6] J.T. Katsikadelis, Nonlinear dynamic analysis of viscoelastic membranes described with fractional differential models, J. Theoret. Appl. Mech. 50 (2012), no. 3, 743–753.

  • [7] A. Khalouta, A. Kadem, A new numerical technique for solving Caputo time-fractional biological population equation, AIMS Mathematics 4 (2019), no. 5, 1307–1319.

  • [8] A. Khalouta and A. Kadem, Fractional natural decomposition method for solving a certain class of nonlinear time-fractional wave-like equations with variable coefficients, Acta Univ. Sapientiae Math. 11 (2019), no. 1, 99–116.

  • [9] A. Khalouta and A. Kadem, An efficient method for solving nonlinear time-fractional wave-like equations with variable coefficients, Tbilisi Math. J. 12 (2019), no. 4, 131–147.

  • [10] A. Khalouta and A. Kadem, A new representation of exact solutions for nonlinear time-fractional wave-like equations with variable coefficients, Nonlinear Dyn. Syst. Theory. 19 (2019), no. 2, 319–330.

  • [11] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

  • [12] Z. Odibat, On the optimal selection of the linear operator and the initial approximation in the application of the homotopy analysis method to nonlinear fractional differential equations, Appl. Numer. Math. 137 (2019), 203–212.

  • [13] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.

  • [14] D. Sharma, P. Singh, and S. Chauhan, Homotopy perturbation transform method with He’s polynomial for solution of coupled nonlinear partial differential equations, Nonlinear Engineering 5 (2016), no. 1, 17–23.

  • [15] B.R. Sontakke, A.S. Shelke, and A.S. Shaikh, Solution of non-linear fractional differential equations by variational iteration method and applications, Far East Journal of Mathematical Sciences 110 (2019), no. 1, 113–129.

  • [16] A. Yıldırım, Analytical approach to fractional partial differential equations in fluid mechanics by means of the homotopy perturbation method, Internat. J. Numer. Methods Heat Fluid Flow 20 (2010), no. 2, 186–200.

  • [17] Y. Zhou and L. Peng, Weak solutions of the time-fractional Navier-Stokes equations and optimal control, Comput. Math. Appl. 73 (2017), no. 6, 1016–1027.


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