Semi-Analytic Scheme for Fractional Insight into Multi-Component KdV System Arises in Water Wave Mechanism

The present study aims to broaden the applications of a newly emerged Laplace-like transform known as the generalized bivariate integral transform. This transform is combined with the homotopy perturbation method to obtain approximate and analytic solutions for the fractional multi-component Korteweg-de Vries system. The time fractional derivative is considered in the conformable sense. The scheme involves the applications of generalized bivariate transform and its inversion to handle fractional operator. He's polynomials, associated with homotopy perturbation method, are employed to decompose the non-linearity of the proposed problem. Through the combination of these two powerful tools, an iterative expression is derived to compute the components of a rapidly convergent series solution. A rigorous analysis is conducted, and sufficient conditions for convergence are presented in detail. The proposed procedure effectively demonstrates distinct soliton mechanisms under various of initial conditions for proposed fractional system. The novelty of this study lies in the first-time application of the proposed scheme to a conformable fractional system. A key highlight of the study is the convergence behavior for non-integer order derivatives to study soliton behavior even in the absence of a known exact solution.