On the solution of intuitionistic fuzzy nonlinear Fredholm integral equation using direct computational method

• Published in: Journal of big data Vol. 12; no. 1; pp. 132 - 23
• Main Authors: Khan, Zain, Abdullah, Saleem, Rahimzai, Ariana Abdul, Khan, Saifullah
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2025; Springer Nature B.V; SpringerOpen
• Subjects: Communications Engineering; Computational Science and Engineering; Computer Science; Data Mining and Knowledge Discovery; Database Management; Decomposition; Direct computational method; Exact solutions; Fredholm equations; Fuzzy sets; Information Storage and Retrieval; Integral equations; Intuitionistic fuzzy set; Mathematical analysis; Mathematical Applications in Computer Science; Networks; Nonlinear Fredholm integral equation; Parametric form of an intuitionistic fuzzy number; Uncertainty; Intuitionistic fuzzy set; Nonlinear Fredholm integral equation; Parametric form of an intuitionistic fuzzy number; Direct computational method
• ISSN: 2196-1115; 2196-1115
• DOI: 10.1186/s40537-025-01168-9

Fuzzy integral equations play an important role in addressing uncertain mathematical problems. There are various techniques present in the literature to solve fuzzy linear integral equations. Different methodologies provide numerical solutions for fuzzy nonlinear integral equations. However, there are few recognized methods for finding an exact solution. The fuzzy set has limitations because it lacks a non-membership degree for investigating uncertainty. To address this limitation, we use an intuitionistic fuzzy set that considers both membership and non-membership degrees together. Using the parametric forms of an intuitionistic fuzzy number, the nonlinear Fredholm integral equation is decomposed into a set of four equations. This set of four equations is then named the intuitionistic fuzzy nonlinear Fredholm integral equation. For an exact solution to the intuitionistic fuzzy nonlinear Fredholm integral equation, we use the Direct Computational Method. We solve two different examples in detail to demonstrate the reliability, effectiveness, and applicability of the proposed methodology. Graphs made using MATLAB represent visual judgments on how uncertainty impacts solutions. The results obtained for both examples are carefully examined and discussed in detail. The proposed method is compared to different decomposition and deep learning methods to ensure its accuracy. It is concluded that the proposed method is valid and reliable to get an exact solution for an intuitionistic fuzzy nonlinear Fredholm integral equation.