Hilbert solution of fuzzy fractional boundary value problems

In this paper, we study two-point boundary value problems which play a vital role in constituting mathematical models for solving real-world problems in which uncertainty pervades. Based on Caputo–Fabrizio approach, we adopt non-singular kernel derivative of order β∈(1,2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \beta \in (1,2]$$\end{document}. To solve fuzzy fractional boundary value problem, we convert it into four equivalent crisp systems in light of the generalized differentiability sense. Then, we solve each of the obtained systems using reproducing kernel functions to build orthonormal set of functions and then to obtain analytical and numerical solutions. A discussion about the accepted type of solution among the four systems is presented in summarized algorithm and carried out in numerical examples. This work proves the simplicity and efficiency of the proposed method, especially when adopting Caputo–Fabrizio derivative. Moreover, the applicability of the reproducing kernel method for solving different types of problems with distinct fractional operators is evident from the appearance of very small error of approximation.