Nonlocal Modeling and Inverse Parameter Estimation of Time-Varying Vehicular Emissions in Urban Pollution Dynamics

Abstract

This paper investigates the dispersion of atmospheric pollutants in urban environments using a fractional-order convection–diffusion-reaction model with dynamic line sources associated with vehicle traffic. The model includes Caputo fractional time derivatives and Riesz fractional space derivatives to account for memory effects and non-local transport phenomena characteristic of complex urban air flows. Vehicle trajectories are generated stochastically on the road network graph using Dijkstra’s algorithm, and each moving vehicle acts as a mobile line source of pollutant emissions. To reflect the daily variability of emissions, a time-dependent modulation function determined by unknown parameters is included in the source composition. These parameters are inferred by solving an inverse problem using synthetic concentration measurements from several fixed observation points throughout the area. The study presents two main contributions. Firstly, a detailed numerical analysis of how fractional derivatives affect pollutant dispersion under realistic time-varying mobile source conditions, and secondly, an evaluation of the performance of the proposed parameter estimation method for reconstructing time-varying emission rates. The results show that fractional-order models provide increased flexibility for representing anomalous transport and retention effects, and the proposed method allows for reliable recovery of emission dynamics from sparse measurements.

1. Introduction

In modern conditions of intensive anthropogenic impact on the environment, modeling the spread of pollutants in the atmosphere of large cities is becoming especially relevant. Increasing traffic flows lead to the formation of line emission sources, having a serious impact on air quality and public health [1,2,3,4,5,6,7].

When describing the transport of pollutants in the atmosphere, the convection–diffusion equation is a traditional tool [8,9,10,11,12,13] that allows for taking into account the features of the movement of the medium and the process of spontaneous equalization of concentrations [14,15]. In addition, it is used to analyze the movement of impurities in water bodies [16,17,18,19] and soil [20,21] and is also in demand in heat exchange tasks [22,23,24] and  fluid dynamics, acting as a basic model for describing the flow of fluid or gas under various disturbances [25,26].

However, the rapid development of the urban environment, intensive transport traffic, and dense development often lead to abnormal processes of transfers of harmful substances [5,27], which go beyond the capabilities of classical convection–diffusion models. In order to adequately describe such phenomena, mathematical models with fractional derivatives have recently been increasingly used, allowing for memory effects, the fractal structure of the medium, and anomalous diffusion to be taken into account [28,29].

Fractional differential equations are actively used in modeling complex systems in fluid dynamics, geophysics, biology and other areas where classical models do not always adequately describe the observed phenomena [29,30,31]. In particular, to describe the phenomena of subdiffusion and superdiffusion, as well as processes that have a “memory” character, fractional derivatives of Caputo and Riesz are often introduced. The fractional Caputo derivative provides a flexible mechanism for accounting for temporal memory effects and historical influence [32,33,34], while the Riesz derivative is often used to describe spatial anomalous diffusion and fractal structures [35,36]. In the context of atmospheric processes, fractional derivatives can account for uneven mixing and turbulent fluctuations caused by the complex structure of urban developments, the presence of high-rise buildings, and the channeling of air flows along streets and avenues [37].

Along with deterministic models, the stochastic component plays an increasingly important role in describing urban processes, allowing for a quantitative assessment of uncertainties and variations in emission sources [38]. Thus, when modeling the spread of pollutants in the atmosphere, it is necessary to take into account fluctuations in the intensity of sources (for example, different traffic at different times of the day) or random selection of vehicle routes. To do this, stochastic terms are introduced into the right-hand side of the transfer equations, reflecting the corresponding disturbances [39,40]. This approach helps to take into account a wide range of uncertainties associated with both natural and anthropogenic factors: changing weather conditions, instability of model parameters, or the random nature of pollution sources [41].

A number of studies [42,43] note that ensemble approaches, coupled with stochastic descriptions of processes, are successfully used to analyze risks, determine probabilistic boundaries for the spread of impurities, and assess worst-case pollution scenarios. The combination of a fractional differential operator with a probabilistic description provides an even more complete understanding of the existing uncertainty [44], which is especially important in an urban environment with its dynamic and heterogeneous conditions. Modern studies demonstrate that the use of ensemble approaches for stochastic uncertainty assessment [45,46] allows for efficient coefficient recovery, which significantly improves the accuracy and reliability of pollutant transport modeling.

The classical approach to air pollution modeling is the combined consideration of convective–diffusion processes and the Navier–Stokes equations describing the dynamics of air flow [47]. At the same time, the real urban landscape, characterized by high-rise buildings and long highways, creates complex geometry for air flows and forms local zones of concentration of pollutants. Taking into account line sources located along highways, as well as cluster emissions in industrial areas, allows for a detailed analysis of the process of formation of spatial–temporal pollution fields [48].

The numerical solution of the Navier–Stokes equations in combination with the fractional differential convection–diffusion equation places increased demands on the methods of discretization in time and space. Finite volume, finite element, and finite difference methods are traditionally used for numerical solutions of hydrodynamic problems [49,50], which allow for modeling the complex geometry of urban development. At the same time, when taking into account anomalous diffusion and memory effect, an approximation of fractional derivatives is necessary, which is often implemented based on the Grünwald–Letnikov formulas [51,52], L1-schemes [53,54,55], or pseudospectral approaches [56,57].

The subject of this paper is a model of a pollutant dispersion in the atmosphere based on a fractional-order generalization of the convection–diffusion-reaction equation. The temporal derivative is formally replaced by the Caputo fractional derivative to take into account the hereditary properties of the process, non-Markovian dynamics, characteristic of the accumulation and retention of pollutants in the atmosphere under changing meteorological conditions. In addition, the first-order and second-order spatial derivatives in the convective and diffusion terms, respectively, are formally replaced by the Riesz fractional derivative to account for anomalous diffusion and non-local transport phenomena.

Unlike prior studies based on static point sources or uniform emission fields, this work models vehicular emissions as a dynamic line source that moves with traffic flow. The movement of vehicles is modeled using a road network represented as a graph, where the trajectory of each vehicle is determined by randomly selected start and end nodes and is calculated using Dijkstra’s shortest path algorithm. To account for the variability of urban traffic, we consider a stochastic ensemble of such vehicle routes.

It is assumed that every vehicle that traverses a city emits pollutants, with emissions varying over time, depending on daily activity cycles. This temporal modulation is described by a parametric function, the parameters of which are usually unknown in practice. To assess them, we formulate and solve an inverse problem using measurements of pollutant concentrations from several monitoring stations distributed throughout the urban area.

The traditional approach to solving inverse problems is to formulate an adjoint problem. However, when identifying a large number of parameters, this becomes impractical. Therefore, in this paper, an alternative approach based on parameter estimation technique is proposed.

The main contributions of this paper are twofold:

(1)

We systematically investigate how replacing the classical time and spatial derivatives in the convection–diffusion-reaction equation with fractional Caputo and Riesz derivatives, respectively, affects the dispersion of atmospheric pollutants. This analysis is carried out in a realistic setting involving dynamic line sources associated with vehicle motion, providing new insights into the memory and non-local transport effects that can be captured by fractional models.

(2)

We propose and evaluate a parameter estimation method for solving an inverse problem aimed at recovering unknown coefficients on the right-hand side of the governing equation. These coefficients modulate time-dependent pollutant emissions and are derived from realistic synthetic concentration measurements from spatially distributed observation points. Our results demonstrate the ability of the method to efficiently recover emission time profiles, even under model uncertainty caused by stochastic vehicle motion and fractional dynamics.

Until now, such a formulation of the problem, which simultaneously takes into account the Caputo and Riesz fractional derivatives, the Navier–Stokes equations, and stochastic disturbances from line pollution sources, has practically not been considered in the available literature. Moreover, the known studies are more often focused either on deterministic models with fractional derivatives or on standard stochastic approaches that do not cover the entire complex of the listed factors. The solution proposed in this paper is the first experience of such a comprehensive analysis of anomalous diffusion in combination with a full hydrodynamic description and probabilistic accounting of uncertainties, which opens up new prospects for modeling pollution in the urban environment.

2. Materials and Methods

2.1. Problem Statement

2.1.1. Statement of the Forward Problem

Let us formulate the fractional differential stochastic model of dispersion of a pollutant in the atmosphere emitted by vehicles moving along random routes. We adopt a two-dimensional model to describe the dispersion of a pollutant in the atmosphere. This simplification is justified by the nature of the emission sources (vehicles emitting pollutants near ground level) and the dominant horizontal transport in urban environments.

In $\mathcal{D}={\left(0,L\right)}^2$, $L>0$, we represent the road network as a directed graph $\mathcal{G}=\left(\mathcal{V},\mathcal{E}\right)$, where $\mathcal{V}\subset \mathcal{D}$ is the set of nodes (intersections) and $\mathcal{E}\subset \mathcal{V}\times \mathcal{V}$ is the set of directed edges (roads) (Figure 1). Let $\left(\mathsf{\Omega },\mathcal{F},\mathbb{P}\right)$ be the complete probability space, where $\mathsf{\Omega }$ is the sample space, i.e., the set of all possible trajectories of a vehicle, $\mathcal{F}$ is the $\sigma$-algebra on $\mathsf{\Omega }$ representing a set of measurable subsets of events, and $\mathbb{P}$ is the probability measure on $\left(\mathsf{\Omega },\mathcal{F}\right)$.

Let the route of the mth vehicle be described by the function ${\mathbf{X}}_m:\left[0,T\right]\times \mathsf{\Omega }\to \mathcal{D}$, where $T>0$ is time, $m\in \left\{1,\dots ,M\right\}$, and M is the number of vehicles. Thus, the sample $\omega \in \mathsf{\Omega }$ defines a set of trajectories $\omega \mapsto \left\{{\mathbf{X}}_1\left(t,\omega \right),\dots ,{\mathbf{X}}_M\left(t,\omega \right)\right\}$, where $t\in \left[0,T\right]$.

In $\mathcal{D}\times \left(0,T\right]\times \mathsf{\Omega }\times {\mathbb{R}}^N$, we consider a fractional-order generalization of the convection–diffusion-reaction equation, which describes the dispersion of a pollutant in the atmosphere:

$$\begin{array}{cc}\hfill & {}_0{}^C\mathbb{D}_t^{\alpha }\varphi \left(\mathbf{x},t,\omega ,\mathbf{p}\right)+{\mathcal{C}}^{\beta }\left(\mathbf{u}\left(\mathbf{x},t\right),\varphi \left(\mathbf{x},t,\omega ,\mathbf{p}\right)\right)-{\mathcal{D}}^{\gamma }\left(\varphi \left(\mathbf{x},t,\omega ,\mathbf{p}\right)\right)\hfill \\ \hfill & +r\left(\mathbf{x}\right)\varphi \left(\mathbf{x},t,\omega ,\mathbf{p}\right)=f\left(\mathbf{x},t,\omega ,\mathbf{p}\right),\hfill \end{array}$$

(1)

where $\varphi$ is the pollutant concentration, $\omega \in \mathsf{\Omega }$, $\mathbf{u}\left(\mathbf{x},t\right)=\left(u_1\left(\mathbf{x},t\right),u_2\left(\mathbf{x},t\right)\right)$ is the wind velocity field, $\mathbf{p}\in {\mathbb{R}}^N$ is the vector of numerical parameters to be defined below, $\mathbf{x}=\left(x_1,x_2\right)\in \mathcal{D}$, and ${}_0{}^C\mathbb{D}_t^{\alpha }$ is the temporal Caputo fractional derivative:

$${}_0{}^C\mathbb{D}_t^{\alpha }\varphi \left(·,t,·,·\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(1-\alpha \right)}}{\int }_0^t{\displaystyle \frac{{\partial }_{\xi }\varphi \left(·,\xi ,·,·\right)}{{\left(t-\xi \right)}^{\alpha }}}d\xi ,t\in \left(0,T\right],\alpha \in \left(0,1\right),$$

and ${\mathcal{C}}^{\beta }$ and ${\mathcal{D}}^{\gamma }$ are defined as follows:

$$\begin{array}{cc}\hfill & {\mathcal{C}}^{\beta }\left(\mathbf{u}\left(\mathbf{x},t\right),\varphi \left(\mathbf{x},t,\omega ,\mathbf{p}\right)\right)=-{\varsigma }_{\beta }\sum _{i=1}^2\left(u_i^{-}\left(\mathbf{x},t\right){}_0{}^{RL}\mathbb{D}_{x_i}^{\beta }\varphi \left(\mathbf{x},t,\omega ,\mathbf{p}\right)-u_i^{+}\left(\mathbf{x},t\right){}_{x_i}{}^{RL}\mathbb{D}_L^{\beta }\varphi \left(\mathbf{x},t,\omega ,\mathbf{p}\right)\right),\hfill \\ \hfill & {\mathcal{D}}^{\gamma }\left(\varphi \left(\mathbf{x},t,\omega ,\mathbf{p}\right)\right)=-{\varsigma }_{\gamma }\kappa \sum _{i=1}^2\left({}_0{}^{RL}\mathbb{D}_{x_i}^{\gamma }\varphi \left(\mathbf{x},t,\omega ,\mathbf{p}\right)+{}_{x_i}{}^{RL}\mathbb{D}_L^{\gamma }\varphi \left(\mathbf{x},t,\omega ,\mathbf{p}\right)\right),\hfill \end{array}$$

where $u_i^{-}\left(\mathbf{x},t\right)=max\left\{u_i\left(\mathbf{x},t\right),0\right\}$, $u_i^{+}\left(\mathbf{x},t\right)=min\left\{u_i\left(\mathbf{x},t\right),0\right\}$, ${\displaystyle {\varsigma }_{\nu }={\left(2cos\frac{\pi \nu }{2}\right)}^{-1}}$, $\kappa >0$ is the atmospheric diffusion coefficient, and ${}_0{}^{RL}\mathbb{D}_{x_i}^{\nu }$ and ${}_{x_i}{}^{RL}\mathbb{D}_L^{\nu }$ are Riemann–Liouville fractional derivatives:

$$\begin{array}{cccc}\hfill & {}_0{}^{RL}\mathbb{D}_{x_i}^{\beta }\varphi \left(\mathbf{x},·,·,·\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(1-\beta \right)}}{\partial }_{x_i}{\int }_0^{x_i}{\displaystyle \frac{{\left.\varphi \left(\mathbf{x},·,·,·\right)\right|}_{x_i=\xi }}{{\left(x_i-\xi \right)}^{\beta }}}d\xi ,\hfill & \hfill & \beta \in \left(0,1\right),\hfill \\ \hfill & {}_{x_i}{}^{RL}\mathbb{D}_L^{\beta }\varphi \left(\mathbf{x},·,·,·\right)=-{\displaystyle \frac{1}{\mathsf{\Gamma }\left(1-\beta \right)}}{\partial }_{x_i}{\int }_{x_i}^L{\displaystyle \frac{{\left.\varphi \left(\mathbf{x},·,·,·\right)\right|}_{x_i=\xi }}{{\left(\xi -x_i\right)}^{\beta }}}d\xi ,\hfill & \hfill & \beta \in \left(0,1\right),\hfill \\ \hfill & {}_0{}^{RL}\mathbb{D}_{x_i}^{\gamma }\varphi \left(\mathbf{x},·,·,·\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(2-\gamma \right)}}{\partial }_{x_i}^2{\int }_0^{x_i}{\displaystyle \frac{{\left.\varphi \left(\mathbf{x},·,·,·\right)\right|}_{x_i=\xi }}{{\left(x_i-\xi \right)}^{\gamma }}}d\xi ,\hfill & \hfill & \gamma \in \left(1,2\right),\hfill \\ \hfill & {}_{x_i}{}^{RL}\mathbb{D}_L^{\gamma }\varphi \left(\mathbf{x},·,·,·\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(2-\gamma \right)}}{\partial }_{x_i}^2{\int }_{x_i}^L{\displaystyle \frac{{\left.\varphi \left(\mathbf{x},·,·,·\right)\right|}_{x_i=\xi }}{{\left(\xi -x_i\right)}^{\gamma }}}d\xi ,\hfill & \hfill & \gamma \in \left(1,2\right).\hfill \end{array}$$

Including fractional derivatives with respect to the spatial variable in Equation (1) ensures that buildings and structures are taken into account, and including a fractional derivative with respect to the temporal variable ensures that the hereditary properties of the process are taken into account.

We note that the orders of fractional derivatives used in our study are representative of anomalous diffusion regimes observed in urban and turbulent atmospheric environments. Values of $\alpha \in \left(0,1\right)$ are commonly associated with subdiffusive memory effects, particularly in low-turbulence zones, where pollutants linger longer than predicted by classical models [36]. However, extremely low values of $\alpha$, particularly $\alpha <0.5$, correspond to excessively strong memory effects and subdiffusive behavior that are not representative of real atmospheric dispersion, even under stable or low-turbulence conditions. Such values tend to overly suppress temporal propagation and would lead to unrealistically slow dissipation of pollutants. Riesz spatial operators with $\gamma \in \left(1.5,1.9\right)$ have been shown to better capture heavy-tailed spatial dispersion patterns in field-scale experiments [58]. The use of $\beta <1$ in the first-order Riesz derivative reflects spatial nonlocality in advective transport, where influence is not limited to nearest neighbors, which is a feature often seen in urban atmospheric dynamics.

Now let us define the right-hand side of Equation (1). Each vehicle emits pollutants into the atmosphere as it moves, but their intensity may not be constant throughout the day. To take into account the time-varying activity of pollutant emissions, we introduce a scalar dimensionless modulation function ${\mathsf{\Theta }}_{\mathbf{p}}\left(t\right)$, constructed as a piecewise linear function interpolating a sequence of points ${\left\{\left(t_i,p_i\right)\right\}}_{i=1}^N$, where $t_i\in \left[0,T\right]$ are discrete moments of time, and $p_i$ is the empirical representation of the temporal change in emission activity. Let $\mathbf{p}=\left(p_1,p_2,\dots ,p_N\right)$ and define the right-hand side of Equation (1) as follows:

$$f\left(\mathbf{x},t,\omega ,\mathbf{p}\right)={\mathsf{\Theta }}_{\mathbf{p}}\left(t\right)\sum _{m=1}^M{\rho }_m\delta \left(\mathbf{x}-{\mathbf{X}}_m\left(t,\omega \right)\right),$$

(2)

where ${\rho }_m$ is the pollution density of the mth vehicle. The physical meaning of the function ${\mathsf{\Theta }}_{\mathbf{p}}\left(t\right)$ is the normalized traffic intensity of vehicles over time (Figure 2). It scales emissions figures for all vehicles equally to reflect typical daily changes in traffic flow, such as morning and evening rush hours, lower traffic volumes at night, or possible temporary increases due to events.

The wind velocity field is determined from the Navier–Stokes equations for an incompressible fluid:

$$\begin{array}{cc}\hfill & {\partial }_t\mathbf{u}\left(\mathbf{x},t\right)+\left(\mathbf{u}\left(\mathbf{x},t\right)·\nabla \right)\mathbf{u}\left(\mathbf{x},t\right)+\nabla \pi \left(\mathbf{x},t\right)={\displaystyle \frac{1}{\mathrm{Re}}}{\nabla }^2\mathbf{u}\left(\mathbf{x},t\right),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & \nabla ·\mathbf{u}\left(\mathbf{x},t\right)=0,\hfill \end{array}$$

(4)

where $\pi$ is air pressure, and $\mathrm{Re}>0$ is the Reynolds number.

Equations (1)–(4) are supplemented with the initial and boundary conditions:

$$\begin{array}{llll}\hfill & \mathbf{u}\left(\mathbf{x},t\right)={\mathbf{u}}_d\left(\mathbf{x},t\right),\hfill & \hfill & \mathbf{x}\in \partial \mathcal{D},t>0,\hfill \end{array}$$

(5)

$$\begin{array}{llll}\hfill & \varphi \left(\mathbf{x},t,\omega ,\mathbf{p}\right)={\varphi }_d\left(\mathbf{x},t\right),\hfill & \hfill & \mathbf{x}\in \partial \mathcal{D},t>0,\omega \in \mathsf{\Omega },\mathbf{p}\in {\mathbb{R}}^N,\hfill \end{array}$$

(6)

$$\begin{array}{llll}\hfill & \mathbf{u}\left(\mathbf{x},t\right)={\mathbf{u}}_0\left(\mathbf{x},t\right),\hfill & \hfill & \mathbf{x}\in \overline{\mathcal{D}},t=0,\hfill \end{array}$$

(7)

$$\begin{array}{llll}\hfill & \varphi \left(\mathbf{x},t,\omega ,\mathbf{p}\right)={\varphi }_0\left(\mathbf{x},t\right),\hfill & \hfill & \mathbf{x}\in \overline{\mathcal{D}},t=0,\omega \in \mathsf{\Omega },\mathbf{p}\in {\mathbb{R}}^N.\hfill \end{array}$$

(8)

The choice of the boundary conditions is due to our use of a horizontal, depth-averaged two-dimensional urban model embedded in a large-scale atmospheric reservoir. We assume that the outer-boundary velocity equals the externally provided mesoscale wind $u_{\infty }$, while the concentration is fixed to the observed background ${\varphi }_d$ at the outer boundary.

Thus, the forward problem is formulated as follows:

Problem 1

(Forward problem). Find the pollutant concentration ϕ, velocity $\mathbf{u}$, and air pressure π from Equations (1), (3) and (4) subject to the initial and boundary conditions (5)–(8).

Due to the stochastic nature of the problem, the statistical moments of the solution, such as expected concentration $\mathbb{E}\left[\varphi \left(\mathbf{x},t,\omega ,\mathbf{p}\right)\right]$, concentration variance $\mathrm{Var}\left[\varphi \left(\mathbf{x},t,\omega ,\mathbf{p}\right)\right]$, or the probability of exceeding a certain threshold concentration $\mathbb{P}\left(\varphi \left(\mathbf{x},t,\omega ,\mathbf{p}\right)>{\varphi }_{\mathrm{threshold}}\right)$, are of interest. This reduces to the need to repeatedly solve Problem 1 for randomly generated samples $\omega$.

2.1.2. Statement of the Inverse Problem

In practice, the parameter vector $\mathbf{p}$ is typically unknown and must be inferred. In this work, $\mathbf{p}$ is estimated by formulating and solving an inverse problem constrained by observed pollutant concentration data.

Let us assume that $\mathcal{D}$ contains S stations monitoring the concentration of a pollutant located at points ${\mathbf{x}}_1,{\mathbf{x}}_2,\dots ,{\mathbf{x}}_S$ (Figure 1). Let ${\widehat{\varphi }}_s\left(t\right)$ denote the concentration value measured by the sth station located at the point ${\mathbf{x}}_s$ at time t.

We introduce the functional for quantitatively assessing the discrepancy between the predicted solution and the reference measurement data:

$$\mathcal{I}\left(\mathbf{p}\right)=\sum _{s=1}^S{\int }_0^T{\left|\mathbb{E}\left[\varphi \left({\mathbf{x}}_s,t,\omega ,\mathbf{p}\right)\right]-{\widehat{\varphi }}_s\left(t\right)\right|}^{2}dt,$$

(9)

where $\mathbb{E}\left[·\right]$ denotes the expected value. Then, the inverse problem is formulated as follows:

Problem 2

(Inverse problem). Determine the parameter vector $\mathbf{p}$ that minimizes the functional given in Equation (9).

2.2. Solution Method

2.2.1. Method of Solving the Deterministic Forward Problem

Let us start by discretizing Equation (1) at fixed $\omega$ and $\mathbf{p}$.

Let ${\mathsf{\Xi }}_h=\left\{x_k=kh,k=0,1,\dots ,N_x,N_{x}h=L\right\}$ for $h>0$, and ${\mathcal{D}}_h={\mathsf{\Xi }}_h\times {\mathsf{\Xi }}_h$. In addition, we introduce a uniform partition of the time interval ${\mathfrak{T}}_{\tau }=\left\{t_n=n\tau ,n=0,1,\dots ,N_t,\right.$ $\left.N_t\tau =T\right\}$ for $\tau >0$. Time $t_n$ will be referred to as the nth time layer. Let us denote by ${\varphi }_{i,j}^n$ the value of the grid function $\varphi$ at the node ${\mathbf{x}}_{ij}=\left(x_i,x_j\right)\in {\mathcal{D}}_h$ at time $t_n\in {\mathfrak{T}}_{\tau }$ for fixed $\omega$ and $\mathbf{p}$. In addition, the vector of internal nodes will be used below:

$$\widehat{\mathbf{x}}={\left({\mathbf{x}}_{1,1},{\mathbf{x}}_{1,2},\dots ,{\mathbf{x}}_{1,N_x-1},{\mathbf{x}}_{2,1},\dots ,{\mathbf{x}}_{2,N_x-1},\dots ,{\mathbf{x}}_{N_x-1,1},\dots ,{\mathbf{x}}_{N_x-1,N_x-1}\right)}^{\top }.$$

To discretize the temporal fractional derivative in Equation (1), we use the approximation formula [59]:

$${}_0{}^C\mathbb{D}_t^{\alpha }\phi \left(t_n\right)={\delta }_{n,n}^{\left(\alpha \right)}\phi \left(t_n\right)-\sum _{s=1}^{n-1}\left({\delta }_{n,s+1}^{\left(\alpha \right)}-{\delta }_{n,s}^{\left(\alpha \right)}\right)\phi \left(t_s\right)-{\delta }_{n,1}^{\left(\alpha \right)}\phi \left(t_0\right)+O\left({\tau }^{2-\alpha }\right),$$

(10)

where

$${\delta }_{n,s}^{\left(\alpha \right)}={\displaystyle \frac{{\tau }^{-\alpha }}{\mathsf{\Gamma }\left(2-\alpha \right)}}\left[{\left(n-s+1\right)}^{1-\alpha }-{\left(n-s\right)}^{1-\alpha }\right].$$

There are several approximation formulas for discretizing the Riemann–Liouville fractional derivative. Since using the same Riemann–Liouville fractional derivative discretization formulas leads to an unstable scheme [35], we use different approximation formulas for the convective and diffusion parts. To discretize the convective term ${\mathcal{C}}^{\beta }\left(\mathbf{u}\left(\mathbf{x},t\right),\varphi \left(\mathbf{x},t,\omega ,\mathbf{p}\right)\right)$, we use the standard Grünwald–Letnikov formula, which for node $x_k\in {\mathsf{\Xi }}_h$ has the following form [28]:

$$\begin{array}{c}{\displaystyle {}_0{}^{RL}\mathbb{D}_x^{\beta }\phi \left(x_k\right)=\frac{1}{h^{\beta }}\sum _{s=0}^k{g}_s^{\left(\beta \right)}\phi \left(x_{k-s}\right)+O\left(h\right),}\hfill \\ {\displaystyle {}_x{}^{RL}\mathbb{D}_L^{\beta }\phi \left(x_k\right)=\frac{1}{h^{\beta }}\sum _{s=0}^{N_x-k}{g}_s^{\left(\beta \right)}\phi \left(x_{k+s}\right)+O\left(h\right),}\hfill \end{array}$$

(11)

where the coefficients $g_s^{\left(\beta \right)}$ are defined as

$$g_0^{\left(\beta \right)}=1,g_s^{\left(\beta \right)}={\displaystyle \frac{{\left(-1\right)}^s}{s!}}\beta \left(\beta -1\right)\dots \left(\beta -s+1\right),s=1,2,\dots ,N_x.$$

To discretize the diffusion term, we use the L2-approximation method, which for node $x_k\in {\mathsf{\Xi }}_h$ has the following form [60]:

$$\begin{array}{c}{\displaystyle {}_0{}^{RL}\mathbb{D}_{x_i}^{\gamma }\phi \left(x_k\right)=\frac{1}{h^{\gamma }\mathsf{\Gamma }\left(3-\gamma \right)}\left[\frac{\left(1-\gamma \right)\left(2-\gamma \right)\phi \left(x_0\right)}{k^{\gamma }}+\frac{\left(2-\gamma \right)\left(\phi \left(x_1\right)-\phi \left(x_0\right)\right)}{k^{\gamma -1}}\right.}\hfill \\ {\displaystyle \left.+\sum _{s=0}^{k-1}\left(\phi \left(x_{k-s+1}\right)-2\phi \left(x_{k-s}\right)+\phi \left(x_{k-s-1}\right)\right)\left({\left(s+1\right)}^{2-\gamma }-s^{2-\gamma }\right)\right]}\hfill \\ {\displaystyle {}_{x_i}{}^{RL}\mathbb{D}_L^{\beta }\phi \left(x_k\right)=\frac{1}{h^{\gamma }\mathsf{\Gamma }\left(3-\gamma \right)}\left[\frac{\left(1-\gamma \right)\left(2-\gamma \right)\phi \left(x_N\right)}{{\left(N-k\right)}^{\gamma }}+\frac{\left(2-\gamma \right)\left(\phi \left(x_N\right)-\phi \left(x_{N-1}\right)\right)}{{\left(N-k\right)}^{\gamma -1}}\right.}\hfill \\ {\displaystyle \left.+\sum _{s=0}^{N_x-k-1}\left(\phi \left(x_{k+s-1}\right)-2\phi \left(x_{k+s}\right)+\phi \left(x_{k+s+1}\right)\right)\left({\left(s+1\right)}^{2-\gamma }-s^{2-\gamma }\right)\right].}\hfill \end{array}$$

(12)

To implement the algorithm numerically, it is convenient to switch to a matrix form. Let $\mathbb{T}\left(\mathbf{c},\mathbf{r}\right)$ denote the Toeplitz matrix with the first column $\mathbf{c}\in {\mathbb{R}}^{N_x-1}$ and the first row $\mathbf{r}\in {\mathbb{R}}^{N_x-1}$. Let us define the matrices

$$\begin{array}{cc}\hfill & {\mathbf{C}}^{+}=\mathbb{T}\left(\left(g_0^{\left(\beta \right)},g_1^{\left(\beta \right)},\dots ,g_{N_x-2}^{\left(\beta \right)}\right),\left(g_0^{\left(\beta \right)},0,\dots ,0\right)\right),\hfill \\ \hfill & {\mathbf{C}}^{-}=\mathbb{T}\left(\left(0,0,\dots ,0\right),\left(0,g_0^{\left(\beta \right)},g_1^{\left(\beta \right)},\dots ,g_{N_x-3}^{\left(\beta \right)}\right)\right),\hfill \\ \hfill & {\mathbf{D}}_1=\mathbb{T}\left(\left(g_1^{\left(\gamma \right)},g_2^{\left(\gamma \right)},\dots ,g_{N_x-1}^{\left(\gamma \right)}\right),\left(g_1^{\left(\gamma \right)},g_0^{\left(\gamma \right)},0,\dots ,0\right)\right),\hfill \\ \hfill & \mathbf{D}={\mathbf{D}}_1+{\mathbf{D}}_1^{\top }.\hfill \end{array}$$

As a result, we arrive at the following matrix equation for the approximate solution of Equation (1) at the nth time layer:

$$\mathbf{A}{\mathsf{\Phi }}^n={\mathbf{b}}^n,$$

(13)

where ${\mathsf{\Phi }}^n\in {\mathbb{R}}^{{\left(N_x-1\right)}^2}$ is the desired solution vector, defined as

$${\mathsf{\Phi }}^n={\left({\varphi }_{1,1}^n,\dots ,{\varphi }_{N_x-1,1}^n,{\varphi }_{1,2}^n,\dots ,{\varphi }_{N_x-1,2},\dots ,{\varphi }_{1,N_x-1}^n,\dots ,{\varphi }_{N_x-1,N_x-1}^n\right)}^{\top },$$

and the matrix $\mathbf{A}\in {\mathbb{R}}^{\left(N_x-1\right)\times \left(N_x-1\right)}$ and the vector ${\mathbf{b}}^n\in {\mathbb{R}}^{{\left(N_x-1\right)}^2}$ are defined as

$$\begin{array}{cc}\hfill & \mathbf{A}={\delta }_{1,1}^{\left(\alpha \right)}\mathbb{I}+{\displaystyle \frac{{\varsigma }_{\beta }{\tau }^{\alpha }}{h^{\beta }}}\left(\left(\mathbb{I}\otimes {\mathbf{C}}^{+}\right){\mathbf{U}}_1^{n,+}-\left(\mathbb{I}\otimes {\mathbf{C}}^{-}\right){\mathbf{U}}_1^{n,-}\right)\hfill \\ \hfill & +{\displaystyle \frac{{\varsigma }_{\beta }{\tau }^{\alpha }}{h^{\beta }}}\left(\left({\mathbf{C}}^{+}\otimes \mathbb{I}\right){\mathbf{U}}_2^{n,+}-\left({\mathbf{C}}^{-}\otimes \mathbb{I}\right){\mathbf{U}}_2^{n,-}\right)-{\displaystyle \frac{\kappa {\tau }^{\alpha }}{h^{\gamma }}}\mathbf{D}\oplus \mathbf{D}+{\tau }^{\alpha }\mathbf{R},\hfill \\ \hfill & {\mathbf{b}}^n={\delta }_{1,1}^{\left(\alpha \right)}{\mathsf{\Phi }}^{n-1}-\sum _{s=1}^{n-1}{\delta }_{n,s}^{\left(\alpha \right)}\left({\mathsf{\Phi }}^s-{\mathsf{\Phi }}^{s-1}\right)+{\tau }^{\alpha }{\mathbf{F}}^n,\hfill \end{array}$$

where the properties of the coefficients ${\delta }_{n,s}^{\left(\alpha \right)}$ are used [30], $\mathbb{I}$ is the unit matrix, ⊗ stands for the Kronecker product, ⊕ denotes the Kronecker sum defined as $A\oplus B=\mathbb{I}\otimes B+A\otimes \mathbb{I}$, and $\mathbf{R}={\left(r\left({\widehat{\mathbf{x}}}_j\right)\right)}_{j=1}^{{\left(N_x-1\right)}^2}$. In addition, we use the upstream partition to discretize the convective term ${\mathcal{C}}^{\beta }\left(\mathbf{u}\left(\mathbf{x},t\right),\varphi \left(\mathbf{x},t,\omega ,\mathbf{p}\right)\right)$. Therefore, diagonal matrices ${\mathbf{U}}_i^{n,\pm }$ are introduced that are formed from the positive and negative parts of the velocity components $u_i$:

$$\begin{array}{cc}\hfill & {\mathbf{U}}_i^{n,+}=\mathrm{diag}{\left(max\left\{u_i\left({\widehat{\mathbf{x}}}_j,t_n\right),0\right\}\right)}_{j=1}^{{\left(N_x-1\right)}^2},\hfill \\ \hfill & {\mathbf{U}}_i^{n,-}=\mathrm{diag}{\left(min\left\{u_i\left({\widehat{\mathbf{x}}}_j,t_n\right),0\right\}\right)}_{j=1}^{{\left(N_x-1\right)}^2}.\hfill \end{array}$$

Let us now dwell on the implementation of the calculation of the right-hand-side vector ${\mathbf{F}}^n$. For each of the M simulated vehicles, we randomly assign a source–destination pair $\left(s_m,d_m\right)\in \mathcal{V}\times \mathcal{V}$, $m=1,2,\dots ,M$, and compute the shortest path ${\mathcal{P}}_m=\left(v_{m,0},v_{m,1},\dots ,v_{m,L_m}\right)$ using Dijkstra’s algorithm.

To model the continuous movement of a vehicle along a discrete route ${\mathcal{P}}_m$, we apply linear interpolation between its successive nodes. Assuming that $\Delta t_m$ is the total duration of the vehicle’s trip, we divide it into $L_m-1$ intervals $\left[t_{m,l},t_{m,l+1}\right]$, where the time of the vehicle’s appearance on the road network $t_{m,0}$ is a random variable, and the subsequent time is determined as

$$t_{m,l+1}=t_{m,l}+\Delta t_{m,l},$$

where $\Delta t_{m,l}$ proportional to the length of the corresponding segment. Then, the trajectory ${\mathbf{X}}_m\left(t_n,\omega \right)$ is linearly defined piecewise as

$${\mathbf{X}}_m\left(t_n,\omega \right)=\left(1-\theta \right)v_{m,l}+\theta v_{m,l+1},t\in \left[t_{m,l},t_{m,l+1}\right],\theta ={\displaystyle \frac{t_n-t_{m,l}}{\Delta t_{m,l}}}.$$

(14)

The pollutant emitted by a vehicle is assumed to be distributed among nearby grid nodes by a discrete approximation of the Dirac delta function defined by a hat function centered at the vehicle position ${\mathbf{X}}_m\left(t_n,\omega \right)$. Thus, the right-hand side function f is approximated as

$$f\left({\mathbf{x}}_{i,j},t_n,\omega ,\mathbf{p}\right)\approx f_{i,j}^n={\mathsf{\Theta }}_{\mathbf{p}}\left(t_n\right)\sum _{m=1}^M{\rho }_{m}exp\left(-{\displaystyle \frac{∥{\mathbf{x}}_{i,j}-{\mathbf{X}}_m\left(t_n,\omega \right)∥}{{\sigma }^2}}\right),$$

and ${\mathbf{F}}^n={\left(f_{i,j}^n\right)}_{j=1}^{{\left(N_x-1\right)}^2}$, and $\sigma$ is a normalization parameter.

To solve Equations (3) and (4) we use the well-known approximation approach on a staggered grid according to the following algorithm, which includes three computational steps [61]:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\mathbf{u}}_{i,j}^{n+\frac{1}{2}}-{\mathbf{u}}_{i,j}^n}{\tau }}=-L_h{\mathbf{u}}_{i,j}^n+{\mathsf{\Lambda }}_h{\mathbf{u}}_{i,j}^n,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & {\mathsf{\Lambda }}_h{\pi }_{i,j}^{n+1}=\frac{1}{\tau }{\nabla }_h·{\mathbf{u}}_{i,j}^{n+\frac{1}{2}},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\mathbf{u}}_{i,j}^{n+1}-{\mathbf{u}}_{i,j}^{n+\frac{1}{2}}}{\tau }}=-{\nabla }_h{\pi }_{i,j}^{n+1},\hfill \end{array}$$

(17)

where $L_h$ and ${\mathsf{\Lambda }}_h$ are difference analogues of convective and turbulent exchange operators, ${\nabla }_h$ is the discrete analogue of the del operator. The first computational step, expressed by Equation (15), takes into account only convective and turbulent exchange and calculates the intermediate velocity field ${\mathbf{u}}_{i,j}^{n+\frac{1}{2}}$. The second step (16) calculates the pressure field from the intermediate velocity field. In the third step (17), the velocity field is corrected taking into account the pressure gradient.

2.2.2. Method for Solving the Stochastic Forward Problem

Now let us fix the parameter vector $\mathbf{p}$. To solve the stochastic problem, we simulate $N_r$ independent realizations of $\omega \in \mathsf{\Omega }$ and solve Problem 1 for each ${\omega }_r$.

Note that the matrix $\mathbf{A}$ in (13) is the same for all samples ${\left\{{\omega }_r\right\}}_{r=1}^{N_r}$ within a single time layer, which provides two key advantages. Firstly, this allows for solving the stochastic problem once for all samples ${\left\{{\omega }_r\right\}}_{r=1}^{N_r}$ at once, rather than solving it separately for each sample [62]. This comes down to solving a large number of systems of linear equations with the same coefficient matrix and different right-hand sides. Secondly, it makes it possible to use efficient factorization methods [44]. For this reason, an incomplete LU factorization preconditioner [63] is used to speed up the calculations.

2.2.3. Method for Solving the Inverse Problem

The parameter vector $\mathbf{p}$ is identified through an iterative procedure based on the repeated solution of the forward problem and successive updates of the parameter values via an optimization algorithm. At each iteration j, the current estimate ${\mathbf{p}}_j$ is used to solve Equation (1) subject to the boundary and initial conditions given in Equations (6) and (8) and with the corresponding right-hand side $f\left(\mathbf{x},t,\omega ,{\mathbf{p}}_j\right)$. The numerical solution obtained for this parameter set is then evaluated at the monitoring-station locations, and its deviation from the observed data is quantified by the discrepancy functional (9). This discrepancy provides a scalar objective value that guides the parameter update step. This process is repeated iteratively, forming a sequence ${\left\{{\mathbf{p}}_j\right\}}_{j=0}^{\infty }$, until a predefined convergence criterion is satisfied. Upon convergence, the final parameter vector ${\mathbf{p}}^{*}$ is taken as the best-fit estimate with respect to the observed data and the given forward model.

In our previous paper [64], the applicability of a number of optimization algorithms, such as the conjugate gradient method [63], NEWUOA [65], BOBYQA [66], L-BFGS [67], and the Nelder–Mead method [68], was studied in detail. Based on numerous numerical tests, we came to the conclusion that the L-BFGS algorithm is more robust and allows identifying parameters in a wide class of functions and achieving greater accuracy. Moreover, its limited-memory structure makes it well-suited for high-dimensional problems, allowing for efficient handling of large parameter vectors without the need to store or invert full Hessian matrices. These advantages make L-BFGS particularly effective for the inverse problem considered in this study, where the parameter space is moderately large and the objective functional may be nonconvex.

Note that Equations (3) and (4) do not depend on the concentration field $\varphi$. This allows us to calculate the velocity field $\mathbf{u}$ and pressure $\pi$ on all time layers in advance and do not include this stage when solving the inverse problem.

2.2.4. Computational Algorithm

Thus, the proposed approach to solving the inverse Problem 2 consists of three stages, shown in Algorithm 1.

The algorithm takes the parameter L defining the domain, the final time T, the atmospheric diffusion coefficient $\kappa$, the Reynolds number $\mathrm{Re}$, the initial and boundary conditions (5)–(8), the orders of fractional derivatives $\alpha ,\beta ,\gamma$, the number of vehicles M, the number of parameters N, the number of realizations of the stochastic problem $N_r$, the accuracy parameter $\epsilon$ and the maximum number of iterations $J_{max}$ as input parameters. We also assume that each component of the vector ${\mathbf{p}}_j$ is bounded by positive numbers $p_{*}$ and $p^{*}$.

Next, the algorithm consists of three main stages. At the first stage, the Navier–Stokes equations are solved to determine the velocity field at all time layers. The second preparatory stage involves generating the stochastic input data necessary for simulating multiple realizations. Specifically, random traffic routes are constructed for each realization ${\omega }_r\in \mathsf{\Omega }$, representing the spatial and temporal distribution of moving vehicles throughout the domain. Each vehicle is assigned a pollution density ${\rho }_m^{\left(r\right)}$. Additionally, for each realization ${\omega }_r$, an initial guess for the unknown parameter vector ${\mathbf{p}}_0^{\left(r\right)}$ is randomly generated within the admissible parameter space. At the third stage, the iterative procedure described in Section 2.2.3 is carried out.

Algorithm 1 Algorithm for solving the inverse problem

Stage 1: Vector field forecast Solve Equations (3)–(5) and (7) using (15)–(17).   Stage 2: Preparatory stage for each realization ${\omega }_r,r\in \left\{1,2,\dots ,N_r\right\}$ do     for each vehicle $m\in \left\{1,2,\dots ,M\right\}$ do         Generate source-destination pair $\left(s_m^{\left(r\right)},d_m^{\left(r\right)}\right)$ of the route         Generate a route using Dijkstra’s algorithm         Generate pollution density ${\rho }_m^{\left(r\right)}$     end for     Assign initial approximations ${\mathbf{p}}_j^{\left(r\right)}$ to the desired coefficients end for   Stage 3: Main stage $j\leftarrow 0$ repeat     $j\leftarrow j+1$     Set the initial condition     for each time layer $t_n,n\in \left\{1,2,\dots ,N_t\right\}$ do         for each realization ${\omega }_r,r\in \left\{1,2,\dots ,N_r\right\}$ do            for each vehicle $m\in \left\{1,2,\dots ,M\right\}$ do                Calculate the vehicle position ${\mathbf{X}}_m\left(t_n,{\omega }_r\right)$ using (14)            end for            Calculate ${\mathsf{\Theta }}_{{\mathbf{p}}_j^{\left(r\right)}}\left(t_n\right)$ using current ${\mathbf{p}}_j^{\left(r\right)}$            Calculate the right-hand side $f\left(\mathbf{x},t_n,{\omega }_r,{\mathbf{p}}_j^{\left(r\right)}\right)$ for all $\mathbf{x}\in \widehat{\mathbf{x}}$            Update the historical part of the fractional derivative            Solve the system of linear Equation (13)         end for     end for     Calculate the mathematical expectation of the concentration     Calculate the functional $\mathcal{I}\left({\mathbf{p}}_j^{\left(r\right)}\right)$     Apply L-BFGS optimization algorithm to update coefficients until $\mathcal{I}\left({\mathbf{p}}_j^{\left(r\right)}\right)<\epsilon$ or $j\ge J_{max}$

3. Results

To analyze the model and computational algorithm for solving the inverse problem, a series of numerical tests has been carried out. All calculations were performed on a workstation with a 16-core AMD Ryzen 9 7950X3D processor with a clock frequency of 4.2 GHz, 128 GB of RAM, and a Gigabyte RTX 4090 graphics card with 24 GB of VRAM.

3.1. Analysis of the Forward Problem

3.1.1. Comparison of the Classical Convection–Diffusion Model with the Fractional Differential Model

In the first computational experiment, an analysis of the forward problem was carried out. The aim of this computational experiment was to analyze the influence of fractional derivatives on the process of pollutant propagation. We considered a model problem in a unit square on a road network, which is represented by a graph shown in Figure 1. The following model parameters were adopted in this computational experiment: $T=1$, $\kappa =0.1$. To focus on the influence of fractional derivatives, we chose a parameter vector in the form $\mathbf{p}=\left(0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5\right)$, which corresponds to the daily intensity profile of the pollutant sources, divided into eight three-hour intervals.

The classical case $\left(\alpha ,\beta ,\gamma \right)=\left(1.0,1.0,2.0\right)$, shown in Figure 3h, serves as a reference, giving a smooth, centrally sharp concentration plume that spreads symmetrically with moderate gradients towards the boundaries. Deviations from this integer-order scenario clearly arise as the orders of fractional derivatives decrease.

In cases where the temporal and spatial fractional orders are lower, the solution behavior reflects enhanced memory and reduced smoothing, which manifest as slower dissipation and more persistent pollutant presence at observation points. This aligns with known characteristics of anomalous transport processes: pollutants remain elevated longer and spread less uniformly, capturing realistic urban atmospheric effects, such as trapping in street canyons or delayed dispersion due to thermal inversions. Figure 3a–g clearly illustrate how variations in fractional orders shift not only the magnitude but also the temporal progression of concentration emphasizing that small deviations from integer-order behavior can lead to qualitatively different pollutant dynamics.

For example, when $\alpha$ decreases to 0.75, the concentration dynamics slow down significantly, reflecting memory effects in time dynamics. Compared to the classical case, the jets for lower $\alpha$ remain less developed at the same finite time, with lower-peak concentrations and smoother gradients. For example, comparing $(0.75,1.0,2.0)$ with $(1.0,1.0,2.0)$ shows that the former retains a broader, more diffuse appearance, indicating a slower accumulation of pollutants.

It can also be noted that decreasing $\beta$ from 1.0 to 0.75 reduces the intensity of convective transport. While the classic plume exhibits a pronounced directional tilt due to convection, a lower $\beta$ value yields a more symmetrical distribution. For example, the shape of the plume at $(1.0,0.75,2.0)$ appears more round and centered compared to the elongated structure in the classical case $(1.0,1.0,2.0)$, which emphasizes the weakening of the directional drift.

Further, reducing $\gamma$ from 2.0 to 1.75 shifts the system towards subdiffusion. This change limits spatial distribution and increases peak concentrations. For example, in the case of $(1.0,1.0,1.75)$, the plume is more compact and maintains higher values in the center compared to the smoother, wider plume in the classical case. Thus, a lower $\gamma$ concentrates the mass closer to the source, reflecting difficult scattering.

The cumulative effect of multiple order declines is particularly significant. In cases such as $(0.75,0.75,1.75)$, highly localized plumes are observed with a noticeably smaller spatial spread and slower development compared to $(1.0,1.0,2.0)$. In contrast, partial reductions, such as $(0.75,1.0,2.0)$ or $(1.0,0.75,1.75)$, exhibit intermediate behavior: slower time evolution, reduced convection, or sharper peaks, depending on which order is decreasing.

3.1.2. Observation of Changes in Concentration at Stations: Transition from the Classical Model to the Fractional-Order Model

To evaluate the influence of integer-order and near-integer order derivatives on pollutant dispersion, we solved the forward problem for seven combinations of parameters $\left(\alpha ,\beta ,\gamma \right)$, where $\alpha \in \left\{0.95,1\right\}$, $\beta \in \left\{0.95,1\right\},$ and $\gamma \in \left\{1.95,2\right\}$. The change in concentration was monitored at four spatial points, $\left(0.1,0.1\right)$, $\left(0.4,0.4\right)$, $\left(0.4,0.7\right)$, and $\left(0.8,0.5\right)$, where stations were installed. The reason for choosing these points for analysis is that the first and fourth points are boundary points, the second is internal, and the third is external.

As can be seen from Figure 4, in all cases, the order $\alpha$ of the fractional time derivative had the greatest impact on concentration levels. Even a moderate decrease in $\alpha$ from 1 to 0.95 resulted in a significant increase in pollutant concentrations, indicating that the transition from fractional to integer time dynamics significantly accelerates pollutant accumulation. Integer-order time derivatives resulted in a two-fold increase in peak concentrations compared to fractional-order cases, especially in the inner region.

Spatial fractional orders $\beta$ and $\gamma$ also influenced concentration profiles, but to a lesser extent. Higher $\beta$ and $\gamma$ values tend to reduce peak concentrations, reflecting increased diffusive transport that more efficiently disperses pollutants. The differences between $\gamma =1.95$ and $\gamma =2$ were small but consistent, with slightly lower concentrations for the integer case $\gamma =2$. Changes in $\beta$ showed similar trends, with $\beta =1$ resulting in slightly less pollutant retention than $\beta =0.95$.

At the boundary point $\left(0.1,0.1\right)$, concentrations generally remained low, although integer-order time derivatives still yielded substantially higher values than fractional cases, demonstrating a persistent $\alpha$ influence even near the domain edges. In the interior regions, such as at $\left(0.4,0.4\right)$ and $\left(0.4,0.7\right)$, the combined effects of $\alpha$, $\beta$, and $\gamma$ become more pronounced, with higher $\alpha$ values and lower spatial orders leading to greater pollutant accumulation. Further from the source, as at $\left(0.8,0.5\right)$, the influence of spatial fractional orders remained evident but was comparatively diminished, while $\alpha$ continued to dominate the concentration levels.

Note that the fluctuations observed in the concentration curves at low fractional orders arise because the fractional derivatives impose a strong memory and non-local transport effect that reduces the natural damping present in classical diffusion. As a result, even small physical or numerical perturbations can persist and propagate, leading to visible temporary fluctuations in predicted concentrations.

3.1.3. Analysis of the Influence of Lower Orders of Fractional Derivative on the Pollutant Dispersion Process

To investigate how lower fractional orders affect pollutant dispersion, we analyzed concentrations at four spatial points for eight combinations of $\left(\alpha ,\beta ,\gamma \right)$ with reduced orders compared to the classical case $(1.0,1.0,2.0)$ and the previously studied slightly fractional case $(0.95,0.95,1.95)$.

It can be seen from Figure 5 that further reduction of the orders of the time derivative $\alpha$ to 0.75 and the spatial derivatives $\beta$ and $\gamma$ to 0.75 and 1.75, respectively, significantly slows down the dispersion process and leads to significantly higher concentrations of pollutants that persist over time.

At all four observation points, a lower $\alpha$ causes a more pronounced memory effect, resulting in concentrations initially rising faster and remaining elevated longer, which is particularly noticeable in the curves for $\alpha =0.75$. Reducing $\gamma$ from 1.95 to 1.75 reduces diffusion, further enhancing the retention of pollutant mass around the source. This is particularly noticeable at points close to the origin, such as $(0.1,0.1)$, where peak concentrations exceed the values observed for $\alpha =0.9$ and $\gamma =1.9$, confirming weaker scattering. Meanwhile, decreasing $\beta$ reduces the effect of convection, slightly delaying the outward transport of the pollutant, resulting in more localized peaks and slower propagation, especially at the periphery, for example, $(0.8,0.5)$.

Overall, compared to the near-classical case $(0.95,0.95,1.95)$, further decreasing orders leads to stronger localization of the pollutant plume and higher residual concentrations at all points, emphasizing that the choice of low fractional orders significantly changes both the dynamics and the possible spread of pollutants in the atmosphere.

Note that low fractional orders (especially $\alpha <0.5$) lead to unrealistic concentrations of pollutants and excessive spatial distribution, which contradicts the physical processes of turbulent diffusion. Therefore, such low orders are not suitable for modeling the dispersion of atmospheric impurities within the framework of this model.

3.1.4. Influence of Parameters on the Spread of a Pollutant

To analyze the influence of the parameters $\mathbf{p}$ on the spread of the pollutant, a series of numerical experiments were carried out. We limited ourselves to considering two parameter vectors, ${\mathbf{p}}_1=\left(0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5\right)$ and ${\mathbf{p}}_2=\left(0.1,0.05,0.3,0.7,0.9,0.5,0.95,0.6\right)$, that define the daily intensity profile of pollutant sources, divided into eight three-hour intervals.

It follows from Figure 6a that in the classical convection–diffusion equation $\left(\alpha ,\beta ,\gamma \right)=\left(1.0,1.0,2.0\right)$ the uniform vector ${\mathbf{p}}_1$ leads to a spatially symmetric concentration distribution with a maximum at the center of the domain and relatively uniform gradients. On the contrary, the use of a non-uniform vector ${\mathbf{p}}_2$ (Figure 6b) induces pronounced spatial inhomogeneities: the maximum concentration shifts to the region corresponding to the total contribution of time intervals with increased emission intensity values, forming local peaks and sharp gradients.

For a fractional differential equation $\left(\alpha ,\beta ,\gamma \right)=\left(0.75,0.75,1.75\right)$ (Figure 6c,d), a characteristic smoothing of the spatial distribution of concentration is observed due to the memory effect and subdiffusion/superdiffusion behavior: maximum concentration values decrease, local extrema are leveled out, and the distribution takes on a more diffuse shape. Thus, fractional derivatives reduce the sensitivity of the solution to daily fluctuations in emission intensity, whereas in the classical formulation, fluctuations in the parameters $\mathbf{p}$ are directly reflected in the spatial structure of the concentration field.

3.2. Stability and Convergence of the Numerical Method

The following numerical tests were aimed at determining the convergence order of the numerical method (13). Since the exact solution is unknown, the empirical convergence order was estimated using a self-consistency method based on successive mesh refinement. Numerical solutions were calculated on five grids with steps $\tau =1/50$, $1/100$, $1/200$, $1/400$, $1/800$ at a fixed spatial step $h=1/1000$. Next, the empirical convergence order was calculated using the following formula:

$$\mathrm{Order}={\displaystyle \frac{1}{log2}}log\left({\displaystyle \frac{{max}_n{∥{\varphi }_{\tau }-{\varphi }_{\tau /2}∥}_{L^2\left({\mathcal{D}}_h\right)}}{{max}_n{∥{\varphi }_{\tau /2}-{\varphi }_{\tau /4}∥}_{L^2\left({\mathcal{D}}_h\right)}}}\right).$$

To determine the convergence order with respect to the spatial variable, a similar technique was used with a fixed time step $\tau =1/1000$.

We limited ourselves to considering one case of orders of fractional derivatives, $\alpha =0.8$, $\beta =0.8$, $\gamma =1.8$. Due to the use of an approximation Formula (10) of order $O\left({\tau }^{2-\alpha }\right)$, the expected convergence order in the temporal variable was 1.2, and due to the use of an approximation Formula (11) of order $O\left(h\right)$, the expected convergence order in the spatial variable was 1.

The results of the computational experiments presented in

Table 1. Convergence analysis of the numerical scheme.

$\mathit{\tau }$	${\mathit{L}}^2$-Error	Order	h	${\mathit{L}}^2$-Error	Order
1/100	$2.1280\times {10}^{-3}$	-	1/20	$1.0374\times {10}^{-2}$	-
1/200	$9.0093\times {10}^{-4}$	1.24	1/40	$5.2593\times {10}^{-3}$	0.98
1/400	$3.8675\times {10}^{-4}$	1.22	1/80	$2.6479\times {10}^{-3}$	0.99
1/800	$1.6718\times {10}^{-4}$	1.21	1/160	$1.3332\times {10}^{-3}$	0.99

are in good agreement with theoretical assumptions.

Since the scheme is fully implicit, it was expected to be unconditionally stable, and no CFL-type coupling between the time step $\tau$ and the grid size h was required. In practice, our calculations were stable over a wide range of $\tau$ and h. We did not observe any unphysical growth or oscillations in all the numerical tests we performed.

3.3. Evaluation of the Efficiency of the Algorithm for Solving the Inverse Problem

To evaluate the efficiency and reliability of the proposed method for solving the inverse problem, we conducted a series of numerical experiments. We generated 1000 random samples of the true parameter vector governing the right-hand side of the convection–diffusion equation modeling the dispersion of pollutants in the atmosphere. For each sample, we solved the forward problem and then applied our iterative inverse method, initialized based on a common initial guess, to infer the unknown coefficients. In this experiment, the vector $\mathbf{p}$ consisted of six unknown coefficients, and the vector ${\mathbf{p}}_0=\left(0.5,0.5,\dots ,0.5\right)$ was chosen as the initial guess.

Our analysis focused on four key performance indicators.

3.3.1. Convergence Behavior

We recorded the number of iterations required for the L-BFGS algorithm to reach convergence for each of the 1000 inverse problems. The histogram shown in Figure 7a displays the frequency distribution of the number of iterations. In most cases, convergence was achieved within 1500–2500 iterations, indicating stable and efficient convergence. Only in a small proportion of cases did the number of iterations exceed 3000, suggesting sometimes slow convergence, possibly due to unfavorable initial approximations or poor conditioning in certain parameter regions.

3.3.2. Accuracy of the Derived Coefficients

To assess the accuracy of the inferred parameters, we calculated the absolute and relative errors between the inferred coefficients and the corresponding true values for each sample. The histogram of relative errors (Figure 7b) shows that for most experiments the absolute error remained below $1\times {10}^{-8}$, demonstrating the good ability of the algorithm to recover parameters. A small number of outliers with higher errors were found, which may indicate local minima or insensitivity of the forward model to certain parameter changes.

3.3.3. Scatter Plot of Actual and Expected Coefficients

Figure 7c shows a scatter plot comparing the true parameter values with the corresponding predicted values for all 1000 samples. Ideally, the points should lie along the identity line $y=x$, indicating perfect parameter recovery. The graph shows a strong linear correlation, confirming that the inverse method reliably recovers the true coefficients in most cases. Deviations from the diagonal highlight the magnitude of the inference errors and reveal areas where the method could benefit from further regularization or more informative initial guesses.

3.3.4. Violin Plot of Accuracy Depending on the Parameter

To examine the variability of inference accuracy under different unknown parameters, we constructed violin plots of the relative errors in estimating each of the six unknown coefficients. The violin plots shown in Figure 7d, illustrate both the density distribution and the error spread for each parameter.

The graphs show that while most parameters are derived with consistently low relative errors, some parameters exhibit wider distributions, indicating higher uncertainty or variability in the estimation process. For example, the first parameters exhibit a somewhat wider distribution, suggesting that they may be less identifiable from the available data or more sensitive to initial guesses and measurement noise.

3.4. Sensitivity to Parameters Initialization

To investigate the sensitivity of the solution of the inverse problem to the choice of initial approximations, we conducted an additional series of experiments. In this case, we fixed a set of true parameters $\mathbf{p}$, shown in

Table 2. The set of parameters used in the computational experiment of Section 3.4.

Parameter No.	Value	Parameter No.	Value
1	0.070064190535697	4	0.1963325132242003
2	0.278814378576473	5	0.4738792967134109
3	0.699921731580088	6	0.8701213766070134

and defining the right-hand side of the convection–diffusion equation, and solved the inverse problem 100 times, each time starting from a random initial guess. Our analysis included three key assessments.

3.4.1. Histogram of the Number of Iterations

Figure 8a shows a histogram showing the number of iterations required to converge over 100 runs. The distribution shows that, while most runs converge in the same number of iterations, in some cases, significantly more iterations are required to achieve convergence. This suggests that while the algorithm is generally stable, some initial guesses may lead to longer optimization paths, possibly due to proximity to local minima or flat regions in the loss landscape.

3.4.2. Histogram of Inference Accuracy

Figure 8b shows a histogram of the relative errors between the inferred parameters and the fixed true parameters. Most runs achieve low levels of inference errors, indicating the ability of the method to recover the true parameters despite changing initial conditions.

3.4.3. Violin Plot of Parameter-Specific Accuracy

To investigate how sensitivity varies with different parameters, we plotted relative errors for each of the six unknown coefficients (Figure 8c). The violin plots show that while some parameters are consistently well recovered regardless of the starting point, others exhibit a wider distribution, indicating greater sensitivity to the initial guess.

4. Discussion

Let us make a few comments regarding the obtained results in the context of previous studies. First of all, our results obtained in Section 3.1 have noticeable similarities with the results of fractional modeling of transport through porous media [30,69,70], where fractional derivatives have been successfully used to account for anomalous diffusion, memory effects, and non-local transport. Just as porous structures result in long residence times and heavy tails in concentration profiles [30], urban atmospheres, characterized by complex geometries, flow barriers, and heterogeneous sources, exhibit slow dispersion and non-Gaussian spread of pollutants. In both cases, the Caputo time derivatives model memory-induced delays in response to sources, while the Riesz spatial derivatives reflect non-local spreading beyond what classical Fickian diffusion can represent.

However, a key difference is that in porous media flow paths are typically static and tortuous, whereas in urban atmospheres pollutant sources are mobile and emission patterns change over time. Such dynamic behavior, in particular the use of time-varying line sources associated with moving vehicles, has been largely absent from existing fractional transport models. Our work extends the applicability of fractional PDEs by including stochastic and time-modulated emissions, providing a more realistic basis for predicting urban air quality.

Our numerical results are consistent with previous studies, demonstrating that fractional derivatives offer a more flexible and accurate framework for modeling anomalous transport phenomena. Similar to the results of porous media flow studies (for example, [71]), we observe that the reduction in time order $\alpha$ leads to memory effects that delay pollutant dispersion and result in maintenance of concentration levels, consistent with the results of [58,72,73]. Similarly, decreasing the spatial orders of $\beta$ and $\gamma$ leads to a non-Gaussian spread and heavier concentration tails, consistent with observations in fractional atmospheric diffusion models [74]. However, unlike most previous studies that assume fixed or point sources of pollutants, our work includes dynamic line sources associated with vehicle traffic, resulting in more realistic and spatially heterogeneous concentration models.

While some earlier works have used fractional models exclusively in forward modeling, our study considers the inverse problem for reconstructing time-dependent emission profiles, an aspect that has been largely unexplored in the context of fractional PDEs for urban air pollution. These differences highlight the novelty of our approach in both source modeling and the integration of fractional dynamics with parameter inference.

Traditional inverse modeling of air pollutant emissions has largely relied on adjoint-based variational methods [75,76], ensemble Kalman filters [77], and Bayesian inference frameworks [78]. These methods are well-suited for large-scale or regional applications where Gaussian assumptions and integer-order transport models are appropriate. However, they require complex adjoint solvers or large ensemble sizes and often assume Markovian dynamics. In contrast, our approach integrates optimization-based parameter estimation with a fractional-order convection–diffusion-reaction model, enabling us to capture nonlocal transport effects and temporal memory without relying on detailed adjoints or covariance tuning.

Compared with previous studies, our approach to solving the inverse problem has a number of clear advantages. Although many studies focus on source identification within classical diffusion models (for example, [79]), they typically assume static or point sources and often use simplified time profiles. In contrast, our method successfully reconstructs the time-varying radiation intensity associated with moving line sources under the additional complexity of fractional-order dynamics. This situation introduces non-locality both in space and time, which makes the inverse problem more difficult. Despite this, our results show that the parameters governing the time-modulated emissions can still be recovered with good accuracy from sparse sensor data, highlighting the robustness of the proposed inference framework. To our knowledge, inverse modeling for such systems, combining fractional convection–diffusion equations with stochastic vehicle emissions, has not been considered in previous studies.

5. Conclusions

In conclusion, our study shows that a fractional-order convection–diffusion-reaction formulation with Caputo temporal memory and Riesz spatial nonlocality provides a physically interpretable representation of anomalous dispersion, heavy tails, and persistence in urban flows, features under-represented by classical integer-order convection–diffusion-reaction models. Our numerical results indicate that the fractional model captures long-range transport and persistent pollutant effects more effectively than the classical model, particularly near sources and at downwind receptors. Although our method is more computationally expensive compared to the integer-order model due to the need to discretize temporal Caputo fractional derivative, which makes identifiability more sensitive to sensor density and placement, it reduces structural model error and improves agreement with observations, yielding more reliable emission-activity estimates. By estimating time-varying emission activity via regularized optimization, we isolate the role of the forward physics and avoid the need to develop an adjoint for nonlocal, history-dependent operators. Overall, our main contributions are a systematic analysis of how fractional orders shape plume propagation and residual structure, and evidence that a compact time–activity parameterization suffices to reconcile observations at city scale.

Our research also contributes to several United Nations Sustainable Development Goals, particularly Goal 3 (Good Health and Well-Being), Goal 11 (Sustainable Cities and Communities), and Goal 13 (Climate Action), by advancing modeling techniques that can support urban air quality management and inform pollution mitigation strategies.

Although this study focuses on synthetic numerical experiments, future work will include validation of the proposed model using real-world data, such as urban air quality measurements and traffic flow records. This will enable a more rigorous assessment of predictive accuracy and parameter identifiability under realistic conditions. The authors also plan to combine the results of this work with the results of a previous paper [64], in which machine learning algorithms were used to predict concentrations at monitoring stations. It is expected that this approach will allow for achieving even more accurate results to solve the inverse problem under consideration. Further, the authors intend to explore the use of physics-informed neural networks [80] as an alternative framework for inferring model parameters, with the goal of improving scalability, accommodating sparse or noisy data, and enabling more flexible representations of time-varying emission activity.