Inducing variables are the parameters or conditions that influence the membership value of an element in a fuzzy set. These variables are often linguistic in nature and represent qualitative aspects of the problem. Thus, the objective of this paper is introduce some aggregation operators based on inducing variable, such as induced complex Polytopic fuzzy ordered weighted averaging aggregation operator (I-CPoFOWAAO) and induced complex Polytopic fuzzy hybrid averaging aggregation operator (I-CPoFHAAO). Induced aggregation operators in decision-making process are indispensable tools for managing uncertainty, integrating multiple criteria, facilitating consensus, and providing a formal and flexible framework for modeling and solving complex decision problems. At the end of the paper, we make an illustrative example to prove the ability and efficiency of the novel proposed aggregation operators.

In this work, we study the solitary wave profiles of the fractional-Sharma–Tasso–Olver equation, which is applicable to particle fission and fusion mechanisms in nuclear physics. In numerical and analytical theories, exact solitary wave solutions are of the uttermost importance for such equations. Improved analytical methods are essential for a deeper understanding of dynamics, despite their widespread implementation. In this study, we use the advanced analytical techniques known as generalized Arnous method, modified generalized Riccati equation mapping technique, and Riccati extended simple equation approach for securing a variety of solutions. This study marks a significant milestone by applying the prescribed techniques to the proposed equation using truncated M-fractional derivatives and providing a significant contribution to the existing literature. This equation is widely regarded as a model that illustrates the propagation of nonlinear dispersive waves in inhomogeneous media. Using the suitable wave transformation with the fractional-derivative, the governing equation is converted into an ordinary differential equation to get the required solutions. Various types of solutions, such as mixed, dark, singular, bright–dark, bright, complex, and combined solitons, are extracted. Moreover, another important aspect of this study is to discuss the multistability and sensitivity analysis of the studied model by the assistance of the Galilean transformation and perturbation term. The utilized methods have strong computing capacity, which helps them effectively handle the exact solutions with high accuracy in these systems. In addition, we depict 3D and 2D phase portrait graphs with appropriate parameters to illustrate the solution’s behavior.

The beta fractional form of the Estevez-Mansfield-Clarkson equation is under consideration and this study is done with the assistance of methods such as modified F-expansion method and the logarithmic transformation. A variety of analytical solutions like bright, dark, mixed, singular, bright-dark, and combined solitons are extracted. Moreover, multi waves structures, interaction with double exponential form, breather waves, mixed type solutions as well as periodic cross kink solutions have been analyzed. The governing equation is converted into an ordinary differential equation by employing an appropriate wave transformation with the β-derivative in order to achieve the desired solutions. The applied approaches have substantial computational capability, enabling them to efficiently address exact solutions with high accuracy in these systems. The results indicate that the equation under investigation theoretically contains a substantial number of soliton solution structures. Additionally, in order to examine the behaviors of the solutions at various parameter values, we plot a variety of graphs that incorporate pertinent parameters. The results of this study have the potential to improve understanding of the nonlinear dynamic characteristics displayed by the specified system and to confirm the effectiveness of the techniques that have been implemented.

This study introduces novel aggregation operators aimed at enhancing data analysis and decision-making processes through the induction of confidence levels into complex polytopic fuzzy systems. Specifically, the induced confidence complex polytopic fuzzy ordered weighted averaging aggregation (ICCPoFOWAA) operator and the induced confidence complex polytopic fuzzy hybrid averaging aggregation (ICCPoFHAA) operator are proposed. By integrating confidence levels into the aggregation process, these operators facilitate a more nuanced interpretation of fuzzy data, allowing for the incorporation of expert judgment and uncertainty in decision-making frameworks. A practical demonstration is provided to validate the efficacy and proficiency of these innovative techniques. Through a comprehensive example, the ability of the ICCPoFOWAA and ICCPoFHAA operators to enhance decision-making accuracy and reliability is substantiated, showcasing their potential as powerful tools in the realms of data analysis and complex decision-making scenarios. The incorporation of confidence levels into fuzzy aggregation processes represents a significant advancement in the field, offering a sophisticated approach to handling uncertainty and expert opinions in multi-criteria decision-making problems. This work not only introduces groundbreaking aggregation operators but also sets a new standard for research in fuzzy decision-making, underscoring the importance of confidence levels in the analytical process.