Natural homotopy perturbation method for solving nonlinear fractional gas dynamics equations

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Thi-Qar, Iraq

Abstract

In this paper, we investigate solutions of nonlinear fractional differential equations by using Natural homotopy perturbation method (NHPM). This method is coupled by the Natural transform (NT) and homotopy perturbation method (HPM). The method in general is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the presented method.

Keywords

[1] W. H. Su, D. Baleanu, X.J. Yang, and H. Jafari, Damped wave equation and dissipative wave equation in fractal strings within the local fractional variational iteration method, Fixed Point Theo. Appl. 2013 (2013) 1–11.
[2] H. Jafari and H.K. Jassim, Local fractional variational iteration method for nonlinear partial differential equations within local fractional operators, Appl. Appl. Math. 10 (2015) 1055–1065.
[3] X.J. Yang, Local Fractional Functional Analysis and its Applications, Asian Academic, Hong Kong, China, 2011.
[4] S. Xu, X. Ling, Y. Zhao, and H.K. Jassim, A novel schedule for solving the two-dimensional diffusion in fractal heat transfer, Therm. Sci. 19 (2015) S99–S103.
[5] X. Yang, Advanced Local Fractional Calculus and its Applications, World Science Publisher, New York, 2012.
[6] H. K. Jassim, J. Vahidiand and V.M. Ariyan, Solving Laplace equation within local fractional operators by using local fractional differential transform and Laplace variational iteration methods, Nonlinear Dyn. Syst. Theo. 20(4) (2020) 388–396.
[7] X.J. Yang, J.A. Machad and H.M. Srivastava, A new numerical technique for solving the local fractional diffusion equation: Two-dimensional extended differential transform approach, Appl. Math. Comput. 274 (2016) 143–151.
[8] A.M. Yang,  X.J. Yang, and Z.B. Li, Local fractional series expansion method for solving wave and diffusion equations Cantor sets, Abstr. Appl. Anal. 2013 (2013) 1–5.
[9] H.K. Jassim and D. Baleanu, A novel approach for Korteweg-de Vries equation of fractional order, J. Appl. Comput. Mech. 5(2) (2019) 192–198.
[10] H.M. Srivastava, A.K. Golmankhaneh and D. Baleanu, Local fractional Sumudu transform with application to IVPs on Cantor set, Abstr. Appl. Anal. 2014 (2014) 1–7.
[11] C.G. Zhao, A.M. Yang, H. Jafari, and A. Haghbin, The Yang-Laplace transform for solving the IVPs with local fractional derivative, Abstr. Appl. Anal. 2014 (2014) 1–5.
[12] H.K. Jassim, The analytical solutions for Volterra integro-differential equations involving local fractional operators by Yang-Laplace transform, Sahand Commun. Math. Anal. 6(1) (2017) 69-76.
[13] Y. Zhang, X.J. Yang, and C. Cattani, Local fractional homotopy perturbation method for solving nonhomogeneous heat conduction equations in fractal domains, Entropy 17 (2015) 6753–6764.
[14] H.K. Jassim, New approaches for solving Fokker Planck equation on Cantor sets within local fractional operators, J. Math. 2015 (2015) 1–8
[15] H.K. Jassim, Local fractional Laplace decomposition method for nonhomogeneous heat equations arising in fractal heat flow with local fractional derivative, Int. J. Adv. Appl. Math. Mech. 2 (2015) 1–7.
[16] M.S. Hu, R.P. Agarwal, and X.J. Yang, Local fractional Fourier series with application to wave equation in fractal vibrating, Abstr. Appl. Anal. 2012 (2012) 1–7.
[17] H.K. Jassim, C. Unlu, S.P. Moshokoa and C.M. Khalique, Local fractional Laplace variational iteration method for solving diffusion and wave equations on Cantor sets within local fractional operators, Math. Prob. Engin. 2015 (2015) 1–7.
[18] H. Jafari and H.K. Jassim, A Coupling method of local fractional variational iteration method and Yang-Laplace transform for solving Laplace equation on Cantor sets, Int. J. Pure Appl. Sci. Tech. 26 (2015) 24–33.
[19] C.F. Liu, S.S. Kong, and J. Zhao, Local fractional Laplace variational iteration method for fractal vehicular traffic flow, Adv. Math. Phys. 2014 (2014) 1–7.
[20] Y. Li, L.F. Wang, and S.J. Yuan, Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem, Therm. Sci. 17 (2013) 715–721.
[21] D. Baleanu and H.K. Jassim, Exact solution of two-dimensional fractional partial differential equations, Fractal Fractional 4 (21) (2020) 1–9.
[22] H.K. Jassim, M.G. Mohammed and H.A. Eaued, A modification fractional homotopy analysis method for solving partial differential equations arising in mathematical physics, IOP Conf. Ser.: Mater. Sci. Engin. 928 (042021) (2020) 1–22.
[23] H.A. Eaued, H.K. Jassim and M.G. Mohammed, A Novel method for the analytical solution of partial differential equations arising in mathematical physics, IOP Conf. Ser.: Mater. Sci. Engin.  928 (042037) (2020) 1-16.
[24] H.K. Jassim and J. Vahidi, A new technique of reduce differential transform method to solve local fractional PDEs in mathematical physics, Int. J. Nonlinear Anal. Appl. 12(1) (2021) 37–44.
[25] S.M. Kadhim, M.G. Mohammad and H.K. Jassim, How to obtain Lie point symmetries of PDEs, J. Math. Comp. Sci. 22 (2021) 306–324.
[26] H.K. Jassim and M.A. Shareef, On approximate solutions for fractional system of differential equations with Caputo-Fabrizio fractional operator, J. Math. Comp. Sci. 23 (2021) 58–66.
[27] H.K. Jassim, S.A. Khafif, SVIM for solving Burger’s and coupled Burger’s equations of fractional order, Prog. Frac. Diff. Appl. 7(1) (2021) 1–6.
[28] H.K. Jassim and H.A. Kadhim, Fractional Sumudu decomposition method for solving PDEs of fractional order, J. Appl. Comput. Mech. 7(1) (2021) 302–311.
[29] H. Jafari, H.K. Jassim, D. Baleanu and Y.M. Chu, On the approximate solutions for a system of coupled Korteweg-de Vries equations with local fractional derivative, Fractals 29(5) (2021) 1–7.
[30] H. Jafari, H.K. Jassim, S.P. Moshokoa, V.M. Ariyan and F. Tchier, Reduced differential transform method for partial differential equations within local fractional derivative operators, Adv. Mech. Engin. 8 (2016)1–6.
[31] D. Baleanu, H.K. Jassim and H. Khan, A modification fractional variational iteration method for solving nonlinear gas dynamic and coupled KdV equations involving local fractional operators, Ther. Sci. 22 (2018) S165–S175.
[32] S. Kumar and M.M. Rashidi, New analytical method for gas dynamics equation arising in shock fronts, Comp. Phy. Commun. 185(7) (2014) 1947–1954.
[33] H. Jafari and H.K. Jassim, Local fractional Laplace variational iteration method for solving nonlinear partial differential equations on Cantor sets within local fractional operators, J. Zankoy Sulaimani-Part A 16 (4) (2014) 49–57.
[34] S. Das and R. Kumar, Approximate analytical solution of fractional gas dynamic equations, Appl. Math. Comput. 217(24) (2011), 9905–9915.
Volume 12, Issue 1
May 2021
Pages 812-820
  • Receive Date: 24 October 2020
  • Revise Date: 05 December 2020
  • Accept Date: 19 January 2021