Convergence Analysis and Analytical Approximation of Fractional Modified Camassa–Holm and Degasperis–Procesi Equations via a Hybrid Transform–Iterative Framework

This paper is devoted to the analytical investigation of fractional nonlinear dispersive equations arising in mathematical physics, namely the fractional modified Camassa–Holm (mCH) and modified Degasperis–Procesi (mDP) models. We develop a hybrid transform–iterative framework based on the Aboodh transform combined with a nonlinear recursive scheme to construct explicit series solutions without linearization. A rigorous convergence analysis is established in a Hilbert space setting, where sufficient conditions ensuring the convergence of the iterative sequence are derived using contraction-type arguments. Error estimates for truncated series solutions are also obtained, providing quantitative bounds on the approximation accuracy. The proposed method is applied to several representative problems with different initial profiles, including smooth and non-smooth data, highlighting its robustness in capturing the nonlocal and memory effects induced by fractional operators. The obtained solutions are shown to converge rapidly to the exact solutions, and numerical comparisons demonstrate improved accuracy relative to existing semi-analytical techniques such as the Adomian decomposition method and the homotopy perturbation method. These results indicate that the proposed framework offers an efficient and reliable analytical tool for studying nonlinear fractional evolution equations, with potential applications in the analysis of dispersive wave phenomena in mathematical physics.