A New Analytical Method for Solving the Fractional Swift-Hohenberg Equation with Atangana-Baleanu Derivative

This work introduces a hybrid analytical technique for solving the Swift–Hohenberg equation by integrating the newly formulated Yasser–Jassim integral transform with the variational iteration method. The proposed framework is designed to efficiently handle both the classical and fractional forms of the equation, providing fast-convergent and highly accurate approximate solutions. To capture memory effects and nonlocal features more realistically, the Atangana–Baleanu fractional derivative in the Caputo sense is incorporated into the model. A rigorous convergence analysis is conducted, establishing sufficient conditions to ensure the stability, reliability, and accuracy of the iterative solutions. The performance of the method is assessed through a series of numerical experiments supported by graphical illustrations and tables of absolute error values, which collectively confirm the method’s superior accuracy and rapid convergence when compared with standard analytical approaches. Additionally, a detailed stability study of the fractional solutions is carried out and clearly verified through analytical arguments and numerical simulations. The results demonstrate that the combined Yasser–Jassim transform and variational iteration method offer a versatile and powerful tool for solving fractional-order partial differential equations. Beyond the Swift–Hohenberg equation, the proposed approach can be extended to a wide range of mathematical models, including nonlinear ordinary differential equations and integro-differential systems. Overall, the findings highlight the potential of this hybrid scheme to advance analytical methodologies within fractional calculus and nonlinear dynamical systems.