Analytical solution of a time-fractional Navier–Stokes equation by Adomian decomposition method Shaher Momani a, * , Zaid Odibat b a Department of Mathematics, Mutah University, P.O. Box 7, Al-Karak 962, Jordan b Prince Abdullah Bin Ghazi Faculty of Science and IT, Al-Balq’a Applied University, Salt, Jordan Abstract The aim of the present analysis is to apply Adomian decomposition method for the solution of a time-fractional Navier–Stokes equation in a tube. By using an initial value, the explicit solution of the equation has been presented in the closed form and then its numerical solution has been represented graphically. The present method performs extremely well in terms of efficiency and simplicity. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Navier–Stokes equations; Caputo fractional derivative; Decomposition method 1. Introduction Large classes of linear and nonlinear differential equations, both ordinary as well as partial, can be solved by the Adomian decomposition method [1,2]. This computational method yields analytical solutions and has certain advantages over standard numerical methods. It is free from rounding off errors as it does not involve discretization, and does not require large computer obtained memory or power. Recently, the solution of frac- tional ordinary differential equations has been obtained through Adomian decomposition method by the researchers [3,4]. The application of Adomian decomposition method for the solution of fractional partial dif- ferential equations has also been established by [5–7]. The fractional derivative has been occurring in many physical and engineering problems such as frequency dependent damping behavior of material, diffusion processes, motion of a large thin plate in a Newtonian fluid, creeping and relaxation functions for viscoelastic materials. For more details on the applications of frac- tional derivatives in continuum and statistical mechanics, see [8,9]. Recently, El-Shahed and Salem [10] have generalized the classical Navier–Stokes equations by replacing the first time derivative by a fractional derivative of order a,0< a 6 1. They used Laplace transform, Fourier sine transform and finite Hankel transforms to obtain exact solutions for three different special cases. 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.11.025 * Corresponding author. E-mail addresses: shahermm@yahoo.com (S. Momani), odibat@bau.edu.jo (Z. Odibat). Applied Mathematics and Computation 177 (2006) 488–494 www.elsevier.com/locate/amc