Galoitica: Journal of Mathematical Structures and Applications

Hypergraphs extend classical graphs by allowing hyperedges to connect arbitrary nonempty subsets of vertices, thereby capturing higher-order, group-level interactions. Superhypergraphs further broaden this setting by iterating the powerset construction, which yields layered supervertices and supports multi-level relational structure. An interval-valued bipolar fuzzy graph assigns positive and negative membership intervals to vertices and edges while satisfying bipolar consistency constraints. In this paper, we extend interval-valued bipolar fuzzy graphs to the settings of hypergraphs and superhypergraphs.

This study focuses on the development of efficient numerical algorithms for solving nonlinear partial differential equations (PDEs). The research integrates theoretical analysis and practical numerical experiments to address the challenges posed by nonlinear PDEs, which often lack closed-form solutions and exhibit sensitivity to initial and boundary conditions. Benchmark models such as Burgers’ Equation, the Korteweg–de Vries (KdV) Equation, and the Navier–Stokes Equations are highlighted due to their significance in physical and engineering applications. Traditional numerical methods—Finite Difference Method (FDM), Finite Element Method (FEM), and Finite Volume Method (FVM)—are reviewed with respect to accuracy, stability, and computational efficiency. Numerical stability concepts, including Von Neumann analysis and the CFL condition, are discussed alongside sources of error and strategies for error reduction. New algorithms were proposed by enhancing traditional schemes, incorporating adaptive mesh refinement, and integrating stability techniques. Numerical experiments on benchmark problems demonstrated improved accuracy, enhanced stability in handling nonlinear terms, and acceptable computational efficiency. The findings emphasize the importance of selecting suitable numerical methods, conducting stability analysis, and applying adaptive techniques. The study recommends higher-order schemes, conservative formulations for fluid dynamics, and double precision when necessary, ensuring reliable and reproducible computational results.

Concepts such as the Fuzzy Set, Neutrosophic Set, and Soft Set are known for handling uncertainty. As extensions of Fuzzy Sets, Neutrosophic Sets, and Soft Sets, concepts such as Bipolar Fuzzy Sets, Bipolar Neutrosophic Sets, and Bipolar Soft Sets have been introduced. In this paper, we further extend these notions and explore Hyperpolar Fuzzy Sets, Hyperpolar Neutrosophic Sets, and Hyperpolar Soft Sets. These structures integrate multi-perspective or multi-agent evaluations into a unified framework by leveraging higher-dimensional mappings and hypercubic representations. This work lays a theoretical foundation for advanced uncertainty modeling in complex, multi-source environments.

Numerous mathematical frameworks have been developed to handle uncertainty, including Fuzzy Sets,1 Intuitionistic Fuzzy Sets,2 Hyperfuzzy Sets,3 Picture Fuzzy Sets,4 Hesitant Fuzzy Sets,5, 6 Neutrosophic Sets,7 Plithogenic Sets,8 and Soft Sets,9 and research in this area continues to evolve rapidly. Rough set theory provides a foundational method for approximating subsets using lower and upper bounds based on equivalence relations, offering an effective approach to modeling uncertainty in classification and data analysis.10, 11 Building upon these foundations, extended models such as HyperRough Sets and SuperHyperRough Sets have been proposed.12 In this paper, we present novel definitions that further generalize Graded Rough Sets and Linguistic Rough Sets—specifically, the Graded HyperRough Set and the Linguistic HyperRough Set. These new frameworks are expected to contribute to the advancement of research in fields such as decision-making, language theory, and artificial intelligence.