Analytical solution of the steady-state atmospheric fractional diffusion equation in a finite domain

Abstract

In this work, an analytical solution for the steady-state fractional advection-diffusion equation was investigated to simulate the dispersion of air pollutants in a finite media. The authors propose a method that uses classic integral transform technique (CITT) to solve the transformed problem with a fractional derivative, resulting in a more general solution. We compare the solutions with data from real experiment. Physical consequences are discussed with the connections to generalised diffusion equations. In the wake of these analysis, the results indicate that the present solutions are in good agreement with those obtained in the literature. This report demonstrates that fractional equations have come of age as a decisive tool to describe anomalous transport processes.

Appendix

To see a possible generalisation of this problem, in the same spirit as in Weiss treatise [58], in Fourier–Laplace space

$$\begin{aligned} u\bar{c}_y (k,u) - \bar{c}_y \left( {k,0} \right)= & {} u\psi (u)\phi \left( k \right) \bar{c}_y \left( {k,u} \right) \nonumber \\&- \psi (u)\bar{c}_y \left( {k,0} \right) , \end{aligned}$$

(45)

the Fourier transform of $\phi \left( k \right)$ is an operator in k. Thus

$$\begin{aligned} \phi \left( k \right) \bar{c}_y \left( {k,x} \right) \equiv \lambda _c (k)\bar{c}_y \left( {k,x} \right) . \end{aligned}$$

(46)

Assume that subdiffusion is characterised by a finite transfer variance $\Sigma ^2$ associated with a diverging characteristic waiting time. Considering the expansions for small k and the usual long-time limit [59], we reach the relation

$$\begin{aligned} \bar{c}_y (k,u) - \frac{{\bar{c}_y (k,0)}}{u} = u^{ - \alpha } L(z)\bar{c}_y (k,u). \end{aligned}$$

(47)

Using the definition of the Riemann–Liouville fractional differentiation of order $1-\alpha$, $0<\alpha <1$ [60]

$$\begin{aligned} _0 D_x^{1 - \alpha } \bar{c}_y (k,u) = \frac{1}{{\Gamma (\alpha )}}\frac{\partial }{{\partial x}}\int _0^x {\text {d}x'} \frac{{\bar{c}_y (k,x')}}{{(x - x')^{1 - \alpha } }}. \end{aligned}$$

(48)

The integration of the corresponding theorem of Laplace transformation can be shown to hold for fractional integration

$$\begin{aligned} \mathscr {L}\{ {_0 D_x^{ - \alpha } c_y (k,x)} \} = u^{ - \alpha } c_y (k,u). \end{aligned}$$

(49)

We obtain the fractional partial equation

$$\begin{aligned} \frac{{\partial \bar{c}_y (k,u)}}{{\partial x}} = _{0}{D}_x^{1 - \alpha } L(z) \bar{c}_y (k,u), \end{aligned}$$

(50)

by considering the operator

$$\begin{aligned} L(z) = \frac{\partial }{{\partial z}}\left[ {K_z (x,z)\frac{\partial }{{\partial z}}}\right] . \end{aligned}$$

Let us consider the fractional diffusion-type eq. (50), in combination with the separation ansatz

$$\begin{aligned} \bar{c}_y {(x,z)}= \sum \limits _{n = 0}^\infty X_{n}(x)Z_{n}(z). \end{aligned}$$

(51)

The resulting equation

$$\begin{aligned} \frac{{\text {d}X(x)}}{{\text {d}x}}(_0 D_x^{1 - \alpha } X)^{ - 1} = \frac{{L(z)}Z}{Z} = - \lambda \end{aligned}$$

(52)

is then separated into the pair of eigenequations [42, 61]

$$\begin{aligned}&_0 D_x^\alpha X(x) - \frac{{x^{ - \alpha } \delta (0)}}{{\Gamma (1 - \alpha )}} = \lambda _{n,\alpha } X(x) \end{aligned}$$

(53)

$$\begin{aligned}&L(z)Z(z)=-\lambda _{n, \alpha }Z \end{aligned}$$

(54)

for an eigenvalue $\lambda _{n,\alpha }$ of L(z).