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

Muath Talal Mahmoud  AlZubi; Ali Fareed  Jameel; Farah Aini  Abdullah; Adila Aida  Azahar

doi:10.6180/jase.202605_29(5).0008

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

Muath Talal Mahmoud AlZubi¹, Ali Fareed Jameel²This email address is being protected from spambots. You need JavaScript enabled to view it., Farah Aini Abdullah³, and Adila Aida Azahar³

¹School of Mathematical Sciences, Universiti Sains Malaysia (USM), Malaysia

²Faculty of Education and Arts, Sohar University, Sultanate of Oman

³Department of Mathematics, Universiti Sains Malaysia (USM), Malaysia

Received: May 16, 2025
Accepted: August 15, 2025
Publication Date: September 6, 2025

Copyright The Author(s). This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.

Download Citation: ||https://doi.org/10.6180/jase.202605_29(5).0008

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.

Keywords: Fuzzy Integral Equations F-IEs, Fredholm Fuzzy Integral Equations FF-IEs, Multistage Optimal Homotopy Asymptotic Methods MOHAM

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