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Discontinuity, Nonlinearity, and Complexity 5(2) (2016) 121--132 | DOI:10.5890/DNC.2016.06.002

Yufeng Zhang

College of Sciences, China University of Mining and Technology, Xuzhou 221116, P.R. China

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Abstract

The noncommutative Bell polynomials and their dual Bell polynomials are presented, respectively, which are extensively applied to mathematics and physics. We make use of them to exhibit a method for generating integrable hierarchies of evolution equations. As applications, we obtain the Burgers hierarchy and a convection-diffusion equation which can be applied to fluid mechanics, specially, be used to represent mass transformations in fluid systems under some constrained conditions. As reduced cases, the Burgers equation which has extensive applications in physics is followed to produce. Furthermore, we obtain a set of nonlinear evolution equations with four potential functions which reduces to a new nonlinear equation similar to the Calogero-Degasperis-Fokas equation. Finally, we discrete the convection-diffusion equation and obtain its three kinds of finite-difference schemes, that is,the weighted implicit difference scheme and the Lax difference scheme. Their some properties including truncation errors, compatibilities and stabilities based on the Von Neumann condition are discussed in detail.

Acknowledgments

This work was supported by the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (2014),the National Natural Science Foundation of China (grant No. 11371361) and the Fundamental Research Funds for the Central Universities (2013XK03) as well as the Natural Science Foundation of Shandong Province (grant No. ZR2013AL016).

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