Mathematical investigation of the Kersten-Krasil’shchik KdV-mKdV system using fractional calculus: a study with q-HATM and VITM

This paper is devoted to explore an extensive comparative investigation between two semi-analytic techniques, the Mohand variational iteration technique (MVIM) and the q-Homotopy Mohand Transform Method (q-HMTM) to handle the coupled nonlinear fractional partial differential equations (NLPDEs). The study is carried out for two important fractional models: the Kersten-Krasil’shchik KdV-mKdV system and homogeneous coupled KdV system, the two systems are addressed for various quantities regarding the fractional order derivative that controls the memory and hereditary characteristics of the system. Both approaches lead to closed-form series solutions whose convergences and accuracies are presented in numerical tables with graphical comparisons in 2D and 3D. The results show that there is great agreement between q-HMTM and MVIM, and that q-HMTM has slightly more precision, especially for low fractional order. The results of the proposed methods were compared with Adomian decomposition method. Besides, the effect of the fractional order, the solution profiles is also investigated in detail. The results validate the fact that q-HMTM and MVIM are stable, efficient and accessible methods for studying nonlinear wave in the complex physical systems based on fractional order derivative. This study offers useful information for fractional parameters impact on the coupling of NLPDEs and acts as an addition to the literature on fractional order modeling in applied science.