Numerical Simulation of the Time-Fractional Caudrey–Dodd–Gibbon Equation in Finding the Exact Traveling Wave Solution of a Nonlinear Evolution Equation

In this study, the Caudrey–Dodd–Gibbon (CDG) equation, a special class of fifth-order nonlinear equation, including nonhomogeneous constraint conditions, is investigated under the reproducing kernel Hilbert space method (RKHSM). The CDG equation holds significance in modeling complex wave phenomena in various physical systems. The key advantages of RKHSM lie in its ability to construct solutions in the form of rapidly convergent series, avoiding the need for the discretization of the domain. The RKHSM is an inherently smooth approach that ensures the preservation of the problem’s structural properties while maintaining numerical stability. Furthermore, the method facilitates an efficient solution strategy with minimal computational overhead, making it suitable for high-dimensional problems and a wide class of fractional differential equations. The efficiency of the proposed method is validated through a numerical experiment. The obtained results are presented in the form of detailed tables and graphical illustrations, which clearly demonstrate the method’s robustness, accuracy, and consistency. Additionally, the convergence behavior of the series solution is rigorously analyzed, thereby reinforcing the theoretical foundation and reliability of the proposed approach.