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Extended Lie symmetry approach for mixed fractional derivatives of magneto-electro-elastic circular rod: innovative reduction, conservation laws, optical solitons and bifurcation analysis

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Abstract

The central objective of this study is to examine how the Lie symmetry technique and conservation law theories can be employed to analyze fractional partial differential equations with a mixed derivative \(\partial _{t}^{2\beta }(u_{x x})\), specifically considering the Riemann–Liouville time-fractional and integer-order x-derivatives. In the beginning, we shed light on the prolongation formula that encompasses the infinitesimal generator, particularly in the presence of mixed derivatives. Following that, through the identification of Lie symmetries, we introduce a comprehensive formula for deriving conservation laws, incorporating the notion of nonlinear self-adjointness. Utilizing the Lie symmetries and an alternative reduction technique, we achieve a novel reduction that eliminates the need for the Erd\(\acute{e}\)lyi-Kober operator. Additionally, family of wave solutions have been obtained corresponding to the longitudinal wave equation in magneto-electro-elastic structures and bifurcation analysis and chaotic analysis has been done for the governing model via Galilean transformation and phase plane portraits have been used to depict the changes in the behavior of the model by changing the values of the potential parameter.

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Acknowledgements

This research has received funding from the Council of Scientific and Industrial Research, New Delhi, India (CSIR file no. 09/1152(0017)/2019 -EMR-1). The support of the CSIR is greatly acknowledged.

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This research has received funding from the Council of Scientific and Industrial Research, New Delhi, India (CSIR file no. 09/1152(0017)/2019 -EMR-1).

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  1. Department of Mathematics, School of Basic Sciences, Central University of Haryana, Mahendergarh, Haryana, 123031, India

    Rajesh Kumar Gupta & Vikash Yadav

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  1. Rajesh Kumar Gupta
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  2. Vikash Yadav
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Gupta, R.K., Yadav, V. Extended Lie symmetry approach for mixed fractional derivatives of magneto-electro-elastic circular rod: innovative reduction, conservation laws, optical solitons and bifurcation analysis. Opt Quant Electron 55, 1088 (2023). https://doi.org/10.1007/s11082-023-05352-0

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