Applied and Computational Mathematics
Volume 3, Issue 6, December 2014, Pages: 323-329
Received: Oct. 24, 2014;
Accepted: Dec. 9, 2014;
Published: Dec. 29, 2014
Views 2611 Downloads 183
Mohamed S. Al-luhaibi, Department of Mathematics, Faculty of Science, Kirkuk University, Iraq
Nahed A. Saker, Department of Mathematics, Faculty of Science, Menoufia University, Egypt
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.
Mohamed S. Al-luhaibi,
Nahed A. Saker,
An Analytical Treatment to Fractional Gas Dynamics Equation, Applied and Computational Mathematics.
Vol. 3, No. 6,
2014, pp. 323-329.
Young GO. Definition of physical consistent damping laws with fractional derivatives. Z Angew Math Mech 1995;75:623–35.
He JH. Some applications of nonlinear fractional differential equations and their approximations. Bull Sci Technol 1999;15(2):86–90.
He JH. Approximate analytic solution for seepage flow with fractional derivatives in porous media. Comput Methods Appl Mech Eng 1998;167:57–68.
Hilfer R, editor. Applications of Fractional Calculus in Physics. Singapore, New Jersey, Hong Kong: World Scientific Publishing Company; 2000. p. 87–130.
Podlubny I. Fractional differential equations. New York: Academic Press; 1999.
Mainardi F, Luchko Y, Pagnini G. The fundamental solution of the space–time fractional diffusion equation. Fract Calc Appl Anal 2001;4:153–92.
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.
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.
Debnath L. Fractional integrals and fractional differential equations in fluid mechanics. Frac Calc Appl Anal 2003;6:119–55.
Caputo M. Elasticita e Dissipazione. Zani-Chelli: Bologna; 1969.
Miller KS, Ross B. An introduction to the fractional calculus and fractional differential equations. New York: Wiley; 1993.
Oldham KB, Spanier J. The fractional calculus theory and applications of differentiation and integration to arbitrary order. New York: Academic Press; 1974.
J. H. He, “Asymptotic methods for solitary solutions and compactons,” Abstract and Applied Analysis, vol. 2012, Article ID 916793, 130 pages, 2012.
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.
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.
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.
S. Bhalekar, V. D. Gejji, New iterative method: application to partial differential equations, Applied Mathematics and Computation. 203 (2) (2008) 778-783.
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.
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.
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.
J.H. He, Variational iteration method-akind of nonlinear analytical technique: some examples, International Journal of Nonlinear Mechanics. 34 (1999) 699-708.
S. Das and R. Kumar, Approximate analytical solutions of fractional gas dynamic equations, Applied Mathematics and Computation, vol. 217, no. 24, pp. 9905–
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.
Y. Cherruault, Convergence of Adomian's method, Kybernetes. 18 (2) (1989) 31- 38
A.J. Jerri, Introduction to Integral Equations with Applications. seconded, Wiley. Interscience. 1999.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
J. Zhang, A Sumudu based algorithm for solving differential equations, Academy of Sciences of Moldova, Vol. 15, No. 3, pp. 303-313, 2007.
A. M. Wazwaz, A new algorithm for calculating Adomian polynomials for nonlinear operators, Applied Mathematics and Computation , Vol. 111, pp. 53-69, 2000.