This study is aimed to perform Lie symmetry analysis of the nonlinear fractional-order conduction-diffusion Buckmaster model (BM), which involves the Riemann-Liouville (R-L) derivative of fractional-order ‘β’. We are going through symmetry reduction to convert the fractional partial differential equation into a fractional ordinary differential equation. The fractional derivatives of the converted differential equations are evaluated with the help of Erdelyi-Kober (E-K) fractional operators. The power series solution and its convergence are analyzed with Implicit theorem. Conservation laws of the physical model are obtained for consistency of system by Noether’s theorem.

The main objective of this research article is to summarize the study of the application of Lie symmetry reduction to the fractional-order coupled nonlinear complex Hirota system of partial differential equations. By the efficient use of symmetries and explicit solutions, this system reducing to nonlinear fractional ordinary differential equations (FODEs) with the application of Erdyli-Kober (E-K) operators for fractional derivatives and integrals depending on real order. Investigating the convergent series solution along with adjoint system and providing the conservation laws by Noether’s theorem.

In this work, we investigated the invariance analysis of fractional-order Hirota-Satsoma coupled Korteveg-de-Vries (HSC-KdV) system of equations based on Riemann-Liouville (RL) derivatives. The Lie Symmetry analysis is considered to obtain infinitesimal generators; we reduced the system of coupled equations into nonlinear fractional ordinary differential equations (FODEs) with the help of Erdelyi’s-Kober (EK) fractional differential and integral operators. The reduced system of FODEs solved by means of the power series technique with its convergence. The conservation laws of the system constructed by Noether’s theorem.