Fuzzy M-fractional integrodifferential models: theoretical existence and uniqueness results, and approximate solutions utilizing the Hilbert reproducing kernel algorithm

Abstract

This article proposes a new approach to solving fuzzy M-fractional integrodifferential models under strongly generalized differentiability using an innovative formulation of the characterization principle. The study presents theoretical effects on the existence-uniqueness of fuzzy two M-solutions and, thus, showcases the solvability of the fuzzy Volterra models. Moreover, the study offers numerical solutions using the Hilbert reproducing kernel algorithm in a new fuzzy look, utilizing two fitting Hilbert spaces. The proposed models and algorithms are under scrutiny, with particular attention given to the analysis of the series solution, the assessment of convergence, and the evaluation of error. The debated Hilbert approach is shown to be effective in solving several fractional Volterra problems under uncertainty, and the numerical impacts manifest the accuracy and competence of the algorithm. Overall, our work contributes to the advancement of mathematical tools for solving complex fractional Volterra problems under uncertainty and shows potential to impact various fields of science and engineering, as depicted in the utilized figures, tables, and comparative analysis. The findings of the study are evaluated based on the analysis conducted, and a numerical algorithm is presented in the final section, along with several suggestions for future research directions.