This research introduced a new method, the Aboodh Tamimi Ansari transform method ( $(AT{)}^2$ method), for solving systems of linear and nonlinear fractional partial differential equations. The method combined the Aboodh transform method and the Tamimi Ansari method, allowing for the simultaneous solution of linear and nonlinear terms without restrictions. The Caputo sense was considered for fractional derivatives. The effectiveness of the proposed method was demonstrated through numerical solutions, graphical representations, and tabular data, showing strong agreement with exact solutions. The approach was deemed precise, easy to apply, and could be extended to address further challenges in fractional-order problems. Computational tasks were carried out using Mathematica 13.

In the field of system analysis, uncertainties pose significant challenges, demanding robust methodologies for decision-making. This study proposes a novel approach that integrates graph theory, fuzzy set theory, and linguistic data to effectively address uncertainties. Motivated by the imperative to enhance decision making authenticity amidst uncertainties, we introduce the hesitant fuzzy linguistic graph (HFLG) model. This model leverages graphical representation to present hesitant fuzzy linguistic values (HFLVs), providing decision-makers with an intuitive framework conducive to informed decision-making. Through a comprehensive examination of the HFLG model’s principles and operations, this study elucidates its utility in navigating uncertainties inherent in decision-making processes. Furthermore, a detailed case study illustrates the practical application of the HFLG model, highlighting its effectiveness across diverse real-world scenarios. The integration of graph theory, fuzzy set theory, and linguistic data in the HFLG model offers a valuable framework for addressing uncertainty, thereby advancing decision-making methodologies in complex environments.