A new truncated M-fractional derivative for air pollutant dispersion
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A new truncated M-fractional derivative for air pollutant dispersion A S T Tagne1,2*, J M Ema’a Ema’a4, G H Ben-Bolie1,2 and D Buske3 1
Laboratory of Nuclear Physics, Department of Physics, Faculty of Science, University of Yaounde I, P. O. Box 812, Yaounde´, Cameroon 2
The African Center of Excellence in Information and Communication Technologies, University of Yaounde I, Yaounde´, Cameroon
3
Laboratory of Pollutant Dispersion Modelling and Nuclear Engineering, Department of Mathematics and Statistics, Institute of Physics and Mathematics, Federal University of Pelotas, P. O. Box 354, Pelotas, Brasil 4
Higher Teachers’ Training College, University of Maroua, P. O. Box 45, Maroua, Cameroon Received: 07 January 2019 / Accepted: 21 August 2019
Abstract: In this paper, we study the potential of fractional derivatives to model air pollution. We introduce an Mfractional truncated derivative type for a-differentiable functions that generalizes other types of fractional derivatives. We denote this new differential operator by i Da;b M , where the parameters a and b, associated with the order of the derivative are such that 0\a\1, b [ 1 and M is the notation to indicate that the function to be derived involves the truncated function of Mittag-Leffler with a parameter. The definition of this type of truncated M-fractional derivative satisfies the properties of the integer calculation. Based on this observation, we solved these models and we compared the solutions with the data obtained from the Copenhagen experiment. Fractional derivative models work much better than the traditional Gaussian model and the computed values are in good agreement with experimental ones. Keywords: M-fractional derivative type; Truncated Mittag-Leffler; Air pollutant PACS Nos.: 82.33.Tb; 47.53.?n; 92.60.Sz
1. Introduction The dispersion of pollutants in the atmosphere is a source of great permanent challenge due to the complexity of the phenomena that occur at this level of the Earth’s texture. An example of non-trivial problems is the non-integer-order calculation or fractional calculus, as it is widely distributed, is also important and ancient as the calculation of integer order, and for many decades, the scientific community has not appropriated it. Diffusion is one of the most fundamental transport processes of biological and physicochemical systems. Unlike normal diffusion, different physical/biological conditions can cause anomalous diffusion [1, 2]. At variance with common diffusion, where the mean square displacement increases linearly with time, in the anomalous diffusion the mean square displacement is not linear. Currently, there are numerous and important definitions of fractional derivatives types, each one of them with its particularity and
application [3–6]. Under these circumstances, the usual differential equations do not adequately describe the problem of turbulent diffusion. Current derivatives are not well defined in the undifferentiated behavior introduced by turbulence. Therefore, it is expected that t
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