Research Article Received 9 July 2016 Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/mma.4144 MOS subject classification: 35J10, 33E12, 35A22, 35Q61, 44A10, 44A30, 81V10, 35C05, 35C10 The fractional natural decomposition method: theories and applications Mahmoud S. Rawashdeh Communicated by C. Cuevas In this paper, we propose a new method called the fractional natural decomposition method (FNDM). We give the proof of new theorems of the FNDM, and we extend the natural transform method to fractional derivatives. We apply the FNDM to construct analytical and approximate solutions of the nonlinear time-fractional Harry Dym equation and the nonlinear time-fractional Fisher’s equation. The fractional derivatives are described in the Caputo sense. The effectiveness of the FNDM is numerically confirmed. Copyright © 2016 John Wiley & Sons, Ltd. Keywords: fractional derivative; Caputo derivative; natural transform; Adomian decomposition method 1. Introduction Fractional derivatives were invented by Leibnitz soon after the more familiar integer-order derivatives [1–8] but have only recently started to attract much more attention of physicists [4] and mathematicians because of its applications. Mathematical modeling of many physical systems leads to linear and nonlinear fractional differential equations in various fields of physics and engineering [9–11]. The use of fractional differentiation for the mathematical modeling of real-world applications has been rapidly growing, such as the modeling of earthquake and the fluid dynamic traffic model with fractional derivatives. Moreover, fractional calculus is used to model anomalous diffusion, in which cloud of particles spreads in a different manner than the traditional diffusion. The famous continuous time random walk model with power law waiting time distribution describes this phenomenon. There are many physical applications in science and engineering [2, 10] that can be represented by models using fractional differen- tial equations, which are quite useful for many physical problems. These equations are represented by linear and nonlinear fractional PDEs, and solving such fractional differential equations is very important. The Harry Dym equation [12] is used in several physical applications, and it was first introduced by Kruskal and Moser [6] and is cred- ited to unpublished paper by Harry Dym in 1973–1974. The Harry Dym equation is related to the Korteweg–de Vries equation, and it is also associated with the Sturm–Liouville operator. The Harry Dym equation represents a system in which dispersion and nonlin- earity are coupled together. Harry Dym is a completely integrable nonlinear evolution equation. It is important because it obeys an infinite number of conservation laws and it does not possess the Painleve property. The Fisher equation was proposed by Fisher in 1937 as a model for the spatial and temporal propagation of a virile gene in an infinite medium. It is skirmished in chemical kinetics, flame propagation, autocatalytic chemical reaction, nuclear reactor theory, neurophysiology, and branching Brownian motion process. Fisher equation combines diffusion with logistic nonlinearity and concludes problems, such as nonlinear evolution of a population in a one-dimensional habitat. Fisher equation models population growth and dispersion. The fractional derivative term models power law delays between movements, and it also models the transmission of nerve impulse. The natural decomposition method (NDM) was first introduced by M. Rawashdeh and S. Maitama in 2014 [13–16], to solve linear and nonlinear ODEs and PDEs that appear in many areas of science. In addition, Baskonus, Bulut, and Pandir [17] also consider the NDM to solve linear and nonlinear PDEs. In [18], Loonker and Banerji gave proofs to two results for the fractional natural transform using the duality of Laplace and N-transform, but in this paper, we give proofs to three major theorems using the duality of Sumudu and N- transform. Recently, several numerical methods were proposed to solve fractional differential equations such as the variational iteration method [3, 19–22], the Sumudu transform method [23, 24], homotopy perturbation method [25–27], Adomian decomposition method Department of Mathematics and Statistics, Jordan University of Science and Technology, PO Box 3030, 22110, Irbid, Jordan * Correspondence to: Mahmoud S. Rawashdeh, Department of Mathematics and Statistics, Jordan University of Science and Technology, PO Box 3030, 22110, Irbid, Jordan. † E-mail: rawashde@msu.edu; msalrawashdeh@just.edu.jo Copyright © 2016 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2016