Invariant Analysis for Space–Time Fractional Three-Field Kaup–Boussinesq Equations

Kaur, Jaskiran; Kumar, Gupta Rajesh; Kumar, Sachin

doi:10.1007/978-981-15-5414-8_28

Cited by 1 publication

(1 citation statement)

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“…By means of the developed frameworks of Lie group, a number of fractional PDEs were investigated such as the fractional dispersive equation 23 , the fractional KdV-type equations 24,25 , the fractional biological population model 26 , the fractional diffusion-type equations 27,28 , and space-time-fractional Kaup-Boussinesq equations. 29 Up to now, the symmetry results of fractional PDEs mainly concentrate on local symmetry, but very little is known about nonlocal symmetry of fractional PDEs. Potential symmetry, the celebrated nonlocal symmetry introduced by Bluman et al 30 , originates from Lie symmetries of the potential system obtained by introducing potential variable to the PDEs having a conserved form, where the compatibility condition of the potential system generates the original PDEs.…”

Section: Introductionmentioning

confidence: 99%

Local symmetry structure and potential symmetries of time‐fractional partial differential equations

Zhang

Lin

2021

Stud Appl Math

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First, we show that the system consisting of integerorder partial differential equations (PDEs) and timefractional PDEs with the Riemann-Liouville fractional derivative has an elegant local symmetry structure. Then with the symmetry structure we consider two particular cases where one is the pure time-fractional PDEs whose symmetry invariant condition is divided into two parts of integer-order and time-fractional, the other is the linear system of time-fractional PDEs, which always admits an infinite-dimensional infinitesimal generator. Second, by considering the composition rules of fractional derivatives we establish a theoretical framework of potential symmetry and construct three potential systems to study potential symmetries of the time-fractional PDEs possessing a divergence form. In particular for a single time-fractional PDE the existence condition of potential symmetries via one typical potential system is presented by means of the local symmetry structure. Finally, local symmetry structure and potential symmetries of a class of time-fractional diffusion equations are studied in detail. Several explicit solutions are constructed by means of the potential symmetries.

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