Fractional-Order Model to Visualize the Effect of Plastic Pollution on Rain

Fractional-Order Model to Visualize the Effect of Plastic Pollution on Rain

Ekta N. Jayswal (Department of Mathematics, Gujarat University, Ahmedabad, India) and Purvi M. Pandya (Department of Mathematics, Gujarat University, Ahmedabad, India)
Copyright: © 2020 |Pages: 18
DOI: 10.4018/978-1-7998-3741-1.ch008

Abstract

In this era, one of the biggest issues faced by humans is due to plastic pollution as it dwells in environment and depletes the ecosystem. This affects the climate and disturbs the chain of rain, which is the common source of obtaining water body. Also, this resulting pollution causes the toxicity in rain. Accordingly, the mathematical model is framed by considering fractional order derivative. Pollution free and endemic equilibrium points are worked out for integer order system of non-linear differential equations. Local stability of equilibrium points brings attention on dynamical behavior of model with sufficient condition. With the help of basic reproduction number, bifurcation is analyzed, which shows the chaotic nature of this model. Providing Caputo derivative of fractional order, a numerical simulation has been done by taking different values of order for the system.
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Introduction

Plastic is most flexible and ubiquitous material hence now a days it is people’s essential need. World produces masses of plastic every day and million tons of plastics are used up every year out of which only one-quarter part is recycled, others going to landfills. This led to a high prominence of plastic pollution in the environment. It affects the environmental key-resources pollution in soil, water and air. Plastic pollution releases harmful toxic chemicals in surroundings and became barrier to the ecosystem via air currents which causes potentially unsafe environment. Rain toxicity is due to access of two gases sulfur dioxide and nitrogen oxides, most of it comes from burning of plastic as it contain fossil fuels. These gases react when it mixes up with water. Upcoming years have solution for this plastic pollution through dumping, burning or recycling according to the plastic category. Some people also use the fund for public health and environmental policy instead of burning and producing of plastic. Vasudevan (2010) (from Madurai, India) has patented a method to reuse plastic wastes mainly municipal solid waste to construct roads and known as “Plastic man”.

Mathematical modeling of dynamical system for integer order is vast branch for formulating the epidemic disease models and its control strategies. Above mentioned details can help to form a fractional order dynamical model for plastic. The word fraction means that any arbitrary non-negative real number. Fractional differential equation is extra ordinary differential equations. This calculus is generalization of ordinary differentiation and integration to a fractional order may be real or complex. For the first time fractional calculus is introduced by Gottfried Wilhelm Leibniz in 1695. Since many years, it is used only for many branches of science and engineering viscoelastic material, electrical networks, fluid flow, rheology, diffusive transport, bioengineering, finance and also in electromagnetic theory, hence become very popular in recent years. This study is used in studying the inconsistent behavior of viscoelasticity, bioengineering chaotic system. Fractional order is used in PID (proportional, integral and derivative) controllers which increase their degree of freedom and also measures its error function. Fractional order derivative is defined using mathematical term known as gamma function. Many mathematicians like Grünwald–Letnikov, Riemann-Liouville (2014), Caputo (1967), Hadamard, Atangana-Baleanu (2016), Riesz (2014) have given the definitions about fractional order derivative in which all are mathematically acceptable as in Ahmed et al. (2007). In signal classification task, Gomolka (2018) has proposed neural network with back propagation rule in which fractional derivative is used. Atangana-Baleanu (2016) has established a new kernel based upon Mitag-Leffler function; this description is filter of fractional regulator and fractional derivative. They all had done very useful work in this field. In this paper, we have used Caputo derivative for observing results. It is prevailing tool which have ability to model traditional phenomenon with long-term memory and long-term spatial interface.

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