Generalized Fractional Mathematical Modelling and Simulation for Dynamical Systems

Abstract

Fractional calculus is an extension of classical calculus, that provides a more nuanced and exible framework for modeling complex phenomena. Its concept is known since 1695 when L'Hospital asked Leibnitz about the fractional power of derivative. By fractional calculus, we may get more accurate results in many physical problems. The notion of fractional operators had not been much worthy for modeling the complex problems of real world. These complex real world problems can be based on those physical occurrences that show fractal behavior. To handle this type of problems, fractal-fractional theory plays a vital role. Malware is a generic issue and many authors have discussed di erent mathematical mod- els to explain its extremities. Due to its complex features involving chaotic behavior, heterogeneities and memory e ect, some authors tried to solve it using the concept of fractional calculus and in advance form of fractal fractional theory. Till now we have seen the models which have a simple nature. So we decided to investigate a more complex mathematical model. This model has a variable infection rate which gives a deep insight of the behavior of malware. Moreover, infection rate is de ned as a nonlinear function of infected nodes. To better understand the behavior of such type of malware and develop antivirus software to overcome the malware, we decided to deal this model by convert- ing it into fractal fractional mathematical models. We also tried to nd the impacts of di erent parameters on malware propagation for integer and non integer orders. We were interested in examining the impact of memory e ects in this dynamical system in the sense of fractal fractional (FF) derivatives with three kernels known as Powerlaw, i Exponential Decay and Mittag-Le er. Initially the models were examined theoretically. Conditions for existence (Leray Schauder criteria), uniqueness (Lipschitz property) and stability (Ulam-Hyers and Ulam-Hyers-Rassias theorems) of the fractal fractional models were examined using concepts of xed point theory. Secondly, numerical schemes were developed using Lagrange interpolation using two point formula and simulations were performed using Matlab codes on R2016a to verify the accuracy of theoretical results. Sensitivity analysis of di erent parameters such as initial infection rate, variable adjust- ment to sensitivity of infected nodes, immune rate of antivirus strategies and loss rate of immunity of removed nodes is investigated under FF model and is compared with classi- cal. We have compared four di erent mathematical models (classical, fractional, fractal, fractal-fractional) so that in di erent forms of malware antivirus strategies could be de- veloped accordingly. Moreover, constant and variable fractional and fractal orders have been compared by graphs. On investigation, we nd that FF model describes the e ects of memory on nodes in detail. Antivirus software can be developed considering the e ect of FF orders and parameters to reduce persistence and eradication of infection. Small changes cause signi cant perturbation in infected nodes and malware can be driven into passive mode by understanding its propagation by FF derivatives and may take necessary actions to prevent the disaster caused by cyber attackers.