Fuzzy Volterra Integral Equation Approximate Solution Via Optimal Homotopy Asymptotic Methods

Abstract

The field of fuzzy integral equations (FIEs) is significant for modeling complex, time-delayed, and uncertain physical phenomena. Nevertheless, the majority of current solutions for FIEs encounter considerable challenges, such as the inability to manage intricate fuzzy functions, stringent assumptions regarding the forms of fuzzy operations utilized, and numerical instability in extremely nonlinear issues. Moreover, the capability of traditional methods in producing precise or reliable outcomes for practical applications is limited, and if they can, will incur substantial computing expenses. These challenges underscore the demand for more effective and efficient methodologies. This study aims to address the demand by developing two approximate analytical techniques to solve the FIEs namely optimal homotopy asymptotic method (OHAM) and the multistage optimal homotopy asymptotic method (MOHAM). A novel iteration of fuzzy OHAM and MOHAM is introduced by integrating the fundamental concepts of these methodologies with fuzzy set theory and optimization techniques. Then, OHAM and MOHAM are further formulated to solve the second-kind linear Volterra fuzzy integral equations (VFIEs). These methods are named fuzzy Volterra optimal homotopy asymptotic method (FV-OHAM) and fuzzy Volterra multistage optimal homotopy asymptotic method (FV-MOHAM), respectively. From two linear examples, FV-MOHAM and FV-OHAM generated significantly more accurate results than other existing methods. A thorough assessment is performed to evaluate their effectiveness and practical use, potentially aiding in solving complex problems across several scientific and engineering fields.