An International Journal of Optimization and Control: Theories & Applications ISSN:2146-0957 eISSN:2146-5703 Vol.8, No.1, pp.63-72 (2018) http://doi.org/10.11121/ijocta.01.2018.00417 RESEARCH ARTICLE New travelling wave solutions for fractional regularized long-wave equation and fractional coupled Nizhnik-Novikov-Veselov equation ¨ Ozkan G¨ uner * Department of International Trade, Faculty of Economics and Administrative Sciences, C ¸ ankırı Karatekin University, C ¸ankırı, Turkey ozkanguner@karatekin.edu.tr ARTICLE INFO ABSTRACT Article History: Received 15 November 2016 Accepted 16 June 2017 Available 23 October 2017 In this paper, solitary-wave ansatz and the (G ′ /G) −expansion methods have been used to obtain exact solutions of the fractional regularized long-wave (RLW) and coupled Nizhnik-Novikov-Veselov (NNV) equation. As a result, three types of exact analytical solutions such as rational function solutions, trigonometric function solutions, hyperbolic function solutions are formally derived from these equations. Proposed methods are more powerful and can be applied to other fractional differential equations arising in mathematical physics. Keywords: Exact solution Ansatz method (G ′ /G) −expansion method Fractional regularized long-wave equation Fractional coupled Nizhnik-Novikov- Veselov equation AMS Classification 2010: 34A08, 35R11, 83C15 1. Introduction Fractional differential equations (FDEs) are the generalized form of classical differential equations of integer order. Researchers especially in applied mathematician and physicist became highly inter- ested in obtaining exact solutions for nonlinear FDEs in recent decades. Nonlinear FDEs are fre- quently used to describe many problems of phys- ical phenomena that may arise in various fields such as biology, physics, chemistry, engineering, heat transfer, applied mathematics, control the- ory, mechanics, signal processing, seismic wave analysis, finance, and many other fractional dy- namical systems [1-3]. In the past several decades, new exact solutions may help to find new phenomena. So, vari- ety of powerful analytical and numerical meth- ods for solving differential equations of fractional order have been suggested such as the adomian decomposition method, the homotopy perturba- tion method, the variational iteration method, the finite difference method, the differential trans- form method, homotopy perturbation method, the homotopy analysis method, the sub-equation method, the first integral method, the (G’/G)- expansion method, the modified trial equation method, the functional variable method, the exp- function method, the simplest equation method, the exponential rational function method, ansatz method and others [4-31]. To solve mathematical problems, the transforms are an important methods. A variety of use- ful transforms for solving different problems ap- peared in the literature, such as the traveling wave transform, the Fourier transform and the others [32-41]. Recently, Li and He [42] suggested a frac- tional complex transform to convert FDEs into ordinary differential equations (ODEs). *Corresponding Author 63