Abstract
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 (G′/G) 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.
Highlights
Nonlinear partial differential equations arise in a large number of physics, mathematics, and engineering problems
It has been found that many physical, chemical, and biological processes are governed by nonlinear partial differential equations of noninteger or fractional order [1,2,3,4]
The results were compared to those obtained by Bulent and Erdal [29] using the direct algebraic method for complex travelling wave solution and those by Yanhong and Baodan [30] using the generalized (G/G) expansion method with the Klein-Gordon equation as an auxiliary equation
Summary
Nonlinear partial differential equations arise in a large number of physics, mathematics, and engineering problems. Various powerful methods have been employed to construct exact travelling wave solutions to nonlinear partial differential equations. We provide a brief explanation of the (G/G) expansion method for finding travelling wave solutions of nonlinear fractional partial differential equations. Αm are constants to be determined and G = G(ξ) is a solution of the linear ordinary differential equation of the following form: G (ξ) + λG (ξ) + μG (ξ) = 0,.
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