Unlocking <i>q</i> -fractional dynamics: breakthrough solutions via homotopy perturbation and new iterative methods

Purpose This study pioneers the application of the homotopy perturbation method (HPM) and the new iterative method (NIM) to unravel solutions for complex q-fractional differential equations (q-FDEs), bridging gaps in quantum mechanics, engineering, and financial modeling. By rigorously testing these methods on pivotal examples, including q-diffusion and nonlinear q-fractional equations, we demonstrate their unparalleled accuracy and computational efficiency. Graphical analyses and convergence proofs validate the methods’ robustness, revealing their capacity to model intricate systems with precision. Comparative studies against existing techniques highlight significant advancements, offering a transformative toolkit for researchers tackling fractional-order phenomena. This work not only resolves longstanding computational challenges but also charts new frontiers in the analysis of q-fractional dynamics, promising broad implications for science and technology. Design/methodology/approach This study pioneers the application of the homotopy perturbation method (HPM) and the new iterative method (NIM) to unravel solutions for complex q-fractional differential equations (q-FDEs), bridging gaps in quantum mechanics, engineering, and financial modeling. By rigorously testing these methods on pivotal examples, including q-diffusion and nonlinear q-fractional equations, we demonstrate their unparalleled accuracy and computational efficiency. Graphical analyses and convergence proofs validate the methods’ robustness, revealing their capacity to model intricate systems with precision. Comparative studies against existing techniques highlight significant advancements, offering a transformative toolkit for researchers tackling fractional-order phenomena. This work not only resolves longstanding computational challenges but also charts new frontiers in the analysis of q-fractional dynamics, promising broad implications for science and technology. Findings This study pioneers the application of the homotopy perturbation method (HPM) and the new iterative method (NIM) to unravel solutions for complex q-fractional differential equations (q-FDEs), bridging gaps in quantum mechanics, engineering, and financial modeling. By rigorously testing these methods on pivotal examples, including q-diffusion and nonlinear q-fractional equations, we demonstrate their unparalleled accuracy and computational efficiency. Graphical analyses and convergence proofs validate the methods’ robustness, revealing their capacity to model intricate systems with precision. Comparative studies against existing techniques highlight significant advancements, offering a transformative toolkit for researchers tackling fractional-order phenomena. This work not only resolves longstanding computational challenges but also charts new frontiers in the analysis of q-fractional dynamics, promising broad implications for science and technology. Originality/value This study pioneers the application of the homotopy perturbation method (HPM) and the new iterative method (NIM) to unravel solutions for complex q-fractional differential equations (q-FDEs), bridging gaps in quantum mechanics, engineering, and financial modeling. By rigorously testing these methods on pivotal examples, including q-diffusion and nonlinear q-fractional equations, we demonstrate their unparalleled accuracy and computational efficiency. Graphical analyses and convergence proofs validate the methods’ robustness, revealing their capacity to model intricate systems with precision. Comparative studies against existing techniques highlight significant advancements, offering a transformative toolkit for researchers tackling fractional-order phenomena. This work not only resolves longstanding computational challenges but also charts new frontiers in the analysis of q-fractional dynamics, promising broad implications for science and technology.