Some Exact Solutions and Conservation Laws of the Coupled Time-Fractional Boussinesq-Burgers System

Abstract

In this paper, we investigate the invariant properties of the coupled time-fractional Boussinesq-Burgers system. The coupled time-fractional Boussinesq-Burgers system is established to study the fluid flow in the power system and describe the propagation of shallow water waves. Firstly, the Lie symmetry analysis method is used to consider the Lie point symmetry, similarity transformation. Using the obtained symmetries, then the coupled time-fractional Boussinesq-Burgers system is reduced to nonlinear fractional ordinary differential equations (FODEs), with 
 
 
 
 E
 r
 d
 
 e
 ´
 
 l
 y
 i
 
 
 
 -
 
 
 
 K
 o
 b
 e
 r
 
 
 
 fractional differential operator. Secondly, we solve the reduced system of FODEs by using a power series expansion method. Meanwhile, the convergence of the power series solution is analyzed. Thirdly, by using the new conservation theorem, the conservation laws of the coupled time-fractional Boussinesq-Burgers system is constructed. In particular, the presentation of the numerical simulations of q-homotopy analysis method of coupled time fractional Boussinesq-Burgers system is dedicated.

Application of the modified simple equation method for solving two nonlinear time-fractional long water wave equations

“…Investigating the solutions of the equation is very significant for civil and coastal engineers to enforce the nonlinear water wave equations to side designs and port construction [42,43]. The nonlinear coupled time-fractional Boussinesq-Burger equation is defined as in the following [33]:…”

“…The MSE method has been used to get exact solutions of various fractional partial differential equations such as the space-time fractional modified regularized long-wave equation, the space-time fractional modified Korteweg-de Vries equation, the space-time fractional coupled Burgers' equations [29], the nonlinear time-fractional Sharma-Tasso-Oliver equation [30], the fractional generalized reaction Duffing equation is and the fractional nonlinear Cahn-Allen equation [31]. We apply this ef-ficient method to the nonlinear conformable time-fractional approximate long water wave equation (ALW) [32] and the nonlinear coupled conformable time-fractional Boussinesq-Burger equation [33] to obtain analytical solutions of physical shallow water equations. There are many effective studies investigating the exact solutions of these equations using various methods.…”

Weak Local Residuals as Smoothness Indicators in Adaptive Mesh Methods for Shallow Water Flows

“…For example, Lee [1] used the forced KdV equation to model dynamics of trapped solitary waves. Shi et al [2] studied the coupled time-fractional Boussinesq-Burgers system and provided some exact solutions to the system. Caputo et al [3] reported some coupling conditions for water waves at forks.…”

Optimal System and Invariant Solutions of a New AKNS Equation with Time-Dependent Coefficients

“…As well as we know, Lie symmetry analysis is a powerful and prolific method for constructing exact solutions for NLPDEs with constant variable [18][19][20]. Recently, the Lie symmetry analysis is extended to find exact solutions of fractional and variable coefficient NLPDEs, such as Time-Fractional Boussinesq-Burgers [21], Gardner equations [22], coupled short pulse equation [23] and so on [24][25][26].…”