New Perspectives on Nonlinear Time-Fractional Wave Systems: Structural Analysis, Numerical Precision, and Fractional Wave Dynamics

This paper develops a Laplace–Residual Power Series Method (L-RPSM) for solving nonlinear time-fractional wave systems in the Caputo sense. The method combines the Laplace transform with a residual power series expansion to construct approximate analytical solutions, where the coefficients are determined recursively through residual conditions. The approach is applied to two benchmark models: the modified Boussinesq system and the approximate long-wave system. The influence of the fractional parameter is examined for different values of α, and convergence behavior is discussed to justify the validity of the method. Numerical results show that the proposed method produces accurate approximations with small absolute errors and stable convergence. Comparisons with established techniques such as ADM, VIM, OHAM, and NIM demonstrate the competitiveness and reliability of the proposed framework for nonlinear fractional wave equations.