Eur. Phys. J. Plus (2019) 134: 402
https://doi.org/10.1140/epjp/i2019-12731-x

Regular Article

Series solutions of nonlinear conformable fractional KdV-Burgers equation with some applications

¹ Department of Mathematics, Faculty of Science, Al Balqa Applied University, 19117, Salt, Jordan
² Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, P. O. Box 1982, 31441, Dammam, Saudi Arabia
³ College of Humanities and Sciences, Ajman University, Ajman, United Arab Emirates
⁴ Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, 21589, Jeddah, Saudi Arabia
⁵ Department of Mathematics, Faculty of Science, University of Jordan, 11942, Amman, Jordan
⁶ Department of Mathematics, Faculty of Science, Quaid-I-Azam University, 45320, Islamabad, Pakistan

^* e-mail: s.momani@ju.edu.jo

Received: 31 March 2019
Accepted: 7 May 2019
Published online: 27 August 2019

Abstract

In this paper, the non-linear fractional KdV-Burgers equation (KdVBE) in terms of conformable fractional derivative (CFD) is reconstituted instead of the Caputo fractional derivative and the series solution of this case is also presented by using the residual power series (RPS) method. Moreover, five important and interesting applications related to the fractional KdVBE are given and discussed in order to show the behavior of the surface graphs of the solutions. More clarifications: Firstly, we compare the solutions of the conformable fractional KdVBE and the Caputo fractional KdVBE. Secondly, in order to demonstrate the generality, potential and superiority of the RPS method, we discuss the simplicity of this method compared with other methods. Thirdly, we present the approximate solutions with graphical results of a time-CFD, space-CFD and time-space-CFD non-linear fractional KdVBEs. Finally, the results indicate that the CFD is very suitable for modeling the KdVBE and computations show that our proposed method for solving the conformable fractional KdVBE does not have mathematical requirements which implies that it is very effective as well as for providing the numerical solutions and more flexible in choosing the initial guesses approximations.