Effective Analytical-Approximate Technique for Caputo Nonlinear Time-Fractional Systems Emerging in Shallow Water Waves and Hydrodynamic Turbulence

Abstract

 This work aims to study and analyze non-linear time-fractional systems that \textcolor{red}{describe} non-linear surface waves propagating and evolution equations utilizing a novel analytic-approximate technique based upon integrating two schemes with adequacy, high accuracy, straightforward of implementation, computations, and elasticity in handling with more sophisticated differential equations, which is named the Laplace transform fractional residual power series method within the Caputo-fractional derivative framework. The proposed technique had been implemented on Drinfeld-Sokolov-Wilson equation (DS-WE) and coupled viscous Burgers’ equation (CVBE). The approximation solutions obtained by the LT-RFPS technique are expressed in an infinite convergent fractional series form toward the exact solution for the integer fractional order. To show the accuracy and efficiency of the proposed method, tabular simulations of the produced approximations and their absolute errors are performed, along with $2D$- and $3D$-representative graphs. The physical interpretation of solution behaviors is also discussed for various $\rho$ values over an adequate duration. Additionally, a numerical comparison is performed with other existing techniques to show the superiority of the LT-RFPS technique. Consequently, the findings of the present work emphasize that the integration between LT and RFPS schemes has led us to a straightforward, effective, and accurate iterative analytical technique for investigating a wide variety of non-linear mathematical fractional models.