Multistage Optimal Homotopy Asymptotic Method to Solve Fuzzy Fredholm Integral Equations of Second Kind

Various complex, unpredictable, and time-delayed physical events rely on fuzzy integral equations (FIEs), playing a crucial role in representing these applications. Despite their importance, current methods to solve FIEs face limitations, such as difficulty in handling nonlinear fuzzy functions, reliance on strict assumptions, and numerical instability, especially for large or ill-defined problems. This research presents new extensions of the multistage optimal homotopy asymptotic method (MOHAM) to address these challenges. Combining OHAM with fuzzy set theory, MOHAM provides open-form solutions to linear and nonlinear second-kind Fredholm fuzzy integral equations (FFIEs), offering high precision, accuracy, and computational efficiency. Comparisons show MOHAM outperforms other methods with improved convergence, reduced complexity, and greater adaptability to complex problems, setting a new benchmark for accuracy and efficiency. This work also fills gaps in the literature, paving the way for future research and practical applications in solving complex, ambiguous, and nonlinear problems.