A comparative study on the analytic solutions of fractional coupled sine–Gordon equations by using two reliable methods

• New modified homotopy analysis method has been proposed.
• The proposed novel method is well suited for present fractional S–G equation.
• It is obvious to see that the MHAM is more accurate, easy and efficient than HPTM.
• Fractional coupled S–G equations have been first time solved by MHAM and HPTM.

In this paper modified homotopy analysis method (MHAM) and homotopy perturbation transform method (HPTM) have been implemented for solving time fractional coupled sine–Gordon equations. We consider fractional coupled sine–Gordon equations which models one-dimensional nonlinear wave processes in two-component media. The results obtained by modified homotopy analysis method (MHAM) and homotopy perturbation transform method (HPTM) are then compared with the modified decomposition method (MDM). By using an initial value system, the numerical solutions of coupled sine–Gordon equations have been represented graphically. Here we obtain the solution of fractional coupled sine–Gordon (S–G) equations, which is obtained by replacing the time derivatives with a fractional derivatives of order α∈(1,2]α∈(1,2] and β∈(1,2]β∈(1,2]. The fractional derivatives here are described in Caputo sense.