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In this paper, the time-fractional coupled Burger equation under Riemann-Liouville derivative is systematically analyzed. Firstly, the Lie point symmetry is obtained by applying the Lie symmetry analysis; then by utilizing the above acquired Lie point symmetry, the similarity reductions are obtained. Through similarity reductions, the coupled time-fractional Boussinesq-Burgers equation is reduced to nonlinear fractional ordinary differential equations (FODEs), with Erdélyi-Kober fractional differential operator. Then the power series method and q-homotopy analysis method are used to obtain the approximate solutions of coupled time-fractional Boussinesq-Burgers equation in the sense of the Caputo fractional derivative. Finally, the conservation laws are derived by using Noether theorem.

On the time-fractional coupled Burger equation: Lie symmetry reductions, approximate solutions and conservation laws