Assessment of the Exact Solutions of the Space and Time Fractional Benjamin-Bona-Mahony Equation via the -Expansion Method, Modified Simple Equation Method, and Liu’s Theorem

Exact travelling wave solutions to the space and time fractional Benjamin-Bona-Mahony (BBM) equation defined in the sense of Jumarie’s modified Riemann-Liouville derivative via the expansion and the modified simple equation methods are presented in this paper. A fractional complex transformation was applied to turn the fractional BBM equation into an equivalent integer order ordinary differential equation. New complex type travelling wave solutions to the space and time fractional BBM equation were obtained with Liu’s theorem. The modified simple equation method is not effective for constructing solutions to the fractional BBM equation. 1. Introduction Nonlinear partial differential equations arise in a large number of physics, mathematics, and engineering problems. In the Soliton theory, the study of exact solutions to these nonlinear equations plays a very germane role, as they provide much information about the physical models they describe. In recent times, it has been found that many physical, chemical, and biological processes are governed by nonlinear partial differential equations of noninteger or fractional order [1–4]. Various powerful methods have been employed to construct exact travelling wave solutions to nonlinear partial differential equations. These methods include the inverse scattering transform [5], the Backlund transform [6, 7], the Darboux transform [8], the Hirota bilinear method [9], the tanh-function method [10, 11], the sine-cosine method [12], the exp-function method [13], the generalized Riccati equation [14], the homogenous balance method [15], the first integral method [16, 17], the expansion method [18, 19], and the modified simple equation method [20–22]. In this paper, we apply the expansion method and the modified simple equation method to construct travelling wave solutions to the space and time fractional Benjamin-Bona-Mahony (BBM) equation in the sense of Jumarie’s modified Riemann-Liouville derivative via a fractional complex transformation and we further get new complex type solutions to the equation by applying Liu’s theorem [23]. The Benjamin-Bona-Mahony equation is of the following form: This equation was introduced in [24] as an improvement of the Korteweg-de Vries equation (KdV equation) for modelling long waves of small amplitude in 1 + 1 dimensions. It is used in modelling surface waves of long wavelength in liquids, acoustic gravity waves in compressible fluids, and acoustic waves in anharmonic crystals. Jumarie’s modified Riemann-Liouville derivative of order with respect to is defined as [25] Some useful