Chaotic finite difference operators.

In this article, we analyze the chaotic behavior of finite difference operators associated with certain differential equations. Our examples range from numerical schemes for a birth-and-death model with proliferation to a class of second-order partial differential equations that includes the hyperbolic heat transfer equation, the telegraph equation, and the wave equation. We provide sufficient conditions on the spatial and time steps of the scheme that guarantee chaos for the corresponding operators, and we compare them with the conditions needed to ensure chaotic analytical solutions.

A case study of 2019-nCOV cases in Argentina with the real data based on daily cases from March 03, 2020 to March 29, 2021 using classical and fractional derivatives.

In this study, our aim is to explore the dynamics of COVID-19 or 2019-nCOV in Argentina considering the parameter values based on the real data of this virus from March 03, 2020 to March 29, 2021 which is a data range of more than one complete year. We propose a Atangana-Baleanu type fractional-order model and simulate it by using predictor-corrector (P-C) method. First we introduce the biological nature of this virus in theoretical way and then formulate a mathematical model to define its dynamics. We use a well-known effective optimization scheme based on the renowned trust-region-reflective (TRR) method to perform the model calibration. We have plotted the real cases of COVID-19 and compared our integer-order model with the simulated data along with the calculation of basic reproductive number. Concerning fractional-order simulations, first we prove the existence and uniqueness of solution and then write the solution along with the stability of the given P-C method. A number of graphs at various fractional-order values are simulated to predict the future dynamics of the virus in Argentina which is the main contribution of this paper.

A new fractional mathematical modelling of COVID-19 with the availability of vaccine.

The most dangerous disease of this decade novel coronavirus or COVID-19 is yet not over. The whole world is facing this threat and trying to stand together to defeat this pandemic. Many countries have defeated this virus by their strong control strategies and many are still trying to do so. To date, some countries have prepared a vaccine against this virus but not in an enough amount. In this research article, we proposed a new SEIRS dynamical model by including the vaccine rate. First we formulate the model with integer order and after that we generalize it in Atangana-Baleanu derivative sense. The high motivation to apply Atangana-Baleanu fractional derivative on our model is to explore the dynamics of the model more clearly. We provide the analysis of the existence of solution for the given fractional SEIRS model. We use the famous Predictor-Corrector algorithm to derive the solution of the model. Also, the analysis for the stability of the given algorithm is established. We simulate number of graphs to see the role of vaccine on the dynamics of the population. For practical simulations, we use the parameter values which are based on real data of Spain. The main motivation or aim of this research study is to justify the role of vaccine in this tough time of COVID-19. A clear role of vaccine at this crucial time can be realized by this study.

Nonlocal operators are chaotic.

We characterize for the first time the chaotic behavior of nonlocal operators that come from a broad class of time-stepping schemes of approximation for fractional differential operators. For that purpose, we use criteria for chaos of Toeplitz operators in Lebesgue spaces of sequences. Surprisingly, this characterization is proved to be-in some cases-dependent of the fractional order of the operator and the step size of the scheme.