Predictor–corrector approach for the numerical solution of fuzzy fractional differential equations and linear multiterm fuzzy fractional equations

Journal article

Recently, the modeling of fuzzy fractional differential equations (FFDEs) has been a very significant issue in many new applications in applied sciences and engineering, while a natural tool for modeling such dynamical systems is to use fuzzy fractional differential equations. We establish the existence and uniqueness of solutions for fuzzy fractional differential equations under sufficient assumptions and contraction principles and study numerical solutions of FFDEs. Our study is based on Caputo’s generalized Hukuhara differentiability. By applying Schauder’s fixed point theorem and a hypothetical condition, we explore the existence of the solutions. In addition, we show the uniqueness of the system's solution by using the contraction mapping theorem. We analyze the predictor–corrector approach (PCA) for FFDEs and multiterm FFDEs. We utilize the PCA to find the approximate solutions to linear multiterm FFDEs under the Caputo fuzzy derivative. After that, we present numerical solutions to initial value problems for solving two families of fuzzy fractional problems: fuzzy fractional differential equations (FFDEs) and multiterm fuzzy fractional differential equations (MFFDEs) utilizing the PCA. The method used in this paper has several advantages; first, it is significant and yields stable results without diverging as well as its ability to solve other mathematical, physical, and engineering problems; second, it is higher accuracy, needs less effort to achieve the results and works to reduces the error between exact and approximate solutions, as depicted in the utilized figures and tables. Finally, the accuracy of our suggested approach is demonstrated by solving some specific examples and analyzing the figures and tables, along with several suggestions for future research directions.

Authors: Al-Sadi, W; Wei, Z; Moroz, I; Abu Arqub, O; Abdullah, TQS

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer Nature
• Journal: Soft Computing
• Volume: 29
• Issue: 3
• Article number: 1347-1368
• Publication date: 2025-02-13
• Acceptance date: 2024-11-08
• DOI: 10.1007/s00500-025-10401-9
• EISSN: 1433-7479
• ISSN: 1432-7643
• Language: English
• Keywords: fuzzy fractional differential equations(FFDEs); predictor-corrector method; Volterra integral equation (VIE); Caputo Hukuhara differentiability; fuzzy fractional multi-term methods