Journal of Mechanical Engineering Research and Developments (JMERD) 42(5) (2019) 132-137 Cite The Article: Ali A. Mustafa, Wafaa M. Taha, A R Seadawy (2019). New Soliton Solutions of Nonlinear Space-Time Fractional (2+1)-Dimensional Ablow Kaup- Newell_Segur (AKNS) Equation by Standard and Improved � ′ �-Expansion Method. Journal of Mechanical Engineering Research and Developments, 42(5) : 132-137 ARTICLE DETAILS Article History: Received 15 July 2019 Accepted 26 August 2019 Available online 29 August 2019 ABSTRACT In this essay, we offer the standard and improved ( ′ )-expansion method to seek right solutions of nonlinear fractional differential equation (2+1)-dimensional Ablow Kaup-Newell_Segur (AKNS) with the fractional complex transform. The fractional derivative is defined in the sense of modified Riemann-Liouville derivative. The solutions of this equation are acquired through of the hyperbolic, trigonometrical and rational functions. It has been shown the expansion method award several novel results which are effectual and easy to calculate by the assist of a symbolic computation system. KEYWORDS (AKNS) equation, standard ( ′ )-expansion method, improved ( ′ )-expansion method, non-linear fractional differential equation. 1. INTRODUCTION During the past few years, the nonlinear differential equations (NLDEs) are vastly utilized as models to depict physical phenomena in the different area of science, especially in fluid mechanics and plasma wave[1–4]. Nowadays, the nonlinear fractional differential equations (FNLDEs) have become the concentrate of many researchers due to their recurrent appearance in different implementation, like in the fluid flow, system identification, and other areas[5–10] . To get the solutions of (FNLDEs) is a significant work. Many active methods have been put for calculating the exact solutions. such as, the modified Kudryashov method [11], (w/g)-- expansion method [12], Lie group analysis method [13], and new fractional sub equation method [14]. Recently, Wang et al. [15] constructed a widely used explicit and brief approach known as ( ′ )- expansion method to get exact travelling wave solutions of (NLDEs). Correspondingly, most authors used this approach to obtain a variety of results for (NLDEs) [16–18]. Additionally, Zhang et al. [19] developed the major ( ′ )-expansion method to produce abundant of travelling wave results. Also, this method has been seem authoritative, active and powerful by them. The main objective of this work is to apply the standard and improved ( ′ )-expansion method to construct new and precise wave solutions. In this paper we will study an important physical model (The AKNS equation). Many researchers have been obtained the solutions of the AKNS equation such as, Hirota’s bilinear method [20], and the ansatz method [21]. The standard and improved ( ′ )-expansion method is one of the efficient method to get right solutions of nonlinear fractional three- dimensional (AKNS) equation in the sense of Modified Riemann-Lioville derivatives and we will supported our results by comparisons with other methods and painting 3D and 2D graphics of the exact solutions. 2. RIEMANN-LIOVILLE DERIVATIVE AND ITS PROPERTIES Definition: The Jumarie’s modified Riemann--Liouville derivative of order is expressed by: () = ⎩ ⎨ ⎧ 1 Γ(1 −) �(−) − �() −(0)� , 0< <1 0 [ () ()] (−) , ≤ < +1 , ≥ 1 . (1) Some necessary properties: = Γ(1 + ) Γ(1 + −) − , >0 (2) [()()] = () ()+ () () (3) [()] = ′ [()] ()= [()]� ′ ()� (4) 3. BASIC IDEA OF FRACTIONAL ( ′ )-EXPANSION METHOD Assume that we have the following (NFPDE): �, , ,… � =0 , 0< ≤ 1 (5) Where is polynomial of (, ) and is an anonymous function. We list the main steps of the method: Firstly. To obtain the solution of (5), we put the variables: (, )= () , , = 1 (1 + ) + 2 (1 + ) , (6) Journal of Mechanical Engineering Research and Developments (JMERD) DOI : http://doi.org/10.26480/jmerd.05.2019.132.137 NEW SOLITON SOLUTIONS OF NONLINEAR SPACE-TIME FRACTIONAL (2+1)- DIMENSIONAL ABLOW KAUP-NEWELL_SEGUR (AKNS) EQUATION BY STANDARD AND IMPROVED � ′ �-EXPANSION METHOD Ali A. Mustafa 1 , Wafaa M. Taha 2 *, and A R Seadawy 3,4 1 Department of Math., College Education for Pure Sciences, Tikrit University 2 Department of Math., College of Sciences, Kirkuk University, Iraq 3 Mathematics Department, Faculty of Science, Taibah University, Medina, Saudi Arabia 4 Mathematics Department, Faculty of Science, Beni-Suef University, Beni Suef, Egypt *Corresponding Author Email: Wafaa_y2005@yahoo.com This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ISSN: 1024-1752 CODEN : JERDFO RESEARCH ARTICLE