Numerical approach in the Hilbert space to solve a fuzzy Atangana-Baleanu fractional hybrid system

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 W₂²[a,b]⊕W₂²[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.