Quasi-Projective Synchronization of Distributed-Order Recurrent Neural Networks

In this paper, the quasi-projective synchronization of distributed-order recurrent neural networks is investigated. Firstly, based on the definition of the distributed-order derivative and metric space theory, two distributed-order differential inequalities are obtained. Then, by employing the Lyapunov method, Laplace transform, Laplace final value theorem, and some inequality techniques, the quasi-projective synchronization sufficient conditions for distributed-order recurrent neural networks are established in cases of feedback control and hybrid control schemes, respectively. Finally, two numerical examples are given to verify the effectiveness of the theoretical results.

Existence, uniqueness, and approximate solutions for the general nonlinear distributed-order fractional differential equations in a Banach space

The purpose of this paper is to provide sufficient conditions for the local and global existence of solutions for the general nonlinear distributed-order fractional differential equations in the time domain. Also, we provide sufficient conditions for the uniqueness of the solutions. Furthermore, we use operational matrices for the fractional integral operator of the second kind Chebyshev wavelets and shifted fractional-order Jacobi polynomials via Gauss–Legendre quadrature formula and collocation methods to reduce the proposed equations into systems of nonlinear equations. Also, error bounds and convergence of the presented methods are investigated. In addition, the presented methods are implemented for two test problems and some famous distributed-order models, such as the model that describes the motion of the oscillator, the distributed-order fractional relaxation equation, and the Bagley–Torvik equation, to demonstrate the desired efficiency and accuracy of the proposed approaches. Comparisons between the methods proposed in this paper and the existing methods are given, which show that our numerical schemes exhibit better performances than the existing ones.