Hybridized HAM and ADM Approaches for Efficient Solutions of Nonlinear Fractional Partial Differential Equation

This paper presents a novel numerical approach for addressing nonlinear dispersive partial differential equations in multi-dimensional spaces. The time-fractional dispersive partial differential equation is a crucial tool in tackling complex problems in ocean science and engineering. This study employs the Homotopy Analysis Method (HAM) and the Adomian Decomposition Method (ADM) to solve nonlinear fractional partial differential equations. A novel scheme is proposed, combining these methods, to derive approximate solutions for the K(2,2) equation with initial conditions. These conditions are modified by substituting certain integer-order time derivatives with fractional derivatives. The HAM and ADM, traditionally used for integer-order partial differential equations, are extended to fractional cases, yielding explicit and analytical solutions. The solutions are represented as convergent series with components that are straightforward to compute. The results demonstrate the high accuracy and efficiency of this approach when applied to the studied models.