<p>This work thoroughly analyses the use of sophisticated mathematical approaches in the solution of fractional-order nonlinear dynamical systems. In particular, we use the Adomian decomposition method and q-homotopy analysis method in conjunction with Laplace transformation to solve the Rosenau–Hyman and Korteweg-de Vries nonlinear equations of fractional order. The order of the fractional derivatives falls inside the closed interval [0,1], and they are calculated in the Caputo sense. The three examples aim to assess the precision, effectiveness, and dependability of the suggested techniques. Furthermore, the results show a good convergence between the numerical and analytical solutions, indicating that the approaches utilised to understand the intricate dynamics of fractional nonlinear systems are effective. There is a strong correlation between the approximated and exact solutions. Moreover, the proposed methods are straightforward to implement and can be extended easily to approximate higher-order problems, opening avenues for further exploration.</p>