Faculty of Natural and Agricultural Sciences, University of the Free State, South Africa.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan.
Math Biosci Eng. 2022 Feb 7;19(4):3526-3563. doi: 10.3934/mbe.2022163.
Many real world problems depict processes following crossover behaviours. Modelling processes following crossover behaviors have been a great challenge to mankind. Indeed real world problems following crossover from Markovian to randomness processes have been observed in many scenarios, for example in epidemiology with spread of infectious diseases and even some chaos. Deterministic and stochastic methods have been developed independently to develop the future state of the system and randomness respectively. Very recently, Atangana and Seda introduced a new concept called piecewise differentiation and integration, this approach helps to capture processes with crossover effects. In this paper, an example of piecewise modelling is presented with illustration to chaos problems. Some important analysis including a piecewise existence and uniqueness and piecewise numerical scheme are presented. Numerical simulations are performed for different cases.
许多实际问题描述了具有交叉行为的过程。对具有交叉行为的过程进行建模一直是人类面临的巨大挑战。事实上，在许多场景中都观察到了从马尔可夫过程到随机性过程的交叉的实际问题，例如在传染病的流行病学中，甚至在某些混沌中。确定性和随机性方法已经分别独立开发，以分别开发系统的未来状态和随机性。最近，Atangana 和 Seda 引入了一个称为分段微分和积分的新概念，这种方法有助于捕捉具有交叉效应的过程。本文提出了一个分段建模的例子，说明了混沌问题。提出了一些重要的分析，包括分段存在性和唯一性以及分段数值方案。针对不同情况进行了数值模拟。
