Analytical solutions of some time-fractional versions of the coupled KdV system of equations via two hybrid transform techniques

In this work, two efficient hybrid transform methods known as the natural homotopy perturbation method (NHPM) and the q-homotopy analysis transform method (q-HATM) are utilised to investigate analytical solutions of the time-fractional coupled Korteweg-de Vries equations and time-fractional Kersten-Krasil’shchik coupled KdV-mKdV equations. These models are considered under the framework of the time-fractional Atangana-Baleanu derivative. The NHPM combines the natural transform method and the homotopy perturbation method, while the q-HATM combines the q-homotopy analysis method and the Laplace transform method. These methods employ an iterative approach to generate rapidly convergent series solutions for the considered problems. We demonstrate the efficiency of both solution methodologies for a variety of test problems. Numerical results are obtained for the considered test problems when the fractional order is equal to 1. These results are then compared with the exact solutions. Error estimates are also obtained for each test problem with respect to the considered methods. The obtained numerical results are close approximations of the exact solutions, thus demonstrating their accuracy and validity. Furthermore, graphical representations in both 2D and 3D are presented for the considered problems to depict the dynamic behaviours of system wave profiles and surface plots for distinct values of the fractional parameter. These results further emphasise the simplicity and straightforwardness of the methods, which make them applicable to complex nonlinear time-fractional systems that model diverse real-life processes.