Eur. Phys. J. Plus (2018) 133: 240
https://doi.org/10.1140/epjp/i2018-12068-0

Regular Article

Time fractional third-order variant Boussinesq system: Symmetry analysis, explicit solutions, conservation laws and numerical approximations

¹ King Saud University, Department of Applied Mathematics, 22452, 11495, Riyadh, Saudi Arabia
² Firat University, Science Faculty, Department of Mathematics, 23119, Elazig, Turkey
³ Federal University Dutse, Science Faculty, Department of Mathematics, 7156, Jigawa, Nigeria
⁴ Cankaya University, Department of Mathematics, öğretmenler Cad., 1406530, Ankara, Turkey
⁵ Institute of Space Sciences, Bucharest, Romania

^* e-mail: minc@firat.edu.tr

Received: 21 February 2018
Accepted: 17 May 2018
Published online: 27 June 2018

Abstract

The current work provides comprehensive investigation for the time fractional third-order variant Boussinesq system (TFTOBS) with Riemann-Liouville (RL) derivative. Firstly, we obtain point symmetries, similarity variables, similarity transformation and reduce the governing equation to a special system of ordinary differential equation (ODE) of fractional order. The reduced equation is in the Erdelyi-Kober (EK) sense. Secondly, we solve the reduced system of ODE using the power series (PS) expansion method. The convergence analysis for the power series solution is analyzed and investigated. Thirdly, the new conservation theorem and the generalization of the Noether operators are applied to construct nonlocal conservation laws (CLs) for the TFTOBS. Finally, we use residual power series (RPS) to extract numerical approximation for the governing equations. Interesting figures that explain the physical understanding for both the explicit and approximate solutions are also presented.