1. bookVolume 34 (2020): Issue 2 (September 2020)
Journal Details
License
Format
Journal
First Published
12 Dec 2015
Publication timeframe
1 time per year
Languages
English
Copyright
© 2020 Sciendo

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

Published Online: 09 Jul 2020
Page range: 203 - 221
Received: 19 Aug 2019
Accepted: 03 Jun 2020
Journal Details
License
Format
Journal
First Published
12 Dec 2015
Publication timeframe
1 time per year
Languages
English
Copyright
© 2020 Sciendo

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.

Keywords

MSC 2010

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

[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.Search in Google Scholar

[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.Search in Google Scholar

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

[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.Search in Google Scholar

[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.Search in Google Scholar

[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.Search in Google Scholar

[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.Search in Google Scholar

[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.Search in Google Scholar

[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.Search in Google Scholar

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

[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.Search in Google Scholar

[13] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.Search in Google Scholar

[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.Search in Google Scholar

[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.Search in Google Scholar

[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.Search in Google Scholar

[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.Search in Google Scholar

Plan your remote conference with Sciendo