Some Qualitative Analyses of Neutral Functional Delay Differential Equation with Generalized Caputo Operator

In this paper, a new class of a neutral functional delay differential equation involving the generalized ψ -Caputo derivative is investigated on a partially ordered Banach space. The existence and uniqueness results to the given boundary value problem are established with the help of the Dhage’s technique and Banach contraction principle. Also, we prove other existence criteria by means of the topological degree method. Finally, Ulam-Hyers type stability and its generalized version are studied. Two illustrative examples are presented to demonstrate the validity of our obtained results.

Crank-Nicolson scheme for two-dimensional in space fractional diffusion equations with functional delay

A two-dimensional in space fractional diffusion equation with functional delay of a general form is considered. For this problem, the Crank-Nicolson method is constructed, based on shifted Grunwald-Letnikov formulas for approximating fractional derivatives with respect to each spatial variable and using piecewise linear interpolation of discrete history with continuation extrapolation to take into account the delay effect. The Douglas scheme is used to reduce the emerging high-dimensional system to tridiagonal systems. The residual of the method is investigated. To obtain the order of the method, we reduce the systems to constructions of the general difference scheme with heredity. A theorem on the second order of convergence of the method in time and space steps is proved. The results of numerical experiments are presented.

Numerical algorithm for fractional order population dynamics model with delay

For a fractional-diffusion equation with nonlinearity in the differentiation operator and with the effect of functional delay, an implicit numerical method is constructed based on the approximation of the fractional derivative and the use of interpolation and extrapolation of discrete history. The source of this problem is a generalized model from population theory. Using a fractional discrete analogue of Gronwall's lemma, the convergence of the method is proved under certain conditions. The resulting system of nonlinear equations using Newton's method is reduced to a sequence of linear systems with tridiagonal matrices. Numerical results are given for a test example with distributed delay and a model example from the theory of population with concentrated constant delay.

Numerical method for fractional diffusion-wave equations with functional delay

For a fractional diffusion-wave equation with a nonlinear effect of functional delay, an implicit numerical method is constructed. The scheme is based on the L2-method of approximation of the fractional derivative of the order from 1 to 2, interpolation and extrapolation with the given properties of discrete prehistory and an analogue of the Crank-Nicolson method. The order of convergence of the method is investigated using the ideas of the general theory of difference schemes with heredity. The order of convergence of the method is more significant than in previously known methods, depending on the order of the starting values. The main point of the proof is the use of the stability of the L2-method. The results of comparing numerical experiments with other schemes are presented: a purely implicit method and a purely explicit method, these results showed, in general, the advantages of the proposed scheme.

Hereditary Evolution Processes Under Impulsive Effects

In this note, we deal with a model of population dynamics with memory effects subject to instantaneous external actions. We interpret the model as an impulsive Cauchy problem driven by a semilinear differential equation with functional delay. The existence of delayed impulsive solutions to the Cauchy problem leads to the presence of hereditary impulsive dynamics for the model. Furthermore, using the same procedure we study a nonlinear reaction–diffusion model.

Solvable delay model for epidemic spreading: the case of Covid-19 in Italy

We study a simple realistic model for describing the diffusion of an infectious disease on a population of individuals. The dynamics is governed by a single functional delay differential equation, which, in the case of a large population, can be solved exactly, even in the presence of a time-dependent infection rate. This delay model has a higher degree of accuracy than that of the so-called SIR model, commonly used in epidemiology, which, instead, is formulated in terms of ordinary differential equations. We apply this model to describe the outbreak of the new infectious disease, Covid-19, in Italy, taking into account the containment measures implemented by the government in order to mitigate the spreading of the virus and the social costs for the population.

Solvable delay model for epidemic spreading: the case of Covid-19 in Italy

We present a simple but realistic model for describing the diffusion of an infectious disease on a population of individuals. The dynamics is governed by a single functional delay differential equation, which, in the case of a large population, can be solved exactly, even in the presence of a time-dependent infection rate. This delay model has a higher degree of accuracy than the so-called SIR model, commonly used in epidemiology, which, instead, is formulated in terms of a set of three ordinary differential equations. We apply our model to describe the outbreak of the new virus COVID-19 in Italy, taking into account the containment measures implemented by the government in order to mitigate the spreading of the virus and the social costs for the population.