The fractional natural decomposition method: theories and applications

Abstract

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.

The approximate solutions of fractional gas dynamics equations by means of fractional natural decomposition method

“…Later, Belgacem and Silambarasan defined inverse of natural transform and studied some additional fundamental properties of its [19]. Recently, N-transform is used to find the solution of differential and integral equations [21,22,23,24,25,26,27,28].…”

Fractional KdV and Boussenisq‐Burger's equations, reduction to PDE and stability approaches

“…Very recently, in studying the fractional equations that arise in shallow water waves, authors are trying to find approximate solutions either analytically or numerically . The present work discusses the details about the results found in these works.…”

Numerical solution for (2 + 1)‐dimensional time‐fractional coupled Burger equations using fractional natural decomposition method

“…The proposed algorithm provides solution in terms infinite series, and the obtained series rapidly converge to a closed form exact solution; for instance, Belgacem and Silambarasan provide a proof for two fundamental results of fractional natural transform by employing duality of Natural and Laplace transform. Further, by applying the duality of Sumudu and N‐transform, Rawashdeh draws the proof for three important results. Moreover, in an attempt, Soliman examined the coupled fractional Burgers (CFB) equations by employing variation iteration method (VIM), and Hızela and Kucukarslan have considered homotopy perturbation method (HPM) to find the approximated solution for CFB equations.…”