Abstract
A new technique has been developed for analytical solutions of fractional order nonlinear ODE system. We propose a reliable method called the fractional natural decomposition method (FNDM). The FNDM is based on the natural transform method (NTM) and the Adomian decomposition method. We use the FNDM to construct new analytical approximate and exact solutions to systems of nonlinear fractional ordinary differential equation (NLFODEs). The fractional derivatives are described in the Caputo sense.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adomian G.: Solving frontier problems of physics: the decomposition method. Kluwer Academic Publisher, Boston (1994)
Adomian G.: A new approach to nonlinear partial differential equations. J. Math. Anal. Appl. 102, 420–434 (1984)
Belgacem F.B.M., Silambarasan R.: Maxwell’s equations by means of the natural transform. Math. Eng. Sci. Aerosp. 3(3), 313–323 (2012)
Caputo M.: Elasticitae Dissipazione. Zanichelli, Bologna (1969)
Caputo M., Mainardi F.: Linear models of dissipation in anelastic solids. Riv. del Nuovo Cimento. 1, 161–198 (1971)
Garg M., Manohar P.: Numerical solution of fractional diffusion-wave equation with two space variables by matrix method. Fract. Calc. Appl. Anal. 13(2), 191–207 (2010)
Garg M., Sharma A.: Solution of space-time fractional telegraph equation by Adomian decomposition method. J. Inequal. Spec. Funct. 2(1), 1–7 (2011)
He J.H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 167, 57–68 (1998)
Hilfer R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Jafari H., Nazari M., Baleanu D., Khalique C.M.: A new approach for solving a system of fractional partial differential equations. Comput. Math. Appl. 66, 838–843 (2013)
Jafari H., Seifi S.: Solving system of nonlinear fractional partial differention equations by homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 14, 1962–1969 (2009)
Katatbeh Q.D., Belgacem F.B.M.: Applications of the Sumudu transform to fractional differential equations. Nonlinear Stud. 18(1), 99–112 (2011)
Khan Z.H., Khan W.A.: N-transform properties and applications. NUST J. Engg. Sci. 1(1), 127–133 (2008)
Kilbas A.A., Srivastava H.M., Trujillo J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Kruskal M.D., Moser J.: Dynamical Systems, Theory and Applications, Lecturer Notes Physics, pp. 3–10. Springer, Berlin (1975)
Kumar, D.; Singh J.; Kiliman A.: An efficient approach for fractional Harry Dym equation by using Sumudu transform, abstract and applied analysis. Article ID 608943, pp. 8 (2013)
Loonker D., Banerji P.K.: Solution of fractional ordinary differential equations by natural transform. Int. J. Math. Eng. Sci. 2(12), 2277–6982 (2013)
Miller K.S., Ross B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)
Momani S., Al–Khaled K.: Numerical solutions for systems of fractional differential equations by the decomposition method. Appl. Math. Comput. 162(3), 1351–1365 (2005)
Podlubny I.: Fractional Differential Equations, Mathematics in Science and Engineering. Academic Press, San Diego (1999)
Rawashdeh M., Maitama S.: Solving nonlinear ordinary differential equations using the NDM. J. Appl. Anal. Comput. 5(1), 77–88 (2015)
Rawashdeh, M.S.: Solving fractional ordinary differential equations using FNTM. Thai J. Math. (2016) (In press)
Rawashdeh M., Maitama S.: Solving coupled system of nonlinear PDEs using the natural decomposition method. Int. J. Pure Appl. Math. 92(5), 757–776 (2014)
Rawashdeh M., Maitama S.: Solving PDEs using the natural decomposition method. Nonlinear Stud. 23(1), 63–72 (2016)
Rawashdeh M.: An efficient approach for time-fractional damped burger and Time–Sharma–Tasso–Olver equations using The FRDTM. Appl. Math. Inf. Sci. 9(3), 1–8 (2015)
Rawashdeh M.: A new approach to solve the fractional Harry Dym equation using the FRDTM. Int. J. Pure Appl. Math. 95(4), 553–566 (2014)
Ray S.S., Bera R.K.: An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method. Appl. Math. Comput. 167, 561–571 (2005)
Wang Q.: Homotopy perturbation method for fractional KdV-Burgers equation. Chaos Solitons Fractals 35, 843–850 (2008)
Yildirim A.: He’s homotopy perturbation method for solving the space and time fractional telegraph equations. Int. J. Comput. Math. 87(13), 2998–3006 (2010)
Yang C., Hou J.: An approximate solution of nonlinear fractional differential equation by Laplace transform and adomian polynomials. J. Inf. Comput. Sci. 10, 213–222 (2013)
Zurigat M., Momani S., Odibat Z., Alawneh A.: The homotopy analysis method for handling systems of fractional differential equations. Appl. Math. Model. 34(1), 24–35 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rawashdeh, M.S., Al-Jammal, H. Numerical Solutions for Systems of Nonlinear Fractional Ordinary Differential Equations Using the FNDM. Mediterr. J. Math. 13, 4661–4677 (2016). https://doi.org/10.1007/s00009-016-0768-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00009-016-0768-7