Convergence Analysis and Numerical Approximation of the Fractional Fornberg–Whitham Equation via the Yasser–Jassim Transform

Abstract

This study introduces an innovative framework for addressing the fractional Fornberg–Whitham equation by melding the Yasser–Jassim integral transform with the Variational Iteration Method, all formulated under the Atangana–Baleanu fractional derivative in the Caputo interpretation. We first derive an explicit series representation of the solution and then rigorously prove that the iterative procedure converges, identifying conditions that guarantee both existence and uniqueness. In addition, we derive a bound on the truncation error to quantify the approximation’s accuracy. To validate the theoretical developments, a detailed computational example is provided, demonstrating rapid convergence and close agreement with the exact solution. The findings highlight the method’s robustness and suggest its broad applicability as an analytical tool for a wide range of nonlinear fractional partial differential equations.