This paper investigated the Schrödinger and Korteweg-de Vries equations within the framework of fractional-order differential equations, utilizing the variational iteration transform method and the q-homotopy analysis transform method. These equations, crucial for modeling wave propagation and nonlinear dispersive systems, were analyzed using the Caputo fractional derivative to explore the influence of non-integer orders on their dynamics. The findings contributed to a deeper understanding of how fractional-order parameters affected the behavior of nonlinear wave models and oscillations, underscoring the growing importance of fractional calculus in mathematical physics and engineering. Both methods presented in this study effectively converted fractional-order problems into iterative schemes that were straightforward to solve, leading to quicker convergence of the analytical solutions. A comparative analysis evaluated the accuracy, computational efficiency, and convergence properties of the variational iteration transform method (VITM) and the q-homotopy analysis transform method q-HATM. The results, supported by numerical simulations and various graphical representations, validated the practicality and effectiveness of these methods for solving complex fractional differential equations. This study not only enhanced our comprehension of fractional wave dynamics but also strengthened the body of knowledge in both analytical and computational methods in mathematical physics and engineering.

In this paper, we explored new families of solitary wave solutions of stochastic Chaffee-Infante equation (SCIE) with Wiener process in Itô sense. SCIE is one of the most important models in mathematical physics used to describe the processes of diffusion and wave propagation. Efficient $(G^{\prime }/G)$-expansion and its version generalized $(r+G^{\prime }/G)$-expansion are applied to obtain explicit solitary wave solutions to the targeted SCIE. The strategic $(G^{\prime }/G)$-expansion method originally converts SCIE to nonlinear ordinary differential equation (NODE) by wave transformation and then converts it to a set of nonlinear algebraic equations on the assumption of finite series-form solutions. Under the analysis of the solutions of the resulting system with Maple, a number of solitary wave solutions in the form of trigonometric, hyperbolic, and rational functions were found. To verify the presence of solitary wave solutions, such as soliton, dark, bright, kink, and anti-kink solitary wave solutions in SCIEs, several solitary wave solutions were evaluated using illustrated 3D visualizations for given parameter values under zero and nonzero noise effects. The implication of our results extends widely across various fields both in stochastic phenomena and nonlinear dynamics and has a contributions to physics and nonlinear science.