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.
Highlights
Fractional differential equations (FDEs) come from the generalization of classical differential equations of integer order
The fractional partial differential equations play an important role in describing physics, engineering and other scientific fields [1,2,3,4]
The famous Noether theorem [23] established a connection between Lie symmetries and conservation laws of differential equations
Summary
Fractional differential equations (FDEs) come from the generalization of classical differential equations of integer order. In 2009, Gazizov and Kasatkin [5] extended Lie symmetry approach to investigate several FDEs. Based on the symmetry, many useful properties of FDEs, such as symmetry generators, similarity transformation, explicit solutions and conservation laws which can be analyzed successively [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. In spite of the symmetry approach and conservation laws have made some progress in FDEs, the research for coupled time fractional FDEs are not very well explored. The main aim of this paper is to investigate the Lie point symmetry, similarity reduction and conservation laws under the definition of Riemann-Liouville fractional differential.
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