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Article Type

Research Article

Abstract

This paper proposes a unified computational framework for solving linear systems under trapezoidal Neutrosophic uncertainty. The system is formulated as A( I )x = b( I ), where both the coefficient matrix and the right-hand side vector incorporate an indeterminacy parameter I, expressed as A( I ) = A0 + IA1 and b( I ) = b0 + Ib1. A decomposition strategy is developed to separate the model into deterministic and indeterminacy components, yielding a solution of the form x( I ) = x0 + Ix1. The deterministic component is obtained via classical linear algebra techniques, while the indeterminacy contribution is derived through a coupled correction system. To preserve uncertainty structure, trapezoidal neutrosophic arithmetic is employed alongside a score-based pivoting mechanism within Gaussian elimination. Furthermore, a stability analysis is conducted to evaluate the sensitivity of the solution with respect to variations in I. It is shown that the solution deviation exhibits linear bounded growth, confirming robustness under indeterminacy. Numerical experiments validate theoretical results and demonstrate the effectiveness of the proposed approach in handling uncertain and indeterminate linear systems. The proposed framework provides a reliable computational tool for modeling and solving real-world problems involving incomplete, inconsistent, and indeterminate data.

Keywords

Neutrosophic linear systems, Trapezoidal numbers, Gaussian elimination, Indeterminacy, Stability analysis, Uncertainty modeling

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