A New Approach for Approximate Solutions of Time–Fractional Coupled KdV–Type Equations Using the Constant Proportional Caputo–Fabrizio Operator

Some investigations in the fields such as fluid mechanics, control theory, quantum mechanics, ocean engineering, non-linear optics, biology, economics, plasma physics and electrodynamics highlights that the non-linear partial fractional differential equations (NPFDEs) are important for modeling and analysing the various real-world issues. Obtaining the solution to these equations will help us for the clear interpretation of the non-linear physical phenomena present in our environment. In this study, we have derived the approximate solutions of the non-linear time-fractional Hirota-Satsuma coupled KdV and coupled MKdV equations using the Iterative Laplace Transform Method (ILTM) in conjunction with the Constant Proportional Caputo-Fabrizio (CPCF) differential operator. The conditions for the solution’s uniqueness are established by the Banach contraction principle. Further, The Picard’s stability conditions are verified to guarantee the stability of obtained approximate solution by the proposed iterative apporoach derived from the fixed-point theorem. These approximate solutions are presented graphically and have compared numerically with the exact solutions. The method ILTM-CPCF serve as effective mathematical tool and seems computationally simple and user-friendly for the analysis of coupled non-linear fractional differential equations.