Novel Approximations to the Third- and Fifth-Order Fractional KdV-Type Equations and Modeling Nonlinear Structures in Plasmas and Fluids

This investigation examines fractional higher-order evolution and fundamental wave equations, namely the third- and fifth-order fractional Korteweg-de Vries (KdV)-type equations, which regulate diverse nonlinear physical processes, especially those occurring in fluids and plasmas. For this purpose, the Aboodh/Laplace residual power series method (ARPSM) and the Aboodh/Laplace transform iterative method (ATIM) are carried out to derive high-accurate approximations. These methods are applied in conjunction with the Caputo operator, which effectively handles the fractional derivatives. The results illustrate the efficacy of both ARPSM and ATIM in analyzing third- and fifth-order time fractional KdV-type equations, providing valuable insights and potential applications in fractional calculus and its applications to complicated physical and engineering issues. The derived approximations are investigated graphically and numerically to understand the effect of the fractional parameter on the properties of the nonlinear phenomena characterized by this family. Furthermore, the precision and efficacy of the suggested techniques are verified by comparing the derived approximations to the exact solutions for the integer-order cases. The findings of this investigation have the potential to benefit a wide range of researchers who are interested in optical physics, fluid physics, and plasma physics. They can be employed to analyze and comprehend the results of their laboratory and space observations.