(8) New exact solutions for the time fractional coupled Boussinesq–Burger equation and approximate long water wave equation in shallow water

Mostafa M.A. Khater ^{a , b , ∗}, Dipankar Kumar ^{c , d}

a Department of Mathematics, Faculty of Science, Jiangsu University, Jiangsu, China

b Faculty of Science, Department of Mathematics, Mansoura University, 35516 Mansoura, Egypt

c Doctoral Student, Division of Engineering Mechanics and Energy, Graduate School of Systems and Information Engineering, University of Tsukuba, Tennodai 1-1-1, Tsukuba, Ibaraki, Japan

d Department of Mathematics, Bangabandhu Sheikh Mujibur Rahman Science, and Technology University, Gopalganj 8100, Bangladesh

Received 2 June 2017; accepted 21 July 2017
Available online 24 August 2017

Abstract
The aim of the article is to construct exact solutions for the time fractional coupled Boussinesq–Burger and approximate long water wave equations by using the generalized Kudryashov method. The fractional differential equation is converted into ordinary differential equations with the help of fractional complex transform and the modified Riemann–Liouville derivative sense. Applying the generalized Kudryashov method through with symbolic computer maple package, numerous new exact solutions are successfully obtained. All calculations in this study have been established and verified back with the aid of the Maple package program. The executed method is powerful, effective and straightforward for solving nonlinear partial differential equations to obtain more and new solutions with the integer and fractional order.
© 2017 Shanghai Jiaotong University. Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )
Keywords: The generalized Kudryashov method; The time fractional coupled Boussinesq–Burger equation; The time fractional approximate long water wave
equation; Exact solutions.