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Dynamical system method for investigating existence and dynamical property of solution of nonlinear time-fractional PDEs

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Abstract

In this paper, based on the dynamical system method, a new approach for investigating solutions of nonlinear time-fractional partial differential equations (PDEs) is introduced. By proposing a novel technique with separation variables, the phase portraits of system derived from the nonlinear time-fractional PDEs are analyzed, and the issue of existence for the solution of the time-fractional PDEs is considered. Moreover, the dynamical properties of the solution of the time-fractional PDEs are studied in detail. As examples, three nonlinear time-fractional models such as the reaction–diffusion model, the biology population model and the fluid model are studied by using this new approach. In some special parametric conditions, exact solutions of these models are obtained. The dynamical properties of some exact solutions are illustrated by graphs. Compared with the results in current studies, the results obtained in this paper are new.

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Acknowledgements

This study was funded by the National Natural Science Foundation of China (Grant No. 11361023), the Natural Science Foundation in Chongqing City of China (Grant No. cstc2018jcyjAX0766) and the Special Support Program for High-level Talents in Chongqing City of China (Grant No. cstc2018kjcxljrc0049).

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  1. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, People’s Republic of China

    Weiguo Rui

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Rui, W. Dynamical system method for investigating existence and dynamical property of solution of nonlinear time-fractional PDEs. Nonlinear Dyn 99, 2421–2440 (2020). https://doi.org/10.1007/s11071-019-05410-x

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