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Classical and non-classical Lie symmetry analysis, conservation laws and exact solutions of the time-fractional Chen–Lee–Liu equation

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Abstract

The current work deals with the classical and non-classical Lie group analysis for the fractional type system of nonlinear Chen–Lee–Liu (CLL) equations. The invariance surface condition is imposed to this system of fractional differential equations to find non-classical generators. Non-classical and classical Lie symmetry analysis are used to similarity reductions of this system. Corresponding exact solutions are obtained for the extracted generators in both classical and non-classical senses. Convergence of the obtained solutions when fractional order tends to the integer one, is graphically checked. Finally, the conservation laws are investigated for the system of time-fractional CLL equations, specially for the non-classical vector fields.

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Notes

  1. Here

    $$\begin{aligned}{} & {} \begin{aligned} \bigg ({\mathcal {P}}^{\tau ,\alpha }_{\beta }{\mathcal {F}}\bigg ):=\prod _{j=0}^{n-1}\bigg (\tau +j-\frac{1}{\beta }\zeta \frac{{\textrm{d}}}{{\textrm{d}}\zeta }\bigg ) \bigg ({\mathcal {K}}^{\tau +\alpha ,n-\alpha }_\beta {\mathcal {F}}\bigg )(\zeta ), \end{aligned}\\{} & {} \begin{aligned} n= {\left\{ \begin{array}{ll} [\alpha ]+1,&{}\quad \alpha \not \in {\mathbb {N}}\\ \alpha ,&{}\quad \alpha \in {\mathbb {N}} \end{array}\right. } \end{aligned} \end{aligned}$$

    and

    $$\begin{aligned} \begin{aligned} \bigg ({\mathcal {K}}^{\tau ,\alpha }_{\beta }{\mathcal {F}}\bigg ):= {\left\{ \begin{array}{ll} \dfrac{1}{\varGamma (\alpha )}\displaystyle \int _1^{\infty }(s-1)^{\alpha -1}s^{-(\tau +\alpha )}{\mathcal {F}}\big (\zeta s^{\frac{1}{\beta }}\big ){\textrm{d}}s, &{}\quad \alpha >0,\\ {\mathcal {F}}(\zeta ),&{}\quad \alpha =0, \end{array}\right. } \end{aligned} \end{aligned}$$

    show the Erdélyi–Kober fractional derivative and integral operators for \( {\mathcal {F}}(\zeta ).\)

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Authors and Affiliations

  1. Department of Computer Engineering, Biruni University, Istanbul, Turkey

    M. S. Hashemi

  2. Department of Mathematics, Basic Sciences Faculty, University of Bonab, P.O. Box 55513-95133, Bonab, Iran

    M. S. Hashemi, A. Haji-Badali & F. Alizadeh

  3. Department of Mathematics, Science Faculty, Firat University, 23119, Elazig, Turkey

    Mustafa Inc

  4. Department of Medical Research, China Medical University, Taichung, 40402, Taiwan

    Mustafa Inc

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  1. M. S. Hashemi
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  2. A. Haji-Badali
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  3. F. Alizadeh
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Correspondence to Mustafa Inc.

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Communicated by Vasily E. Tarasov.

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Hashemi, M.S., Haji-Badali, A., Alizadeh, F. et al. Classical and non-classical Lie symmetry analysis, conservation laws and exact solutions of the time-fractional Chen–Lee–Liu equation. Comp. Appl. Math. 42, 73 (2023). https://doi.org/10.1007/s40314-023-02217-w

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