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Exact Traveling Wave Solutions and Bifurcations of the Time-Fractional Differential Equations with Applications

Wenjing Zhu

Department of Mathematics, China Jiliang University, Hangzhou, Zhejiang 310018, P. R. China

E-mail Address: wenjingzhu122@163.com

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Yonghui Xia

http://orcid.org/0000-0001-8918-3509

Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, P. R. China

E-mail Address: yhxia@zjnu.cn

E-mail Address: xiadoc@163.com

Corresponding author

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Bei Zhang

School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, P. R. China

E-mail Address: zhangbei1929@163.com

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, and

Yuzhen Bai

School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, P. R. China

E-mail Address: baiyu99@126.com

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https://doi.org/10.1142/S021812741950041XCited by:36 (Source: Crossref)

Abstract

This paper presents a method to investigate exact traveling wave solutions and bifurcations of the nonlinear time-fractional partial differential equations with the conformable fractional derivative proposed by [Khalil et al., 2014]. The method is based on employing the bifurcation theory of planar dynamical systems proposed by [Li, 2013]. For the fractional PDEs, up till now, there is no related paper to obtain the exact solutions by applying bifurcation theory. We show how to use this method with applications to two fractional PDEs: the fractional Klein–Gordon equation and the fractional generalized Hirota–Satsuma coupled KdV system, respectively. We find the new exact solutions including periodic wave solution, kink wave solution, anti-kink wave solution and solitary wave solution (bright and dark), which are different from previous works in the literature. This approach can also be extended to other nonlinear time-fractional differential equations with the conformable fractional derivative.

Keywords:

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