Conservation laws and exact series solution of fractional‐order Hirota–Satsuma‐coupled Korteveg–de Vries system by symmetry analysis

Abstract

The main goal of the paper is to obtain invariance analysis of fractional-order Hirota-Satsuma-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-Kober (EK) fractional differential and integral operators. The reduced system of FODEs was solved by means of power series technique with its convergence. The conservation laws of the system were constructed by the Noether's theorem.

Invariant analysis, approximate solutions, and conservation laws for the time fractional higher order Boussinesq–Burgers system

“…By using the power series method [19,20,39,40], we intend to construct analytic approximate solutions of the time fractional higher order B-B system.…”

“…Gazizov et al [17] have adapted this method to solve fractional differential equations, where the considered derivative operator is in the sense of Riemann-Liouville and Caputo. This proper extension of the prolongation formulas has inspired and motivated many researchers to investigate some fractional differential equations [18][19][20][21][22][23][24][25]. Furthermore, one of the most interesting consequences of finding Lie symmetries is building conservation laws.…”

Application of Symmetry Analysis and Conservation Laws to Fractional-Order Nonlinear Conduction-Diffusion Model

“…The concept of nonlinear self-adjointness to time-fractional Kompaneets equation has been obtained by Gazizov et al [44]. In addition, recently, Gandhi et al [45] focused on invariant analysis, exact series solution, convergence of solution by Implicit theorem and conservation laws by Noether's theorem on fractional-order Hirota-Satsuma Coupled KdV system. The comparative study for solving Laplace fractional equation has been produced by Dubey et al [46].…”

Conservation Laws and Explicit Solution of system of Fractional-Order Coupled Nonlinear Hirota Equations by Lie Symmetry Analysis

“…Chauhan and Arora [29] has obtained the complete analysis of time fractional Kupershmidt equation. Recently, Gandhi et al [40,41,50] have applied symmetry reduction on multi-ordered time-fractional KdV equations and Hirota-Satsoma-coupled Korteveg-de-Vries equations to obtain the explicit solutions with convergence and conservation laws; he concluded that the fractional-order parameter  can control the output of solution of fractional mathematical models and in physical and mathematical aspects, the conservation laws play very crucial role to discuss the consistency of system. Zhang et al [45] promoted the 3 conservation laws of Fokkar-Plank equation with power diffusion.…”