Numerical approach in the Hilbert space to solve a fuzzy Atangana-Baleanu fractional hybrid system. (February 2021) - Hasan, Shatha; Al-Smadi, Mohammed; El-Ajou, Ahmad; Momani, Shaher; Hadid, Samir; Al-Zhour, Zeyad
Abstract: The pivotal aim of this paper is to investigate analytical and numerical solutions of fractional fuzzy hybrid system in Hilbert space. Such fuzzy systems are devoted to model control systems that are capable of controlling complex systems that have discrete events with continuous time dynamics. The fractional derivative is described in Atangana-Baleanu Caputo (ABC) sense, which is distinguished by its non-local and non-singular kernel. In this orientation, the main contribution of the current numerical investigation is to generalize the characterization theory of integer fuzzy IVP to the ABC-fractional derivative under a strongly generalized differentiability, and then apply the proposed method to deal with the fuzzy hybrid system numerically. This method optimized the approximate solutions based on orthogonalization Schmidt process on Sobolev spaces, which can be straightway employed in generating Fourier expansion within a sensible convergence rate. The reproducing kernel theory is employed to construct a series solution with parametric form for the considered model in the space of direct sum W 2 2 [ a, b ] ⊕ W 2 2 [ a, b ] . Some theorems related to convergence analysis and approximation error are also proved. Moreover, we obtain the exact solution for the fuzzy model by applying Laplace transform method. So, the results obtained using the proposed method are compared with those of exact solution. To show the effect of Atangana-Baleanu fractional operator, we compare the numerical solution of fractional fuzzy hybrid system with those of integer order. Two numerical examples are carried out to illustrate that such dynamical processes noticeably depend on time instant and time history, which can be efficiently modeled by employing the fractional calculus theory. Finally, the accuracy, efficiency, and simplicity of the proposed method are evident in both classical and fractional cases. Highlights: Based on Atangana-Baleanu fractional derivatives, an attractive numerical method, the reproducing kernel method, is considered for solving fractional fuzzy hybrid models. In this analysis, the fractional hybrid fuzzy system has been studied in Atangana-Baleanu gH-differentiability, which has nonsingular and nonlocal kernel. An algorithm based on the characterization theorem has been presented to give us a systematic manner to solve such fractional fuzzy system. The series form in term of their parametric form are presented in the space W 2 2 [ a, b ] ⊕ W 2 2 [ a, b ] . Two numerical simulations have been carried out to show the effectiveness and credibility of the proposed method. Numerical and graphical results are also provided and conferred quantitatively to clarify the required solutions. Anyhow, the results obtained are like those in previous studies that used Caputo type of fractional derivatives. So, since the ABC definition has a non-singular kernel, the use of ABC in DE modeling can be an appropriate substitute for Caputo fractional derivative and other fractional derivatives. We observed that the RKM method is very suitable, easy and effective to solve such a class of fuzzy fractional DEs and can be used to solve other types of differential equations.