The aim of this research is to demonstrate a novel scheme for approximating the Riemann-Liouville fractional integral operator.This would be achieved by first establishing a fractional-order version of the 2-point Tra...The aim of this research is to demonstrate a novel scheme for approximating the Riemann-Liouville fractional integral operator.This would be achieved by first establishing a fractional-order version of the 2-point Trapezoidal rule and then by proposing another fractional-order version of the(n+1)-composite Trapezoidal rule.In particular,the so-called divided-difference formula is typically employed to derive the 2-point Trapezoidal rule,which has accordingly been used to derive a more accurate fractional-order formula called the(n+1)-composite Trapezoidal rule.Additionally,in order to increase the accuracy of the proposed approximations by reducing the true errors,we incorporate the so-called Romberg integration,which is an extrapolation formula of the Trapezoidal rule for integration,into our proposed approaches.Several numerical examples are provided and compared with a modern definition of the Riemann-Liouville fractional integral operator to illustrate the efficacy of our scheme.展开更多
This paper proposes fractional-order systems for Hopfield Neural Network(HNN).The so-called Predictor Corrector Adams Bashforth Moulton Method(PCABMM)has been implemented for solving such systems.Graphical comparisons...This paper proposes fractional-order systems for Hopfield Neural Network(HNN).The so-called Predictor Corrector Adams Bashforth Moulton Method(PCABMM)has been implemented for solving such systems.Graphical comparisons between the PCABMM and the Runge-Kutla Method(RKM)solutions for the classical HNN reveal that the proposed technique is one of the powerful tools for handling these systems.To determine all Lyapunov exponents for them,the Benettin-Wolf algorithm has been involved in the PCABMM.leased on such algorithm,the Lyapunov exponents as a function of a given parameter and as another function of the fractional-order have been described,the intermittent chaos for these systems has been explored.A new result related to the Mittag-Leffler stability of some nonlinear Fractional-order Hopfield Neural Network(FoHNN)systems has been shown.Besides,the description and the dynamic analysis of those phenomena have been discussed and verified theoretically and numerically via illustrating the phase portraits and the Lyapunov exponents'diagrams.展开更多
文摘The aim of this research is to demonstrate a novel scheme for approximating the Riemann-Liouville fractional integral operator.This would be achieved by first establishing a fractional-order version of the 2-point Trapezoidal rule and then by proposing another fractional-order version of the(n+1)-composite Trapezoidal rule.In particular,the so-called divided-difference formula is typically employed to derive the 2-point Trapezoidal rule,which has accordingly been used to derive a more accurate fractional-order formula called the(n+1)-composite Trapezoidal rule.Additionally,in order to increase the accuracy of the proposed approximations by reducing the true errors,we incorporate the so-called Romberg integration,which is an extrapolation formula of the Trapezoidal rule for integration,into our proposed approaches.Several numerical examples are provided and compared with a modern definition of the Riemann-Liouville fractional integral operator to illustrate the efficacy of our scheme.
基金supporting this work by the University Ajman Grant:2Q20-COVID-19-08.
文摘This paper proposes fractional-order systems for Hopfield Neural Network(HNN).The so-called Predictor Corrector Adams Bashforth Moulton Method(PCABMM)has been implemented for solving such systems.Graphical comparisons between the PCABMM and the Runge-Kutla Method(RKM)solutions for the classical HNN reveal that the proposed technique is one of the powerful tools for handling these systems.To determine all Lyapunov exponents for them,the Benettin-Wolf algorithm has been involved in the PCABMM.leased on such algorithm,the Lyapunov exponents as a function of a given parameter and as another function of the fractional-order have been described,the intermittent chaos for these systems has been explored.A new result related to the Mittag-Leffler stability of some nonlinear Fractional-order Hopfield Neural Network(FoHNN)systems has been shown.Besides,the description and the dynamic analysis of those phenomena have been discussed and verified theoretically and numerically via illustrating the phase portraits and the Lyapunov exponents'diagrams.
