A new truncated M-fractional derivative for air pollutant dispersion

Abstract

In this paper, we study the potential of fractional derivatives to model air pollution. We introduce an M-fractional truncated derivative type for $\alpha$-differentiable functions that generalizes other types of fractional derivatives. We denote this new differential operator by $_{i}D_{\mathrm{M}}^{\alpha ,\beta }$, where the parameters $\alpha$ and $\beta$, associated with the order of the derivative are such that $0<\alpha <1$, $\beta >1$ 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.

Appendices

Appendix 1: Solution of the M-Gaussian model

We start considering the so-called M-fractional linear differential equation with constant coefficients Eq.(14), here $\lambda ^{2}$ is a positive constant. Using item 5 in Theorem, the above Equation can be written as follows:

$$\begin{aligned} \frac{x^{1-\alpha }}{\Gamma (\beta +1)}\frac{{\mathrm{d}}P(x)}{{\mathrm{d}}x}\pm \kappa \lambda ^{2}P(x)=0, \end{aligned}$$

(19)

using chain rule from point 1 of theorem part, Eq. (19) can be expressed in the form:

$$\begin{aligned} {}_{i}D^{\alpha ,\beta }_{N}P(x)-\kappa \lambda ^{2}P(x)=0, \end{aligned}$$

(20)

with $\kappa \sim \gamma \cdot x^{\alpha }$

whose solution is:

$$\begin{aligned} P(x)=C\cdot {\mathbb {E}}_{\beta }(-\lambda _{n}^{2}\kappa x^{\alpha }). \end{aligned}$$

(21)

In the same line, the solutions $Q_{n}(z) (n = 0, 1, 2, 3, \ldots )$ of (12) that satisfy the boundary conditions (9) with $z_{0} = 0$ are obtained at the same time. using the equation: $Q_{n} (z) = Q_{n} \cos (\lambda _{n}z)$ where $\lambda _{n} = \frac{n \pi }{h}$, and assuming that $Q_{n}$ is a constant, from the superposition principle, we arrive at the following formula:

$$\begin{aligned} \bar{c}_{y}=Q_{0}+\sum _{n=1}^{+\infty }Q_{n} \cos (\lambda _{n}z) {\mathbb {E}}_{\beta }(-\lambda _{n}^{2}\kappa x^{\alpha }). \end{aligned}$$

(22)

Finally, by introducing the boundary condition (8) and using the identity

$$\begin{aligned} \delta (z-h_{s})\times h=1+2\sum _{n=1}^{+\infty }Q_{n} \cos (\lambda _{n}z)\cos (\lambda _{n}h_{s}), \end{aligned}$$

(23)

we end up at the final equation (13).

Appendix 2

We are now studying a particular case involving a fractional derivative. Choosing $\beta = 1$ and applying the limit $i\rightarrow 0$ on either side of Eq. (1), we have:

$$\begin{aligned} {}_{i}D^{\alpha ,\beta }_{N}f(x)=\lim _{h\rightarrow 0}\frac{f(x_{i}{\mathbb {E}}(h x^{-\alpha }))-f(x)}{h}, \end{aligned}$$

(24)

it is also known that

$$\begin{aligned} {}_{1}{\mathbb {E}}_{1}(h x^{-\alpha }))=\sum _{k=0}^{1}\frac{(h x^{-\alpha })^{k}}{\Gamma (k+1)}=1+h x^{-\alpha }, \end{aligned}$$

(25)

so, we conclude that

$$\begin{aligned} {}_{i}D^{\alpha ,\beta }_{N}f(x)=\lim _{h\rightarrow 0}\frac{f(x_{i}{\mathbb {E}}(h x^{-\alpha }))-f(x)}{h}=f^{\alpha }(x). \end{aligned}$$

(26)

so that a trivial solution of Eq. (11) can be expressed in the form:

$$\begin{aligned} P(x^{\alpha })=P_{0}{\mathbb {E}}_{\alpha }\left( -\lambda _{n}^{2}\kappa x^{\alpha }\right) , \end{aligned}$$

(27)

where ${\mathbb {E}}_{\alpha }$ represents the Mittag-Leffler function.