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.

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References

  1. [1]

    J P Bouchaud and A Georges Phys. Rep. 195 127 (1990)

    ADS  MathSciNet  Article  Google Scholar 

  2. [2]

    R Metzler and J Klafter Phys. Rep. 339(1) 1 (2000)

    ADS  Article  Google Scholar 

  3. [3]

    R Herrmann Fractional calculus: An Introduction for Physicists (Singapore : World Scientific Publishing Company) p 528 (2011)

    Google Scholar 

  4. [4]

    A A Kilbas, M Srivastava and J J Trujillo Theory and Applications of Fractional Differential Equations (Amsterdam : Elsevier) 204, p 102 (2006)

  5. [5]

    D Kai The Analysis of Fractional Differential Equations (New York : Springer) 2004, p 77 (2004)

  6. [6]

    M Caputo Elasticita e Dissipazione (Bologna : Zani-Chelli) (1969)

  7. [7]

    J Vanterler and E C Oliveira Progr. Fract. Differ. Appl. 4(4) 479 (2018)

    Google Scholar 

  8. [8]

    C Oliveira and J A T Machado Math. Probl. Eng. 2014 1 (2014)

  9. [9]

    R Gorenflo and A A Kilbas, F Mainardi and S V Rogosin Mittag-Leffler Functions, Related Topics and Applications (Berlin: Springer) p 39 (2014)

  10. [10]

    R F Camargo PhD Thesis (UNICAMP, Campinas) (2009)

  11. [11]

    R Gorenflo, A A Kilbas, F Mainardi and S V Rogosin Mittag-Leffler Functions, Related Topics and Applications (Berlin : Springer) (2014)

  12. [12]

    A Giusti Nonlinear Dyn. 93(3) 1757 (2018)

    Article  Google Scholar 

  13. [13]

    M Ortigueira and J Machado Fractal Fract. 1(3) 1 (2017)

    Google Scholar 

  14. [14]

    M Ortigueira and J A T Machado J. Comput. Phys. 293 4 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  15. [15]

    D Del-Castillo-Negrete Phys. Rev. E. Stat. Nonlinear Soft Matter Phys. 79 031120 (2009)

    ADS  Article  Google Scholar 

  16. [16]

    K S Miller and B Ross An Introduction to the Fractional Calculus and Fractional Differential Equations (New York : Wiley) 1, p 257 (1993)

  17. [17]

    K B Oldham and J Spanier The Fractional Calculus (New York : Academic Press) 111, p 96 (1974)

  18. [18]

    S G Samko, A A Kilbas and O I Marichev Fractional Integrals and Derivatives - Theory and Applications (Yverdon : Gordon and Beach) 1, Ch 5, Sec 24, p 457 (1993)

  19. [19]

    I Podlubny J. Fract. Calc. Appl. Anal. 5(4) 367 (2002)

    Google Scholar 

  20. [20]

    M A M Ghandehari and M RanjbarKhaled Comput. Math. Appl. 65 975 (2013)

    MathSciNet  Article  Google Scholar 

  21. [21]

    R J LeVeque Finite Difference Methods for Differential Equations (Washington : A Math) Sec 15, p 201 (2004)

  22. [22]

    B Ross A Brief History and Exposition of the Fundamental Theory of Fractional Calculus (Berlin : Frac. Calc. and its Appl.) p 5 (1975)

  23. [23]

    A G O Goulart, M J Lazo, J M S Suarez and D M Moreira Phys. A Stat. Mech. Appl. 477 9 (2017)

  24. [24]

    S M Khaled Essa, F Mubarak and A Abo Bakr World Appl. Sci. J. 34(10) 1399 (2016)

    Google Scholar 

  25. [25]

    J Sabatier and C Farges J. Comput. Appl. Math. 339 30 (2018)

    MathSciNet  Article  Google Scholar 

  26. [26]

    M D Ortigueira and F J Coito Comput. Math. Appl. 59 1782 (2010)

    MathSciNet  Article  Google Scholar 

  27. [27]

    S P Arya Proc. Indian Natl. Sci. Acad. 69A(6) 709 (2003)

    Google Scholar 

  28. [28]

    P Kumar and M Sharan Proc. R. Soc. A466 383 (2010)

    ADS  Article  Google Scholar 

  29. [29]

    Y Luchko Math. Model. Nat. Phenom. 11(3) 1 (2016)

    MathSciNet  Article  Google Scholar 

  30. [30]

    K H Hoffmann, K Kulmus, C Essex and J Prehl Entropy 20 881 (2018)

    ADS  Article  Google Scholar 

  31. [31]

    F Mainardi Appl. Math. Lett. 9(6) 23 (1996)

    MathSciNet  Article  Google Scholar 

  32. [32]

    G A Briggs Diffusion Estimation for Small Emissions (Tennessee : NOAA Oak Ridge) p 12 (1973)

  33. [33]

    S E Gryning and E Lyck J. Clim. Appl. Meteorol. 23 651 (1984)

    ADS  Article  Google Scholar 

  34. [34]

    W M Cox and J Tikvart Atmos. Environ. 24A(9) 2387 (1990)

    ADS  Article  Google Scholar 

  35. [35]

    M Sharan and P Kumar Atmos. Environ. 43 2268 (2009)

    ADS  Article  Google Scholar 

  36. [36]

    S E Gryning, A A M Holtslag, Atmos. Environ. 21 79 (1987)

    ADS  Article  Google Scholar 

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Correspondence to A S Tankou Tagne.

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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.

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Tagne, A.S.T., Ema’a Ema’a, J.M., Ben-Bolie, G.H. et al. A new truncated M-fractional derivative for air pollutant dispersion. Indian J Phys 94, 1777–1784 (2020). https://doi.org/10.1007/s12648-019-01619-z

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Keywords

  • M-fractional derivative type
  • Truncated Mittag-Leffler
  • Air pollutant

PACS Nos.

  • 82.33.Tb
  • 47.53.+n
  • 92.60.Sz