| Peer-Reviewed

An Analytical Treatment to Fractional Gas Dynamics Equation

Received: 24 October 2014     Accepted: 9 December 2014     Published: 29 December 2014
Views:       Downloads:
Abstract

In this paper, the new iterative method (NIM) is applied to solve nonlinear fractional gas dynamics equation. Further, a coupling of the Sumudu transform and Adomian decomposion (STADM) is used to get an approximate solution of the same problem. The results obtained by the two methods are found to be in agreement. Therefore, the NIM may be considered efficient method for finding approximate solutions of both linear and nonlinear fractional differential equations.

Published in Applied and Computational Mathematics (Volume 3, Issue 6)
DOI 10.11648/j.acm.20140306.16
Page(s) 323-329
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2014. Published by Science Publishing Group

Keywords

Sumudu Transform Method, Adomian Decomposion Method, New Iterative Method,Fractional Gas Dynamics Equation

References
[1] Young GO. Definition of physical consistent damping laws with fractional derivatives. Z Angew Math Mech 1995;75:623–35.
[2] He JH. Some applications of nonlinear fractional differential equations and their approximations. Bull Sci Technol 1999;15(2):86–90.
[3] He JH. Approximate analytic solution for seepage flow with fractional derivatives in porous media. Comput Methods Appl Mech Eng 1998;167:57–68.
[4] Hilfer R, editor. Applications of Fractional Calculus in Physics. Singapore, New Jersey, Hong Kong: World Scientific Publishing Company; 2000. p. 87–130.
[5] Podlubny I. Fractional differential equations. New York: Academic Press; 1999.
[6] Mainardi F, Luchko Y, Pagnini G. The fundamental solution of the space–time fractional diffusion equation. Fract Calc Appl Anal 2001;4:153–92.
[7] Rida SZ, El-Sayed AMA, Arafa AAM. On the solutions of timefractional reaction–diffusion equations. Commun Nonlinear Sci Numer Simul 2010;15(2):3847–54.
[8] Yildirim A. He’s homotopy perturbation method for solving the space- and time- fractional telegraph equations. Int J Comput Math 2010;87(13):2998–3006.
[9] Debnath L. Fractional integrals and fractional differential equations in fluid mechanics. Frac Calc Appl Anal 2003;6:119–55.
[10] Caputo M. Elasticita e Dissipazione. Zani-Chelli: Bologna; 1969.
[11] Miller KS, Ross B. An introduction to the fractional calculus and fractional differential equations. New York: Wiley; 1993.
[12] Oldham KB, Spanier J. The fractional calculus theory and applications of differentiation and integration to arbitrary order. New York: Academic Press; 1974.
[13] J. H. He, “Asymptotic methods for solitary solutions and compactons,” Abstract and Applied Analysis, vol. 2012, Article ID 916793, 130 pages, 2012.
[14] V. D. Gejji, H. Jafari, An iterative method for solving nonlinear functional equations, Journal of Mathematical Analysis and Applications, Vol. 316, No. 2 , pp. 753-763, 2006.
[15] V. D. Gejji, S. Bhalekar, Solving fractional boundary value problems with Dirichlet boundary conditions using a new iterative method, Computers & Mathematics with Applications. 59 (5) (2010) 1801-1809.
[16] A. Bibi, A. Kamran, U. Hayat, S. Mohyud-Din, new iterative method for time- fractional schrodinger equations, World Journal of Modelling and Simulation. 9 (2) (2013) 89-95.
[17] S. Bhalekar, V. D. Gejji, New iterative method: application to partial differential equations, Applied Mathematics and Computation. 203 (2) (2008) 778-783.
[18] A. A. Hemeda, New iterative method: an application for solving fractional physical differential equations, Journal of Abstract and Applied Analysis. Vol. 2013, Article ID 617010, 9 pages, 2013.
[19] H. Eltayeb and A. Kilicman, Application of Sumudu Decomposition Method to Solve Nonlinear System of Partial Differential Equations, Journal of Abstract and Applied Analysis, Vol. 2012, Article ID 412948, 13 pages, 2012.
[20] M. A. Ramadan and M. S. Al-luhaibi, Application of Sumudu Decomposition Method for Solving Linear and Nonlinear Klein-Gordon Equations, International Journal of Soft Computing and Engineering, Vol. 3, No. 6, 2014.
[21] J.H. He, Variational iteration method-akind of nonlinear analytical technique: some examples, International Journal of Nonlinear Mechanics. 34 (1999) 699-708.
[22] S. Das and R. Kumar, Approximate analytical solutions of fractional gas dynamic equations, Applied Mathematics and Computation, vol. 217, no. 24, pp. 9905–
[23] V. B. L. Chaurasia and J. Singh, Application of Sumudu transform in Schrödinger equation occurring in quantum mechanics, Applied Mathematical Sciences, vol. 4, no. 57–60, pp. 2843–2850, 2010. 9915, 2011.
[24] Y. Cherruault, Convergence of Adomian's method, Kybernetes. 18 (2) (1989) 31- 38
[25] A.J. Jerri, Introduction to Integral Equations with Applications. seconded, Wiley. Interscience. 1999.
[26] G. K.Watugala, Sumudu transform-a new integral transform to solve differential equations and control engineering problems, Mathematical Engineering in ndustry,Vol. 6, No. 4, pp. 319-329, 1998.
[27] S. Weerakoon, Application of Sumudu transform to partial differential equations, International Journal of Mathematical Education in Science and Technology, Vol. 25, No. 2, pp. 277-283, 1994.
[28] S. Weerakoon, Complex inversion formula for Sumudu transform", International Journal of Mathematical Education in Science and Technology, Vol. 29, No. 4, pp. 618-621, 1998.
[29] M. A. Asiru, Further properties of the Sumudu transform and its applications, International Journal of Mathematical Education in Science and Technology, Vol. 33, No. 3, pp. 441-449, 2002.
[30] A. Kadem, Solving the one-dimensional neutron transport equation using Chebyshev polynomials and the Sumudu transform, Analele Universitatii dinOradea, Vol. 12, pp. 153-171, 2005.
[31] A. Kilicman, H. Eltayeb, and K. A. M. Atan, A note on the comparison between Laplace and Sumudu transforms, Iranian Mathematical Society, Vol. 37, No. 1, pp. 131-141, 2011.
[32] A. Kilicman and H. E. Gadain, On the applications of Laplace and Sumudu transforms, Journal of the Franklin Institute, Vol. 347, No. 5, pp. 848-862, 2010.
[33] H. Eltayeb, A. Kilicman, and B. Fisher, A new integral transform and associated distributions, Integral Transforms and Special Functions, Vol. 21, No. 5-6, pp. 367- 379, 2010.
[34] A. Kilicman and H. Eltayeb, A note on integral transforms and partial differential equations, Applied Mathematical Sciences, Vol. 4, No. 1-4, pp. 109-118, 2010.
[35] A. Kilicman, H. Eltayeb, and R. P. Agarwal, On Sumudu transform and system of differential equations, Abstract and Applied Analysis, Article ID598702, 11 pages, 2010.
[36] J. Zhang, A Sumudu based algorithm for solving differential equations, Academy of Sciences of Moldova, Vol. 15, No. 3, pp. 303-313, 2007.
[37] A. M. Wazwaz, A new algorithm for calculating Adomian polynomials for nonlinear operators, Applied Mathematics and Computation , Vol. 111, pp. 53-69, 2000.
Cite This Article
  • APA Style

    Mohamed S. Al-luhaibi, Nahed A. Saker. (2014). An Analytical Treatment to Fractional Gas Dynamics Equation. Applied and Computational Mathematics, 3(6), 323-329. https://doi.org/10.11648/j.acm.20140306.16

    Copy | Download

    ACS Style

    Mohamed S. Al-luhaibi; Nahed A. Saker. An Analytical Treatment to Fractional Gas Dynamics Equation. Appl. Comput. Math. 2014, 3(6), 323-329. doi: 10.11648/j.acm.20140306.16

    Copy | Download

    AMA Style

    Mohamed S. Al-luhaibi, Nahed A. Saker. An Analytical Treatment to Fractional Gas Dynamics Equation. Appl Comput Math. 2014;3(6):323-329. doi: 10.11648/j.acm.20140306.16

    Copy | Download

  • @article{10.11648/j.acm.20140306.16,
      author = {Mohamed S. Al-luhaibi and Nahed A. Saker},
      title = {An Analytical Treatment to Fractional Gas Dynamics Equation},
      journal = {Applied and Computational Mathematics},
      volume = {3},
      number = {6},
      pages = {323-329},
      doi = {10.11648/j.acm.20140306.16},
      url = {https://doi.org/10.11648/j.acm.20140306.16},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140306.16},
      abstract = {In this paper, the new iterative method (NIM) is applied to solve nonlinear fractional gas dynamics equation. Further, a coupling of the Sumudu transform and   Adomian decomposion (STADM) is used to get an approximate solution of the same problem. The results obtained by the two methods are found to be in agreement. Therefore, the NIM may be considered efficient method for finding approximate solutions of both linear and nonlinear fractional differential equations.},
     year = {2014}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - An Analytical Treatment to Fractional Gas Dynamics Equation
    AU  - Mohamed S. Al-luhaibi
    AU  - Nahed A. Saker
    Y1  - 2014/12/29
    PY  - 2014
    N1  - https://doi.org/10.11648/j.acm.20140306.16
    DO  - 10.11648/j.acm.20140306.16
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
    SP  - 323
    EP  - 329
    PB  - Science Publishing Group
    SN  - 2328-5613
    UR  - https://doi.org/10.11648/j.acm.20140306.16
    AB  - In this paper, the new iterative method (NIM) is applied to solve nonlinear fractional gas dynamics equation. Further, a coupling of the Sumudu transform and   Adomian decomposion (STADM) is used to get an approximate solution of the same problem. The results obtained by the two methods are found to be in agreement. Therefore, the NIM may be considered efficient method for finding approximate solutions of both linear and nonlinear fractional differential equations.
    VL  - 3
    IS  - 6
    ER  - 

    Copy | Download

Author Information
  • Department of Mathematics, Faculty of Science, Kirkuk University, Iraq

  • Department of Mathematics, Faculty of Science, Menoufia University, Egypt

  • Sections