Long-term behavior of nonautonomous neutral compartmental systems

The asymptotic behavior of the trajectories of compartmental systems with a general set of admissible initial data is studied. More precisely, these systems are described by families of monotone nonautonomous neutral functional differential equations with nonautonomous operator. We show that the solutions asymptotically exhibit the same recurrence properties as the transport functions and the coefficients of the neutral operator. Conditions for the cases in which the delays in the neutral and non neutral parts are different, as well as for other cases unaddressed in the previous literature are also obtained

Hybrid multistep block method for solving neutral volterra integro-differential equation with proportional and mixed delays

The neutral Volterra integro-differential equation with proportional and mixed delays (NDVIDE) is being solved by a newly proposed technique in numerical method, namely, the two-point one off-point block multistep method (1OBM3). The method is also known as a hybrid multistep block method. Subsequently, Lagrange interpolating polynomial is utilized in order to develop the hybrid block method. The foundation of the technique is taken from predictor and corrector formulae. The proposed method will solve NDVIDE in two steps simultaneously, with three predictor formulae including one off-point. The NDVIDE problems are solved via the constant step size technique. In order to solve the integral and differential parts of the problems, two alternative numerical approaches are applied. The differentiation part is approximated by deriving the divided difference formula, while the integration part is interpolated using composite Simpson’s rule. Note that the proposed method has been analysed thoroughly regarding its order, consistency, zero stability and convergence of the method. The stability region for 1OBM3 has been constructed based on the stability polynomial obtained. Consequently, numerical results are presented to demonstrate the effectiveness of the proposed method, 1OBM3