Two-Dimensional Time Fractional River-Pollution Model and Its Remediation by Unsteady Aeration

Abstract

This study contains a mathematical model for river pollution and its remediation for an unsteady state and investigates the effect of aeration on the degradation of pollutants. The governing equation is a pair of nonlinear time-fractional two-dimensional advection-diffusion equations for pollutant and dissolved oxygen (DO) concentration. The coupling of these equations arises due to the chemical interactions between oxygen and pollutants, forming harmless chemicals. The Fractional Reduced Differential Transform Method (FRDTM) is applied to provide approximate solutions for the given model. Also, the convergence of solutions is checked for efficacy and accuracy. The effect of longitudinal and transverse diffusion coefficients of pollutant and DO on the concentration of pollutant and DO is analyzed numerically and graphically. Also, we checked the effect of change in the river’s longitudinal and transverse seepage velocity on pollutant and DO concentration numerically and graphically. We analyzed the comparison of change in the value of half-saturated oxygen demand concentration for pollutant decay on pollutant and DO concentration numerically and graphically. Also, numerical and graphical analysis examined the effect of fractional parameters on pollution levels.

1. Introduction

Several factors must be considered when evaluating river water quality, including the concentration of dissolved oxygen, the presence of nitrates, chlorides, and phosphates, the amount of suspended solids, the occurrence of environmental hormones, the chemical oxygen demand, heavy metals, and the presence of bacteria. Pollutants originating from agricultural activities can be considered a substantial factor in the degradation of surface and groundwater quality. The advection-dispersion equations are relevant in several fields, such as water pollution, groundwater hydrology, chemical engineering, biosciences, environmental sciences, and petroleum engineering [1]. Huang et al. built a mathematical model to study the movement of solutes in porous materials with varying properties, considering the effect of scale-dependent dispersion. They hypothesized that the dispersion coefficient is directly proportional to the velocity of the pore water, having a linear relationship [2]. Pimpunchat et al. developed a fundamental mathematical model to study river pollution and examined the effects of aeration on reducing pollutants [3].

Hussain et al. examined a mathematical model that enables estimating river-pollutant concentration levels for the steady-state in a one-dimensional region without dispersion. The concentrations of pollutants and DO are determined by a set of coupled equations that include the processes of reaction, diffusion, and advection [4]. Ibrahim et al. analyzed a study on river-pollution cleanup. They examined the use of unstable aeration with arbitrary initial and boundary conditions [5]. On a finite domain, Liu et al. solved a space-time fractional advection-dispersion equation using an implicit difference method (IDM) and an explicit difference method (EDM). They also verified the convergence and stability of the methods [6]. Attia et al. used a numerical approach to find an approximate solution of the time-fractional advection–diffusion equation (FADE) under the Atangana–Baleanu derivative in Caputo sense (ABC) with Mittag–Leffler kernel by using reproducing kernel Hilbert space method (RKHSM) [7].

Mardani et al. used a very effective and precise meshless technique to solve the time fractional advection-diffusion equation with variable coefficients. This approach is based on the moving least square (MLS) approximation [8]. Maisuria et al. [9] applied the fractional reduced differential transform method (FRDTM) to derive a solution for the time-fractional two-dimensional advection-dispersion equation. The equation considered variable dispersion coefficients, velocities, decay constant of the first order, production rate coefficient for the solute at the zero-order level, and retardation factor. Fractional-order models provide more versatility in describing complex dynamic phenomena because of their continuous nature which can more accurately represent real-world dynamics. Nonlinear fractional partial differential equations are often more challenging to solve than linear ones. Furthermore, no technique provides a precise solution for the fractional-order differential equation. Therefore, the presence of fractional-order nonlinear PDEs improves the significance of the study. They can provide more precise analyses of systems that possess memory and hereditary features. Fractional-order models explicitly include non-local memory effects, while integer-order models assume just local memory. Nevertheless, fractional-order models have some disadvantages. They may possess more mathematical complexity and provide a more demanding analysis compared to models based on integer order. Solving fractional-order differential equations needs the use of specific methods.

In this work, we have used the latest method available, the FRDTM—an effective and reliable approach to solving the coupled non-linear time-fractional two-dimensional advection-diffusion equations that characterize the concentration of dissolved oxygen (DO) and the amount of pollution. For nonlinear time-fractional differential equations, the FRDTM produces very accurate numerical answers without needing spatial discretization, linearization, transformation, or perturbation. The originality of our work is that the FRDTM can be used to generate a wide range of coupled fractional-order nonlinear equations with different physical structures that arise in science.

The structure of the manuscript is as follows. Section 2 presents the mathematical formulation of the problem. In Section 3, the core concepts of the fractional reduced differential transform method and the associated convergence analysis are introduced. Section 4 provides a detailed discussion of the numerical results, including three-dimensional graphical comparisons and tabulated data. Finally, Section 5 offers a summary of the conclusions.

2. Mathematical Formulation of the Problem

In this study, a two-dimensional mathematical model is discussed to represent turbulent river flow. The advection-diffusion equation can be the best fit to approximate river pollution. The cross-sectional area of river is assumed to be constant. The two-dimensional nature of the unstable flow in the river defined by distances $x^{*}\left(m\right)$ and $y^{*}\left(m\right)$ measured from the origin with $x^{*}=0$ and $y^{*}=0$ have been considered for the present study. The governing equations have been described as follow [1,10,11]:

$${\displaystyle \frac{\partial \left(A^{*}{p}^{*}\right)}{\partial t^{*}}}=D_{p_x}^{*}{\displaystyle \frac{{\partial }^2\left(A^{*}{p}^{*}\right)}{\partial x^{{*}^2}}}+D_{p_y}^{*}{\displaystyle \frac{{\partial }^2\left(A^{*}{p}^{*}\right)}{\partial y^{{*}^2}}}-{\displaystyle \frac{\partial \left(u_x^{*}{A}^{*}{p}^{*}\right)}{\partial x^{*}}}-{\displaystyle \frac{\partial \left(u_y^{*}{A}^{*}{p}^{*}\right)}{\partial y^{*}}}-k_1^{*}{\displaystyle \frac{q^{*}}{q^{*}+k^{*}}}{A}^{*}{p}^{*}+f^{*}$$

(1)

$${\displaystyle \frac{\partial \left(A^{*}{q}^{*}\right)}{\partial t^{*}}}=D_{q_x}^{*}{\displaystyle \frac{{\partial }^2\left(A^{*}{q}^{*}\right)}{\partial x^{{*}^2}}}+D_{q_y}^{*}{\displaystyle \frac{{\partial }^2\left(A^{*}{q}^{*}\right)}{\partial y^{{*}^2}}}-{\displaystyle \frac{\partial \left(u_x^{*}{A}^{*}{q}^{*}\right)}{\partial x^{*}}}-{\displaystyle \frac{\partial \left(u_y^{*}{A}^{*}{q}^{*}\right)}{\partial y^{*}}}-k_2^{*}{\displaystyle \frac{q^{*}}{q^{*}+k^{*}}}{A}^{*}{p}^{*}+{\beta }^{*}\left(K^{*}-q^{*}\right)$$

(2)

where $0\le x^{*}\le L^{*}$, $0\le y^{*}\le W^{*}$, $t^{*}\ge 0$

Let us consider the following non-dimensional variables and parameters [1,12]

$$\begin{array}{c}x={\displaystyle \frac{x^{*}}{L^{*}}}, y={\displaystyle \frac{y^{*}}{L^{*}}}, t=k_2^{*}{t}^{*}, p={\displaystyle \frac{p^{*}}{p_0^{*}}}, q={\displaystyle \frac{q^{*}}{q_0^{*}}}, W={\displaystyle \frac{W^{*}}{L^{*}}}, D_{p_x}={\displaystyle \frac{D_{p_x}^{*}}{L^{{*}^2}{k}_1^{*}}}, D_{p_y}={\displaystyle \frac{D_{p_y}^{*}}{L^{{*}^2}{k}_1^{*}}}{u}_x={\displaystyle \frac{u_x^{*}}{L^{*}{k}_1^{*}}}, u_y={\displaystyle \frac{u_y^{*}}{L^{*}{k}_1^{*}}}, k={\displaystyle \frac{k^{*}}{q_0^{*}}}, f={\displaystyle \frac{f^{*}}{A^{*}{k}_1^{*}{p}_0^{*}}}, \hfill \\ D_{q_x}={\displaystyle \frac{D_{q_x}^{*}}{L^{{*}^2}{k}_1^{*}}}, D_{q_y}={\displaystyle \frac{D_{q_y}^{*}}{L^{{*}^2}{k}_1^{*}}}, k_3={\displaystyle \frac{k_2^{*}}{k_1^{*}}}, \beta ={\displaystyle \frac{{\beta }^{*}}{A^{*}{k}_1^{*}}}, K={\displaystyle \frac{K^{*}}{q_0^{*}}}\hfill \end{array}$$

Hence, governing Equations (1) and (2) have been converted in the given form:

$${\displaystyle \frac{\partial p}{\partial t}}=D_{p_x}{\displaystyle \frac{{\partial }^{2}p}{\partial x^2}}+D_{p_y}{\displaystyle \frac{{\partial }^{2}p}{\partial y^2}}-{\displaystyle \frac{\partial \left(u_{x}p\right)}{\partial x}}-{\displaystyle \frac{\partial \left(u_{y}p\right)}{\partial y}}-{\displaystyle \frac{q}{q+k}}p+f$$

(3)

$${\displaystyle \frac{\partial q}{\partial t}}=D_{q_x}{\displaystyle \frac{{\partial }^{2}q}{\partial x^2}}+D_{q_y}{\displaystyle \frac{{\partial }^{2}q}{\partial y^2}}-{\displaystyle \frac{\partial \left(u_{x}q\right)}{\partial x}}-{\displaystyle \frac{\partial \left(u_{y}q\right)}{\partial y}}-k_3{\displaystyle \frac{q}{q+k}}p+\beta \left(K-q\right)$$

(4)

where $0\le x\le 1$, $0\le y\le W$, $t\ge 0$

The initial conditions of Equations (3) and (4) are [12,13]:

$$p(x,y,0)=C_1{e}^{-\left(x+y\sqrt{{\displaystyle \frac{D_{p_y}}{D_{p_x}}}}\right)}$$

(5)

$$q(x,y,0)=C_2{e}^{-\left(x+y\sqrt{{\displaystyle \frac{D_{q_y}}{D_{q_x}}}}\right)}$$

(6)

3. Fractional Reduced Differential Transform Method (FRDTM)

This section contains an explanation of several fundamental concepts and properties of fractional calculus theory [14,15].

Definition 1.

The fractional derivative of the function $m\in {\mathbb{C}}_{\mu }$ of order $\alpha \ge 0$, in the sense of Caputo, is denoted by $D^{\alpha }m\left(\tau \right)$ and is defined as

$$\begin{array}{c}\hfill D^{\alpha }m\left(\tau \right)={\displaystyle \frac{1}{\mathsf{\Gamma }(k-\alpha )}}{\int }_0^1(\tau -t)m^k\left(t\right)dt,\\ \hfill fork-1<\alpha \le k,k\in \mathbb{N},\tau >0,m\in {\mathbb{C}}_{\mu }^k,\mu \ge -1\end{array}$$

(7)

Let $\psi (\phi ,\varkappa ,\tau )$ be a function of three variables, which can be expressed as $\psi (\phi ,\varkappa ,\tau )=a\left(\phi \right)b\left(\varkappa \right)c\left(\tau \right)$. The differential transform’s characteristics can be used to determine the representation of the function $\psi (\phi ,\varkappa ,\tau )$.

$$\psi (\phi ,\varkappa ,\tau )=\sum _{k=0}^{\infty }{\Psi }_k(\phi ,\varkappa ){\tau }^{\alpha k}$$

(8)

where ${\Psi }_k(\phi ,\varkappa )$ is a t-dimensional spectrum function. The transformed function of $\psi (\phi ,\varkappa ,\tau )$ is ${\Psi }_k(\phi ,\varkappa )$. The fundamental concepts and operations of FRDTM are as outlined below:

The operator $R_D$ represents the reduced differential transform, whereas $R_D^{-1}$ represents the inverse reduced differential transform.

Definition 2.

Assuming that $\psi (\phi ,\varkappa ,\tau )$ is analytic and continuous differentiable with respect to φ, ϰ, and τ inside the domain of interest. The reduced differential transformations for the function $\psi (\phi ,\varkappa ,\tau )$ is defined as follows [16,17]:

$$R_D\left[\psi (\phi ,\varkappa ,\tau )\right]\approx {\Psi }_k(\phi ,\varkappa )={\displaystyle \frac{1}{\mathsf{\Gamma }(\alpha k+1)}}{\left[{\displaystyle \frac{{\partial }^{\alpha k}}{\partial {\tau }^{\alpha k}}}\psi (\phi ,\varkappa ,\tau )\right]}_{\tau =0}$$

(9)

The time-fractional order derivative is represented by the symbol α.

Definition 3.

The inverse differential transformations of the function ${\Psi }_k(\phi ,\varkappa )$ is presented as:

$$R_D^{-1}\left[{\Psi }_k(\phi ,\varkappa )\right]\approx \psi (\phi ,\varkappa ,\tau )=\sum _{k=0}^{\infty }{\Psi }_k(\phi ,\varkappa ){\tau }^{\alpha k}$$

(10)

From Equations (9) and (10), we obtain

$$\psi (\phi ,\varkappa ,\tau )=\sum _{k=0}^{\infty }{\displaystyle \frac{1}{\mathsf{\Gamma }(\alpha k+1)}}{\left[{\displaystyle \frac{{\partial }^{\alpha k}}{\partial {\tau }^{\alpha k}}}\psi (\phi ,\varkappa ,\tau )\right]}_{\tau =0}{\tau }^{\alpha k}$$

(11)

We examine the general fractional non-linear non-homogeneous partial differential equation in order to understand the operation of the FRDTM [18].

$$M\psi (\phi ,\varkappa ,\tau )+A\psi (\phi ,\varkappa ,\tau )+N\psi (\phi ,\varkappa ,\tau )=g(\phi ,\varkappa ,\tau )$$

(12)

with initial condition

$$\psi (\phi ,\varkappa ,0)=\Xi (\phi ,\varkappa )$$

(13)

where $M={\displaystyle \frac{{\partial }^{\alpha }}{\partial {\tau }^{\alpha }}}$, the linear operator A has partial derivatives, whereas the operator $N\psi (\phi ,\varkappa ,\tau )$ is non-linear, and $g(\phi ,\varkappa ,\tau )$ represents an inhomogeneous term.

We can derive the following iteration formula from the concepts of the FRDTM:

$${\displaystyle \frac{\mathsf{\Gamma }\left(\alpha \right(k+1)+1)}{\mathsf{\Gamma }(\alpha k+1)}}{\Psi }_{k+1}(\phi ,\varkappa )=G_k(\phi ,\varkappa )-A{\Psi }_k(\phi ,\varkappa )-N{\Psi }_k(\phi ,\varkappa )$$

(14)

The functions ${\Psi }_k(\phi ,\varkappa )$, $A{\Psi }_k(\phi ,\varkappa )$, $N{\Psi }_k(\phi ,\varkappa )$, and $G_k(\phi ,\varkappa )$ represent the transformations functions of the $\psi (\phi ,\varkappa ,\tau )$, $A\psi (\phi ,\varkappa ,\tau )$, $N\psi (\phi ,\varkappa ,\tau )$, and $g(\phi ,\varkappa ,\tau )$ functions, respectively.

From the initial condition (13), we obtain

$${\Psi }_0(\phi ,\varkappa )=\Xi (\phi ,\varkappa )$$

(15)

By substituting Equation (15) into Equation (14) and carrying out a basic iterative computation, we obtain the values of ${\Psi }_k(\phi ,\varkappa )$ as follows. Subsequently, the inverse translation of the set of values ${\left\{{\Psi }_k(\phi ,\varkappa )\right\}}_{k=0}^n$ yields the n-terms approximation solution in the following manner.

$${\psi }_n(\phi ,\varkappa ,\tau )=\sum _{k=0}^n{\Psi }_k(\phi ,\varkappa ){\tau }^{\alpha k}$$

(16)

Therefore, the exact answer is provided by

$$\psi (\phi ,\varkappa ,\tau )=\underset{n\to \infty }{lim}{\psi }_n(\phi ,\varkappa ,\tau )$$

(17)

Theorem 1.

If $\psi (\phi ,\varkappa ,\tau )={\sum }_{k=0}^{\infty }{\Psi }_k(\phi ,\varkappa ){\tau }^{\alpha k}$ is a given series [19,20,21];

$1.$ If $\exists 0<\lambda <1$ such that ${\displaystyle \frac{\parallel \Psi k+1\parallel }{\parallel {\Psi }_k\parallel }}\le \lambda$ then the given series solution is convergent.

$2.$ If $\exists \lambda >1$ such that ${\displaystyle \frac{\parallel {\Psi }_{k+1}\parallel }{\parallel {\Psi }_k\parallel }}\ge \lambda$ then the given series solution is divergent.

The proof of Theorem 1, which is a particular case of Banach’s fixed point theorem, can be found in the literature.

Corollary 1.

The series solution ${\sum }_{k=0}^{\infty }{\Psi }_k(\phi ,\varkappa ,\tau )$ exhibits convergence towards the exact solution $\psi (\phi ,\varkappa ,\tau )$ with the condition that $0\le {\chi }_n<1$, where n is a whole number and for every n [19,20,21]:

$${\chi }_n=\left\{\begin{array}{cc}\hfill {\displaystyle \frac{\parallel {\Psi }_{n+1}\parallel }{\parallel {\Psi }_n\parallel }},\hfill & \hfill \parallel {\Psi }_n\parallel \ne 0\hfill \\ \hfill 0,\hfill & \hfill \parallel {\Psi }_n\parallel =0\hfill \end{array}\right.$$

The fundamental mathematical operations of the FRDTM are shown in Table 1.

Table 1. Transform table [22,23,24,25].

Function	Transformation
$\psi (\phi ,\varkappa ,\tau )$	${\Psi }_i(\phi ,\varkappa )={\displaystyle \frac{1}{\mathsf{\Gamma }(\alpha i+1)}}{\left[{\displaystyle \frac{{\partial }^{\alpha i}}{\partial {\tau }^{\alpha i}}}\psi (\phi ,\varkappa ,\tau )\right]}_{\tau =0}$
$co(\phi ,\varkappa ,\tau )\pm dp(\phi ,\varkappa ,\tau )$	$c{O}_i(\phi ,\varkappa )\pm d{P}_i(\phi ,\varkappa )$, c and d are constant.
$o(\phi ,\varkappa ,\tau )p(\phi ,\varkappa ,\tau )$	${\sum }_{k=0}^i{O}_k(\phi ,\varkappa )P_{i-k}(\phi ,\varkappa )$
${\displaystyle \frac{{\partial }^{\alpha k}}{\partial {\tau }^{\alpha k}}}\psi (\phi ,\varkappa ,\tau )$	${\displaystyle \frac{\mathsf{\Gamma }(i\alpha +k\alpha +1)}{\mathsf{\Gamma }(i\alpha +1)}}{\Psi }_{i+k}(\phi ,\varkappa )$
${\displaystyle \frac{{\partial }^k}{\partial {\phi }^k}}\psi (\phi ,\varkappa ,\tau )$	${\displaystyle \frac{{\partial }^k}{\partial {\phi }^k}}{\Psi }_i(\phi ,\varkappa )$.
${\displaystyle \frac{{\partial }^k}{\partial {\varkappa }^k}}\psi (\phi ,\varkappa ,\tau )$	${\displaystyle \frac{{\partial }^k}{\partial {\varkappa }^k}}{\Psi }_i(\phi ,\varkappa )$.
${\phi }^m{\varkappa }^j{\tau }^k$	${\phi }^m{\varkappa }^j\delta (i\alpha -k)$ where $\delta (i\alpha -k)=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill i\alpha =k\hfill \\ \hfill 0,\hfill & \hfill i\alpha \ne k\hfill \end{array}\right.$.

4. Results and Discussion

This section shows the effect of various parameters on concenteration functions as well as the numerical and graphical outcomes of the obtained solutions.

We fractionalized Equations (3) and (4) into time-fractional partial differential equations in order to comprehend the anomalous behavior of the coupled system of equations. These equations are provided by

$${\displaystyle \frac{{\partial }^{\lambda }p}{\partial t^{\lambda }}}=D_{p_x}{\displaystyle \frac{{\partial }^{2}p}{\partial x^2}}+D_{p_y}{\displaystyle \frac{{\partial }^{2}p}{\partial y^2}}-{\displaystyle \frac{\partial \left(u_{x}p\right)}{\partial x}}-{\displaystyle \frac{\partial \left(u_{y}p\right)}{\partial y}}-{\displaystyle \frac{q}{q+k}}p+f$$

(18)

$${\displaystyle \frac{{\partial }^{\lambda }q}{\partial t^{\lambda }}}=D_{q_x}{\displaystyle \frac{{\partial }^{2}q}{\partial x^2}}+D_{q_y}{\displaystyle \frac{{\partial }^{2}q}{\partial y^2}}-{\displaystyle \frac{\partial \left(u_{x}q\right)}{\partial x}}-{\displaystyle \frac{\partial \left(u_{y}q\right)}{\partial y}}-k_3{\displaystyle \frac{q}{q+k}}p+\beta \left(K-q\right)$$

(19)

where $0\le \lambda \le 1$.

Applying FRDTM to both sides of Equations (18) and (19), we obtain

$${\displaystyle \frac{\mathsf{\Gamma }\left(\lambda \right((k+1)+1\left)\right)}{\mathsf{\Gamma }(\lambda k+1)}}{P}_{k+1}=D_{p_x}{\displaystyle \frac{{\partial }^2{P}_k}{\partial x^2}}+D_{p_y}{\displaystyle \frac{{\partial }^2{P}_k}{\partial y^2}}-u_x{\displaystyle \frac{\partial P_k}{\partial x}}-u_y{\displaystyle \frac{\partial P_k}{\partial y}}-\sum _{r=0}^k{G}_r{P}_{k-r}+f\delta \left(k\lambda \right)$$

(20)

$${\displaystyle \frac{\mathsf{\Gamma }\left(\lambda \right((k+1)+1\left)\right)}{\mathsf{\Gamma }(\lambda k+1)}}{Q}_{k+1}=D_{q_x}{\displaystyle \frac{{\partial }^2{Q}_k}{\partial x^2}}+D_{q_y}{\displaystyle \frac{{\partial }^2{Q}_k}{\partial y^2}}-u_x{\displaystyle \frac{\partial Q_k}{\partial x}}-u_y{\displaystyle \frac{\partial Q_k}{\partial y}}-k_3\sum _{r=0}^k{G}_r{P}_{k-r}+\beta (K\delta \left(k\lambda \right)-Q_k)$$

(21)

$$P_{k+1}={\displaystyle \frac{\mathsf{\Gamma }(\lambda k+1)}{\mathsf{\Gamma }\left(\lambda \right((k+1)+1\left)\right)}}\left(D_{p_x}{\displaystyle \frac{{\partial }^2{P}_k}{\partial x^2}}+D_{p_y}{\displaystyle \frac{{\partial }^2{P}_k}{\partial y^2}}-u_x{\displaystyle \frac{\partial P_k}{\partial x}}-u_y{\displaystyle \frac{\partial P_k}{\partial y}}-\sum _{r=0}^k{G}_r{P}_{k-r}+f\delta \left(k\lambda \right)\right)$$

(22)

$$Q_{k+1}={\displaystyle \frac{\mathsf{\Gamma }(\lambda k+1)}{\mathsf{\Gamma }\left(\lambda \right((k+1)+1\left)\right)}}\left(D_{q_x}{\displaystyle \frac{{\partial }^2{Q}_k}{\partial x^2}}+D_{q_y}{\displaystyle \frac{{\partial }^2{Q}_k}{\partial y^2}}-u_x{\displaystyle \frac{\partial Q_k}{\partial x}}-u_y{\displaystyle \frac{\partial Q_k}{\partial y}}-k_3\sum _{r=0}^k{G}_r{P}_{k-r}+\beta (K\delta \left(k\lambda \right)-Q_k)\right)$$

(23)

where $G_r$ is the transformation function of $g={\displaystyle \frac{q}{q+k}}$. From the initial conditions (5) and (6), we write

$$P_0(x,y)=C_1{e}^{-\left(x+y\sqrt{{\displaystyle \frac{D_{p_y}}{D_{p_x}}}}\right)}$$

(24)

$$Q_0(x,y)=C_2{e}^{-\left(x+y\sqrt{{\displaystyle \frac{D_{q_y}}{D_{q_x}}}}\right)}$$

(25)

Putting (24) and (25) into (22) and (23), we obtain the approximate analytical solution as follows:

$$p(x,y,t)=P_0(x,y)+P_1(x,y)t^{\lambda }+P_2(x,y)t^{2\lambda }+P_3(x,y)t^{3\lambda }+...={\chi }_0+{\chi }_1+{\chi }_2+{\chi }_3+...$$

(26)

$$q(x,y,t)=Q_0(x,y)+Q_1(x,y)t^{\lambda }+Q_2(x,y)t^{2\lambda }+Q_3(x,y)t^{3\lambda }+...={\psi }_0+{\psi }_1+{\psi }_2+{\psi }_3+...$$

(27)

where

$$\begin{array}{ccc}\hfill P_1(x,y)\hfill & \hfill =\hfill & \hfill {\displaystyle \frac{\left(\begin{array}{c}f+D_{p_x}{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}+u_x{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}+u_y{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}\sqrt{\frac{D_{p_y}}{D_{p_x}}}\\ +\frac{D_{p_y}^2{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}}{D_{p_x}}+C_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(k+C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-2\right)\end{array}\right)}{\mathsf{\Gamma }\left(\lambda +1\right)}}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill P_2(x,y)\hfill & \hfill =\hfill & \hfill {\displaystyle \frac{u_y\left(\begin{array}{c}{D}_{p_x}p{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}\sqrt{\frac{D_{p_y}}{D_{p_x}}}+C_1{C}_2^2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-2x-2y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}\\ +u_x{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}\sqrt{\frac{D_{p_y}}{D_{p_x}}}+\frac{D_{p_y}^2{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{D_{p_x}}+\frac{D_{p_y}{u}_y{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}}{D_{p_x}}\\ +C_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{p_y}}{D_{p_x}}}\left(k+C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-2\right)\\ +C_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}\left(k+C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-2\right)\end{array}\right)}{\mathsf{\Gamma }\left(2\lambda +1\right)}}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \hfill & & \hfill +{\displaystyle \frac{D_{p_y}\left(\begin{array}{c}{D}_{p_y}{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}+2C_1{C}_2^2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-2x-2y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{p_y}}{D_{p_x}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}\\ +\frac{D_{p_y}^3{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}}{D_{p_x}^2}+\frac{D_{p_y}{u}_x{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}}{D_{p_x}}+\frac{3D_{q_y}{C}_1{C}_2^2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-2x-2y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}}{D_{q_x}}\\ +\frac{D_{p_y}{u}_y{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{D_{p_x}}+\frac{D_{p_y}{C}_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(k+C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-2\right)}{D_{p_x}}\\ +\frac{D_{q_y}{C}_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(k+C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-2\right)}{D_{q_x}}\\ +2C_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{p_y}}{D_{p_x}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}\left(k+C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-2\right)\end{array}\right)}{\mathsf{\Gamma }\left(2\lambda +1\right)}}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \hfill & & \hfill +{\displaystyle \frac{D_{p_x}\left(\begin{array}{c}{D}_{p_x}{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}+u_x{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}+5C_1{C}_2^2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-2x-2y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\\ +u_y{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}\sqrt{\frac{D_{p_y}}{D_{p_x}}}+\frac{D_{p_y}^2{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}}{D_{p_x}}\\ +4C_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(k+C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-2\right)\end{array}\right)}{\mathsf{\Gamma }\left(2\lambda +1\right)}}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \hfill & & \hfill +{\displaystyle \frac{u_x\left(\begin{array}{c}{u}_x{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}+C_1{C}_2^2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-2x-2y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}+u_y{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}\sqrt{\frac{D_{p_y}}{D_{p_x}}}\\ +\frac{D_{p_y}^2{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}}{D_{p_x}}+2C_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(k+C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-2\right)\\ +D_{p_x}{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}\end{array}\right)}{\mathsf{\Gamma }\left(2\lambda +1\right)}}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \hfill & & \hfill +{\displaystyle \frac{C_1\mathsf{\Gamma }\left(\lambda +1\right){\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}\left(\begin{array}{c}\frac{\left(\begin{array}{c}{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\\ +k-2\end{array}\right)\left(\begin{array}{c}\beta \left(K-C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\right)+u_x{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\\ +D_{q_x}{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}+u_y{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}\\ +\frac{D_{q_y}^2{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}}{D_{q_x}}\\ +k_3{C}_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(k+C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-2\right)\end{array}\right)}{\mathsf{\Gamma }\left(\lambda +1\right)}\\ +\frac{C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(\begin{array}{c}\beta \left(K-C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\right)+u_x{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\\ +D_{q_x}{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\\ +u_y{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}+\frac{D_{q_y}^2{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}}{D_{q_x}}\\ +k_3{C}_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(k+C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-2\right)\end{array}\right)}{\mathsf{\Gamma }\left(\lambda +1\right)}\end{array}\right)}{\mathsf{\Gamma }\left(2\lambda +1\right)}}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \hfill & & \hfill +{\displaystyle \frac{C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(\begin{array}{c}{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\\ +k-2\end{array}\right)\left(\begin{array}{c}f+D_{p_x}{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}+u_x{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}\\ +u_y{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}\sqrt{\frac{D_{p_y}}{D_{p_x}}}+\frac{D_{p_y}^2{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}}{D_{p_x}}\\ +C_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(k+C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-2\right)\end{array}\right)}{\mathsf{\Gamma }\left(2\lambda +1\right)}}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \hfill & & \hfill \hfill \\ \hfill Q_1(x,y)\hfill & \hfill =\hfill & \hfill {\displaystyle \frac{\left(\begin{array}{c}\beta \left(K-C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\right)+u_x{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}+D_{q_x}{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\\ +u_y{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}+\frac{D_{q_y}^2{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}}{D_{q_x}}\\ +k_3{C}_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(k+C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-2\right)\end{array}\right)}{\mathsf{\Gamma }\left(\lambda +1\right)}}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill Q_2(x,y)\hfill & \hfill =\hfill & \hfill {\displaystyle \frac{u_x\left(\begin{array}{c}{u}_x{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}+D_{q_x}{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-\beta C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}+u_y{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}\\ +\frac{D_{q_y}^2{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}}{D_{q_x}}+k_3{C}_1{C}_2^2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-2x-2y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\\ +2k_3{C}_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(k+C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-2\right)\end{array}\right)}{\mathsf{\Gamma }\left(2\lambda +1\right)}}\hfill \\ \hfill \hfill & & \hfill +{\displaystyle \frac{u_y\left(\begin{array}{c}{u}_x{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}+D_{q_x}{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}-\beta C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}\\ +\frac{D_{q_y}^2{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}}{D_{q_x}}+\frac{u_y{D}_{q_y}{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}}{D_{q_x}}+k_3{C}_1{C}_2^2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-2x-2y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}\\ +k_3{C}_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{p_y}}{D_{p_x}}}\left(k+C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-2\right)\\ +k_3{C}_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}\left(k+C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-2\right)\end{array}\right)}{\mathsf{\Gamma }\left(2\lambda +1\right)}}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \hfill & & \hfill +{\displaystyle \frac{D_{q_x}\left(\begin{array}{c}{u}_x{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}+D_{q_x}{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-\beta C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}+u_y{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}\\ +\frac{D_{q_y}^2{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}}{D_{q_x}}+5k_3{C}_1{C}_2^2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-2x-2y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\\ +4k_3{C}_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(k+C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-2\right)\end{array}\right)}{\mathsf{\Gamma }\left(2\lambda +1\right)}}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \hfill & & \hfill +{\displaystyle \frac{C_2\left(\begin{array}{c}{D}_{q_y}{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}+\frac{D_{q_y}^3{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}}{D_{q_x}^2}+\frac{u_x{D}_{q_y}{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}}{D_{q_x}}-\frac{D_{q_y}\beta C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}}{D_{q_x}}\\ +\frac{u_y{D}_{q_y}{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}}{D_{q_x}}+2k_3{C}_1{C}_2^2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-2x-2y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{p_y}}{D_{p_x}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}\\ +2k_3{C}_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{p_y}}{D_{p_x}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}\left(k+C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-2\right)\\ +\frac{D_{p_y}{k}_3{C}_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(k+C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-2\right)}{D_{p_x}}+\frac{3D_{q_y}{k}_3{C}_1{C}_2^2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-2x-2y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}}{D_{q_x}}\\ +\frac{D_{q_y}{k}_3{C}_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(k+C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-2\right)}{D_{q_x}}\end{array}\right)}{\mathsf{\Gamma }\left(2\lambda +1\right)}}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \hfill & & \hfill -{\displaystyle \frac{\beta \left(\begin{array}{c}\beta \left(K-C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\right)+u_x{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}+D_{q_x}{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\\ +u_y{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}\\ +\frac{D_{q_y}^2{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}}{D_{q_x}}+k_3{C}_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(k+C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}-2\right)\end{array}\right)}{\mathsf{\Gamma }\left(2\lambda +1\right)}}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \hfill & & \hfill +{\displaystyle \frac{k_3\mathsf{\Gamma }\left(\lambda +1\right)\left(\begin{array}{c}{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}\left(\begin{array}{c}\frac{\left(\begin{array}{c}{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\\ +k-2\end{array}\right)\left(\begin{array}{c}\beta \left(K-C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\right)+u_x{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\\ +D_{q_x}{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}+u_y{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}\\ +\frac{D_{q_y}^2{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}}{D_{q_x}}\\ +k_3{C}_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(\begin{array}{c}{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\\ +k-2\end{array}\right)\end{array}\right)}{\mathsf{\Gamma }\left(\lambda +1\right)}\\ +\frac{C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(\begin{array}{c}\beta \left(K-C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\right)+u_x{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\\ +D_{q_x}{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}+u_y{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\sqrt{\frac{D_{q_y}}{D_{q_x}}}\\ +\frac{D_{q_y}^2{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}}{D_{q_x}}\\ +k_3{C}_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(\begin{array}{c}{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\\ +k-2\end{array}\right)\end{array}\right)}{\mathsf{\Gamma }\left(\lambda +1\right)}\end{array}\right)\\ +\frac{C_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(\begin{array}{c}{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\\ +k-2\end{array}\right)\left(\begin{array}{c}f+D_{p_x}{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}+u_x{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}\\ +u_y{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}\sqrt{\frac{D_{p_y}}{D_{p_x}}}+\frac{D_{p_y}^2{C}_1{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}}{D_{p_x}}\\ +C_1{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{p_y}}{D_{p_x}}}}{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\left(\begin{array}{c}{C}_2{\mathrm{e}}^{-x-y\sqrt{\frac{D_{q_y}}{D_{q_x}}}}\\ +k-2\end{array}\right)\end{array}\right)}{\mathsf{\Gamma }\left(\lambda +1\right)}\end{array}\right)}{\mathsf{\Gamma }\left(2\lambda +1\right)}}\hfill \end{array}$$

In addition, to analyze the convergence of an individual series solution, we compute the terms ${\xi }_k$ using Corollary 1. For the series solution of $p(x,y,t)$, we obtain ${\xi }_0={\displaystyle \frac{\parallel {\chi }_1\parallel }{\parallel {\chi }_0\parallel }}=0.91992<1$,${\xi }_1={\displaystyle \frac{\parallel {\chi }_2\parallel }{\parallel {\chi }_1\parallel }}=0.75091<1$. And for the series solution of $q(x,y,t)$, we obtain ${\xi }_0={\displaystyle \frac{\parallel {\psi }_1\parallel }{\parallel {\psi }_0\parallel }}=0.23513<1$,${\xi }_1={\displaystyle \frac{\parallel {\psi }_2\parallel }{\parallel {\psi }_1\parallel }}=0.05883<1$. This observation verifies that the FRDTM produces a series solution for p and q, demonstrating convergence towards the exact answer.

Now, we have obtained the following tables of numerical values and graphs of the concentration functions using MATLAB 2023 for fixed parameters $D_{p_x}=1$, $D_{p_y}=0.1$, $u_x=1$, $u_y=0.1$, $f=1$, $D_{q_x}=1$, $D_{q_y}=0.1$, $k_3=0.5$, $\beta =2$, $K=2$, $k=1$.

The numerical values of pollutant concentration for fixed $t=0.1,0.5$ are in Table 2 and Table 3. The numerical values of DO concentration for fixed $t=0.1,0.5$, respectively, are given in Table 4 and Table 5. The impact of change in time on pollutant and DO concentration can be seen graphically in Figure 1 and Figure 2, respectively. The numerical values of p and q for fixed $x=0.4,0.8$ are given in Table 6, Table 7, Table 8 and Table 9. Figure 3 and Figure 4 give the three-dimensional graphical comparison of pollutant and DO concentration function for fixed x, respectively. The numerical values of p and q for fixed $y=0.01,0.05$ are given in Table 10, Table 11, Table 12 and Table 13. The impact of change in y on pollutant and DO concentration functions can be seen graphically in Figure 5 and Figure 6, respectively. It has been seen that the amount of pollution and DO concentration rises as t rises. And the levels of p and q decrease when x and y rise.

Table 2. $p(x,y,0.1)$.

x∖y	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1
0.1	0.413736	0.411974	0.410227	0.408494	0.406775	0.405069	0.403378	0.4017	0.400035	0.398383
0.2	0.364269	0.362893	0.361528	0.360172	0.358827	0.357491	0.356166	0.35485	0.353544	0.352248
0.3	0.325284	0.324189	0.323101	0.322021	0.320949	0.319883	0.318825	0.317774	0.31673	0.315694
0.4	0.293999	0.293112	0.29223	0.291355	0.290485	0.28962	0.288761	0.287907	0.287059	0.286216
0.5	0.268475	0.267745	0.26702	0.266299	0.265583	0.26487	0.264162	0.263458	0.262758	0.262062
0.6	0.247342	0.246734	0.246129	0.245527	0.244929	0.244334	0.243742	0.243153	0.242568	0.241985
0.7	0.229613	0.2291	0.228589	0.22808	0.227575	0.227071	0.226571	0.226073	0.225577	0.225084
0.8	0.214567	0.214129	0.213692	0.213258	0.212826	0.212396	0.211968	0.211543	0.211119	0.210697
0.9	0.201667	0.20129	0.200914	0.20054	0.200167	0.199796	0.199427	0.19906	0.198694	0.19833
1	0.190511	0.190183	0.189856	0.189531	0.189207	0.188885	0.188564	0.188245	0.187926	0.18761

Table 3. $p(x,y,0.5)$.

x∖y	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1
0.1	5.732374	5.67944	5.627131	5.575438	5.524354	5.473869	5.423977	5.374668	5.325936	5.277773
0.2	4.324727	4.288016	4.251715	4.21582	4.180325	4.145226	4.110516	4.076192	4.042247	4.008678
0.3	3.339795	3.313837	3.288154	3.262742	3.237599	3.212721	3.188105	3.163747	3.139645	3.115795
0.4	2.637396	2.618698	2.600187	2.581862	2.56372	2.54576	2.527978	2.510373	2.492942	2.475685
0.5	2.12733	2.113624	2.100048	2.0866	2.07328	2.060086	2.047016	2.034069	2.021245	2.00854
0.6	1.750571	1.740357	1.730236	1.720205	1.710265	1.700414	1.690651	1.680975	1.671385	1.661881
0.7	1.467825	1.460098	1.452437	1.444841	1.43731	1.429843	1.422439	1.415098	1.407819	1.400602
0.8	1.25249	1.246561	1.24068	1.234847	1.22906	1.223321	1.217628	1.211981	1.206379	1.200823
0.9	1.086249	1.08164	1.077066	1.072528	1.068025	1.063556	1.059122	1.054722	1.050355	1.046022
1	0.956285	0.952658	0.949059	0.945485	0.941938	0.938417	0.934922	0.931452	0.928008	0.924589

Table 4. $q(x,y,0.1)$.

x∖y	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1
0.1	2.126643	2.120715	2.114811	2.108929	2.103069	2.097232	2.091417	2.085624	2.079854	2.074105
0.2	1.949676	1.944415	1.939172	1.933949	1.928745	1.923561	1.918395	1.913248	1.90812	1.903011
0.3	1.792241	1.78755	1.782875	1.778217	1.773575	1.76895	1.764342	1.75975	1.755174	1.750614
0.4	1.65164	1.647444	1.643261	1.639094	1.634941	1.630802	1.626678	1.622568	1.618472	1.61439
0.5	1.52571	1.521946	1.518196	1.514458	1.510732	1.50702	1.50332	1.499633	1.495958	1.492296
0.6	1.41267	1.409289	1.405919	1.40256	1.399212	1.395876	1.392551	1.389237	1.385934	1.382642
0.7	1.311031	1.307989	1.304956	1.301934	1.298921	1.295918	1.292926	1.289943	1.28697	1.284007
0.8	1.219524	1.216783	1.214051	1.211328	1.208614	1.205909	1.203212	1.200525	1.197846	1.195176
0.9	1.137054	1.134582	1.132119	1.129664	1.127217	1.124777	1.122346	1.119922	1.117507	1.115099
1	1.06267	1.06044	1.058217	1.056002	1.053793	1.051592	1.049398	1.047211	1.045031	1.042859

Table 5. $q(x,y,0.5)$.

x∖y	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1
0.1	5.166659	5.138586	5.110808	5.083321	5.05612	5.029204	5.002566	4.976204	4.950115	4.924294
0.2	4.404801	4.384408	4.364212	4.344208	4.324395	4.30477	4.285331	4.266076	4.247003	4.228108
0.3	3.84405	3.82881	3.813702	3.798727	3.783881	3.769163	3.754572	3.740107	3.725765	3.711546
0.4	3.419727	3.408031	3.396426	3.384914	3.373493	3.362161	3.350918	3.339763	3.328694	3.317712
0.5	3.09033	3.081134	3.072003	3.062939	3.05394	3.045005	3.036133	3.027325	3.018579	3.009895
0.6	2.8287	2.821313	2.813975	2.806686	2.799444	2.79225	2.785102	2.778001	2.770946	2.763937
0.7	2.616693	2.610649	2.604643	2.598672	2.592738	2.586839	2.580976	2.575147	2.569354	2.563595
0.8	2.44192	2.436898	2.431903	2.426937	2.421998	2.417087	2.412203	2.407346	2.402516	2.397712
0.9	2.295735	2.291505	2.287297	2.283111	2.278948	2.274805	2.270684	2.266585	2.262506	2.258449
1	2.171961	2.168359	2.164775	2.161209	2.157659	2.154128	2.150613	2.147116	2.143635	2.140172

Figure 1. Comparison of pollutant concentration for fixed t.

Figure 2. Comparison of DO concentration for fixed t.

Table 6. $p(0.4,y,t)$.

t∖y	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1
0.1	0.293999	0.293112	0.29223	0.291355	0.290485	0.28962	0.288761	0.287907	0.287059	0.286216
0.2	0.556634	0.554305	0.551997	0.549707	0.547438	0.545188	0.542957	0.540744	0.538551	0.536376
0.3	0.989466	0.984101	0.978785	0.973521	0.968305	0.963139	0.958021	0.952951	0.947929	0.942953
0.4	1.660414	1.649798	1.639286	1.628878	1.618571	1.608365	1.59826	1.588252	1.578342	1.568528
0.5	2.637396	2.618698	2.600187	2.581862	2.56372	2.54576	2.527978	2.510373	2.492942	2.475685
0.6	3.988331	3.958101	3.928178	3.898558	3.869237	3.840213	3.811481	3.783038	3.75488	3.727005
0.7	5.781137	5.735308	5.689947	5.645048	5.600607	5.556618	5.513074	5.469972	5.427305	5.38507
0.8	8.083734	8.017619	7.952183	7.887417	7.823314	7.759865	7.697064	7.634901	7.573369	7.512462
0.9	10.96404	10.87233	10.78157	10.69175	10.60284	10.51485	10.42775	10.34155	10.25622	10.17176
1	14.48997	14.36676	14.24481	14.12412	14.00468	13.88646	13.76945	13.65364	13.53901	13.42556

Table 7. $q(0.4,y,t)$.

t∖y	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1
0.1	1.65164	1.647444	1.643261	1.639094	1.634941	1.630802	1.626678	1.622568	1.618472	1.61439
0.2	2.012667	2.007864	2.003082	1.998321	1.99358	1.98886	1.98416	1.97948	1.97482	1.97018
0.3	2.39768	2.391597	2.385547	2.379532	2.37355	2.367601	2.361685	2.355802	2.349951	2.344132
0.4	2.851696	2.8434	2.835161	2.826979	2.818853	2.810782	2.802767	2.794805	2.786898	2.779043
0.5	3.419727	3.408031	3.396426	3.384914	3.373493	3.362161	3.350918	3.339763	3.328694	3.317712
0.6	4.146789	4.130246	4.113847	4.097589	4.081472	4.065493	4.049652	4.033946	4.018375	4.002935
0.7	5.077895	5.054804	5.031925	5.009256	4.986794	4.964537	4.942483	4.920629	4.898973	4.877512
0.8	6.25806	6.226461	6.195164	6.164165	6.133461	6.103049	6.072924	6.043083	6.013523	5.98424
0.9	7.732297	7.689975	7.648068	7.60657	7.565478	7.524785	7.484487	7.44458	7.405058	7.365918
1	9.545622	9.490104	9.435139	9.380722	9.326846	9.273503	9.220688	9.168393	9.116613	9.065342

Table 8. $p(0.8,y,t)$.

t∖y	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1
0.1	0.214567	0.214129	0.213692	0.213258	0.212826	0.212396	0.211968	0.211543	0.211119	0.210697
0.2	0.370469	0.369579	0.368695	0.367816	0.366944	0.366077	0.365216	0.36436	0.363511	0.362666
0.3	0.580169	0.578341	0.576526	0.574724	0.572937	0.571162	0.569401	0.567654	0.565919	0.564197
0.4	0.866549	0.863104	0.859687	0.856296	0.852933	0.849596	0.846285	0.843001	0.839742	0.836509
0.5	1.25249	1.246561	1.24068	1.234847	1.22906	1.223321	1.217628	1.211981	1.206379	1.200823
0.6	1.760874	1.751401	1.742007	1.732689	1.723448	1.714282	1.70519	1.696173	1.687229	1.678357
0.7	2.41458	2.400316	2.386169	2.372138	2.358223	2.344421	2.330733	2.317156	2.30369	2.290334
0.8	3.236492	3.215995	3.195668	3.175508	3.155513	3.135683	3.116015	3.096508	3.077161	3.057972
0.9	4.24949	4.221131	4.193005	4.165111	4.137447	4.11001	4.082799	4.05581	4.029041	4.002491
1	5.476456	5.438412	5.400683	5.363264	5.326153	5.289347	5.252843	5.216638	5.180729	5.145112

Table 9. $q(0.8,y,t)$.

t∖y	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1
0.1	1.219524	1.216783	1.214051	1.211328	1.208614	1.205909	1.203212	1.200525	1.197846	1.195176
0.2	1.536579	1.533657	1.530747	1.527847	1.524958	1.522079	1.51921	1.516352	1.513504	1.510666
0.3	1.828475	1.825168	1.821874	1.818594	1.815327	1.812075	1.808836	1.80561	1.802398	1.799199
0.4	2.121495	2.117514	2.113552	2.10961	2.105686	2.101782	2.097897	2.094031	2.090184	2.086355
0.5	2.44192	2.436898	2.431903	2.426937	2.421998	2.417087	2.412203	2.407346	2.402516	2.397712
0.6	2.816031	2.80952	2.803049	2.796617	2.790226	2.783873	2.77756	2.771285	2.765049	2.75885
0.7	3.27011	3.261581	3.253109	3.244693	3.236332	3.228027	3.219777	3.21158	3.203438	3.195349
0.8	3.830438	3.819283	3.808205	3.797205	3.786281	3.775434	3.764661	3.753963	3.743338	3.732786
0.9	4.523296	4.508825	4.494459	4.480196	4.466037	4.451978	4.438021	4.424163	4.410404	4.396743
1	5.374966	5.35641	5.337991	5.319709	5.301561	5.283546	5.265664	5.247912	5.23029	5.212797

Figure 3. Comparison of pollutant concentration for fixed x.

Figure 4. Comparison of DO concentration for fixed x.

Table 10. $p(x,0.01,t)$.

x∖t	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
0.1	0.413736	0.911818	1.848428	3.397352	5.732374	9.027281	13.45586	19.19189	26.40916	35.28146
0.2	0.364269	0.756092	1.463546	2.611475	4.324727	6.728149	9.946587	14.10489	19.3279	25.74046
0.3	0.325284	0.641936	1.189003	2.057839	3.339795	5.126225	7.50848	10.57791	14.42588	19.14372
0.4	0.293999	0.556634	0.989466	1.660414	2.637396	3.988331	5.781137	8.083734	10.96404	14.48997
0.5	0.268475	0.491729	0.841848	1.369994	2.12733	3.165018	4.534219	6.286097	8.471814	11.14253
0.6	0.247342	0.441496	0.730803	1.154186	1.750571	2.558881	3.618042	4.966978	6.644613	8.689872
0.7	0.229613	0.401995	0.645954	0.991296	1.467825	2.105348	2.93367	3.982595	5.28193	6.86148
0.8	0.214567	0.370469	0.580169	0.866549	1.25249	1.760874	2.41458	3.236492	4.24949	5.476456
0.9	0.201667	0.344959	0.528465	0.769718	1.086249	1.495592	2.015278	2.66284	3.45581	4.411722
1	0.190511	0.324054	0.487307	0.693606	0.956285	1.288678	1.704121	2.215948	2.837494	3.582093

Table 11. $q(x,0.01,t)$.

x∖t	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
0.1	2.126643	2.588718	3.189644	4.019073	5.166659	6.722056	8.774918	11.4149	14.73165	18.81483
0.2	1.949676	2.365732	2.8677	3.524438	4.404801	5.577646	7.111831	9.07621	11.53964	14.57098
0.3	1.792241	2.176119	2.609525	3.147242	3.84405	4.754731	5.934066	7.436837	9.317824	11.63181
0.4	1.65164	2.012667	2.39768	2.851696	3.419727	4.146789	5.077895	6.25806	7.732297	9.545622
0.5	1.52571	1.870274	2.220525	2.614523	3.09033	3.686007	4.439616	5.389219	6.572876	8.028648
0.6	1.41267	1.745221	2.070085	2.42025	2.8287	3.32842	3.952395	4.73361	5.705051	6.899702
0.7	1.311031	1.634709	1.940753	2.258352	2.616693	3.044963	3.57235	4.228043	5.041227	6.041092
0.8	1.219524	1.536579	1.828475	2.121495	2.44192	2.816031	3.27011	3.830438	4.523296	5.374966
0.9	1.137054	1.449117	1.730245	2.004447	2.295735	2.628119	3.025611	3.512221	4.111961	4.84884
1	1.06267	1.370938	1.643773	1.90338	2.171961	2.471721	2.824864	3.253594	3.780116	4.426632

Table 12. $p(x,0.05,t)$.

x∖t	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
0.1	0.406775	0.889236	1.791998	3.281542	5.524354	8.686916	12.93571	18.43723	25.35795	33.86436
0.2	0.358827	0.739673	1.423609	2.530528	4.180325	6.492893	9.588126	13.58592	18.60616	24.76875
0.3	0.320949	0.629761	1.160195	2.000168	3.237599	4.960406	7.256507	10.21382	13.92026	18.46376
0.4	0.290485	0.547438	0.968305	1.618571	2.56372	3.869237	5.600607	7.823314	10.60284	14.00468
0.5	0.265583	0.484661	0.826036	1.339109	2.07328	3.077947	4.402511	6.096371	8.208927	10.78958
0.6	0.244929	0.435974	0.718797	1.13102	1.710265	2.494153	3.520306	4.826346	6.449893	8.42857
0.7	0.227575	0.397614	0.636701	0.973658	1.43731	2.056479	2.85999	3.876666	5.13533	6.664806
0.8	0.212826	0.366944	0.572937	0.852933	1.22906	1.723448	2.358223	3.155513	4.137447	5.326153
0.9	0.200167	0.342085	0.522738	0.759069	1.068025	1.46655	1.971589	2.600087	3.368989	4.29524
1	0.189207	0.321682	0.482716	0.685178	0.941938	1.265867	1.669834	2.166708	2.769361	3.490661

Table 13. $q(x,0.05,t)$.

x∖t	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
0.1	2.103069	2.558347	3.144592	3.948338	5.05612	6.554475	8.529936	11.06904	14.25832	18.18431
0.2	1.928745	2.34008	2.831955	3.471138	4.324395	5.458493	6.940198	8.836278	11.2135	14.13863
0.3	1.773575	2.154121	2.580456	3.105927	3.783881	4.667662	5.810616	7.266091	9.087431	11.32798
0.4	1.634941	1.99358	2.37355	2.818853	3.373493	4.081472	4.986794	6.133461	7.565478	9.326846
0.5	1.510732	1.853564	2.200155	2.587837	3.05394	3.635795	4.370732	5.296083	6.449178	7.867347
0.6	1.399212	1.730489	2.052657	2.398161	2.799444	3.288951	3.899125	4.66241	5.61125	6.778089
0.7	1.298921	1.621652	1.92568	2.239783	2.592738	3.013322	3.530312	4.172485	4.968618	5.947489
0.8	1.208614	1.524958	1.815327	2.105686	2.421998	2.790226	3.236332	3.786281	4.466037	5.301561
0.9	1.127217	1.438741	1.718698	1.990847	2.278948	2.606759	2.998041	3.476553	4.066054	4.790305
1	1.053793	1.36165	1.633579	1.891581	2.157659	2.453816	2.802052	3.22437	3.742772	4.379261

Figure 5. Comparison of pollutant concentration for fixed y.

Figure 6. Comparison of DO concentration for fixed y.

Table 14 and Table 15 provide the numerical values of pollutant and DO concentration for various values of $\lambda$ ($\lambda =0.1,0.3,0.5,0.7,1$). The graphical representation of Figure 7 and Figure 8 illustrates the effect of variations in $\lambda$ on the functions of pollutant and DO concentration. The numerical tables and figures provided show that when the value of $\lambda$ has increased from 0 to 1, then the values of p and q are reduced.

Table 14. $p(x,0.07,0.4)$ for different values of $\lambda$.

x	$\mathit{\lambda }=0.1$	$\mathit{\lambda }=0.3$	$\mathit{\lambda }=0.5$	$\mathit{\lambda }=0.7$	$\mathit{\lambda }=1$
0.1	131.948316	73.8004016	33.1716726	13.2108399	3.225622782
0.2	94.13683331	52.8718564	23.96775981	9.699857389	2.491370321
0.3	68.29978316	38.5362045	17.6351535	7.266718406	1.972220367
0.4	50.30194048	28.5254298	13.19330039	5.547936266	1.59825953
0.5	37.52960574	21.4036321	10.01941793	4.31135652	1.32409346
0.6	28.304331	16.2470678	7.711488848	3.406233193	1.119740513
0.7	21.53037552	12.4515558	6.005632159	2.733013382	0.96505829
0.8	16.48021325	9.61516849	4.725662367	2.224819191	0.846285232
0.9	12.66246199	7.46590809	3.751914664	1.835952131	0.753864312
1	9.73954369	5.81655292	3.001728331	1.534661489	0.681053708

Table 15. $q(x,0.07,0.4)$ for different values of $\lambda$.

x	$\mathit{\lambda }=0.1$	$\mathit{\lambda }=0.3$	$\mathit{\lambda }=0.5$	$\mathit{\lambda }=0.7$	$\mathit{\lambda }=1$
0.1	66.05390856	37.4465357	17.84592924	8.485860191	3.913953704
0.2	50.03237409	28.4863522	13.80185495	6.839155968	3.445155514
0.3	39.19137003	22.3902622	11.01742358	5.676722457	3.085736626
0.4	31.68921312	18.146462	9.053560949	4.834434204	2.8027665
0.5	26.37960345	15.1237465	7.635089859	4.208517426	2.574740807
0.6	22.53836044	12.922309	6.586790145	3.732217372	2.387303087
0.7	19.70063084	11.2847595	5.795182815	3.361797688	2.230643742
0.8	17.56290471	10.0424983	5.185440381	3.068042559	2.097897281
0.9	15.92342254	9.08305228	4.707281924	2.831038014	1.984140789
1	14.64553736	8.32995883	4.326257377	2.636927524	1.885758787

Figure 7. Comparison of pollutant concentration for different values of $\lambda$.

Figure 8. Comparison of DO concentration for different values of $\lambda$.

The numerical values of the pollutant and DO concentration for various values of $D_{p_x},D_{q_x},D_{p_y},D_{q_y}$ ($D_{p_x}=D_{q_x}=1,3,5\&D_{p_y}=D_{q_y}=0.1,0.3,0.5$) are provided in Table 16 and Table 17. The three-dimensional graphical comparison of p and q for various values of $D_{p_x},D_{q_x},D_{p_y},D_{q_y}$ is shown in Figure 9 and Figure 10. If the values of $D_{p_x},D_{q_x},D_{p_y},D_{q_y}$ are increased, then the values of p and q are raised.

Table 16. $p(x,0.07,0.4)$ for different values of $D_{p_x},D_{q_x},D_{p_y},D_{q_y}$.

x	${\mathit{D}}_{{\mathit{p}}_{\mathit{x}}}={\mathit{D}}_{{\mathit{q}}_{\mathit{x}}}=1\&{\mathit{D}}_{{\mathit{p}}_{\mathit{y}}}={\mathit{D}}_{{\mathit{q}}_{\mathit{y}}}=0.1$	${\mathit{D}}_{{\mathit{p}}_{\mathit{x}}}={\mathit{D}}_{{\mathit{q}}_{\mathit{x}}}=3\&{\mathit{D}}_{{\mathit{p}}_{\mathit{y}}}={\mathit{D}}_{{\mathit{q}}_{\mathit{y}}}=0.3$	${\mathit{D}}_{{\mathit{p}}_{\mathit{x}}}={\mathit{D}}_{{\mathit{q}}_{\mathit{x}}}=5\&{\mathit{D}}_{{\mathit{p}}_{\mathit{y}}}={\mathit{D}}_{{\mathit{q}}_{\mathit{y}}}=0.5$
0.1	3.225622782	10.35100308	22.15170425
0.2	2.491370321	7.686782206	16.33898209
0.3	1.972220367	5.784754721	12.15687743
0.4	1.59825953	4.412625224	9.126022393
0.5	1.32409346	3.413322842	6.915223144
0.6	1.119740513	2.679198634	5.29313882
0.7	0.96505829	2.135536472	4.096574258
0.8	0.846285232	1.729871087	3.209395814
0.9	0.753864312	1.424971585	2.548326985
1	0.681053708	1.194168312	2.053258012

Table 17. $q(x,0.07,0.4)$ for different values of $D_{p_x},D_{q_x},D_{p_y},D_{q_y}$.

x	${\mathit{D}}_{{\mathit{p}}_{\mathit{x}}}={\mathit{D}}_{{\mathit{q}}_{\mathit{x}}}=1\&{\mathit{D}}_{{\mathit{p}}_{\mathit{y}}}={\mathit{D}}_{{\mathit{q}}_{\mathit{y}}}=0.1$	${\mathit{D}}_{{\mathit{p}}_{\mathit{x}}}={\mathit{D}}_{{\mathit{q}}_{\mathit{x}}}=3\&{\mathit{D}}_{{\mathit{p}}_{\mathit{y}}}={\mathit{D}}_{{\mathit{q}}_{\mathit{y}}}=0.3$	${\mathit{D}}_{{\mathit{p}}_{\mathit{x}}}={\mathit{D}}_{{\mathit{q}}_{\mathit{x}}}=5\&{\mathit{D}}_{{\mathit{p}}_{\mathit{y}}}={\mathit{D}}_{{\mathit{q}}_{\mathit{y}}}=0.5$
0.1	3.913953704	9.187685447	18.65057115
0.2	3.445155514	7.622698858	15.20393681
0.3	3.085736626	6.444496397	12.59892502
0.4	2.8027665	5.540903037	10.60017544
0.5	2.574740807	4.835723633	9.044135138
0.6	2.387303087	4.276354163	7.815633855
0.7	2.230643742	3.825895594	6.832548434
0.8	2.097897281	3.45806434	6.035625304
0.9	1.984140789	3.153860602	5.381634143
1	1.885758787	2.899355674	4.838701197

Figure 9. Comparison of pollutant concentration for fixed $D_{p_x},D_{q_x},D_{p_y},D_{q_y}$.

Figure 10. Comparison of DO concentration for fixed $D_{p_x},D_{q_x},D_{p_y},D_{q_y}$.

The numerical values of the pollutant and DO concentrations for the different $u_x,u_y$ values ($u_x=1,3,5$and$u_y=0.1,0.3,0.5$) are given in Table 18 and Table 19. The impact of variations in $u_x,u_y$ on the functions of pollutant and DO concentration is shown graphically in Figure 11 and Figure 12. We observe that when the values of $u_x,u_y$ rise, the values of p and q also rise.

Table 18. $p(x,0.07,0.4)$ for distinct values of $u_x,u_y$.

x	${\mathit{u}}_{\mathit{x}}=1,{\mathit{u}}_{\mathit{y}}=0.1$	${\mathit{u}}_{\mathit{x}}=3,{\mathit{u}}_{\mathit{y}}=0.3$	${\mathit{u}}_{\mathit{x}}=5,{\mathit{u}}_{\mathit{y}}=0.5$
0.1	3.225622782	6.184395	10.36456421
0.2	2.491370321	4.69464747	7.829728301
0.3	1.972220367	3.627437831	5.99953251
0.4	1.59825953	2.852467117	4.663582708
0.5	1.32409346	2.282575405	3.678457678
0.6	1.119740513	1.858551313	2.945060114
0.7	0.96505829	1.539573101	2.394058674
0.8	0.846285232	1.297105722	1.976393285
0.9	0.753864312	1.110946437	1.656992113
1	0.681053708	0.966624183	1.410554156

Table 19. $q(x,0.07,0.4)$ for distinct values of $u_x,u_y$.

x	${\mathit{u}}_{\mathit{x}}=1,{\mathit{u}}_{\mathit{y}}=0.1$	${\mathit{u}}_{\mathit{x}}=3,{\mathit{u}}_{\mathit{y}}=0.3$	${\mathit{u}}_{\mathit{x}}=5,{\mathit{u}}_{\mathit{y}}=0.5$
0.1	3.913953704	7.26423765	13.17560781
0.2	3.445155514	6.252528358	11.29058683
0.3	3.085736626	5.466032573	9.801609533
0.4	2.8027665	4.841455011	8.603480282
0.5	2.574740807	4.335990014	7.623323371
0.6	2.387303087	3.920094326	6.809704515
0.7	2.230643742	3.572959316	6.125655747
0.8	2.097897281	3.279632101	5.544134335
0.9	1.984140789	3.029156722	5.045020123
1	1.885758787	2.813354725	4.613098884

Figure 11. Comparison of pollutant concentration for fixed $u_x,u_y$.

Figure 12. Comparison of DO concentration for fixed $u_x,u_y$.

Table 20 and Table 21 provide the numerical values of the pollutant and DO concentrations for the various k values ($k=1,3,5$). In Figure 13 and Figure 14, the effects of changes in k on the functions of pollutant and DO concentration are graphically displayed. We can observe that when the value of k increases, the values of p and q rise.

Table 20. $p(x,0.07,0.4)$ for distinct values of k.

x	$\mathit{k}=1$	$\mathit{k}=3$	$\mathit{k}=5$
0.1	3.225623	7.292483	13.93789
0.2	2.49137	5.561109	10.50596
0.3	1.97222	4.33521	8.074585
0.4	1.59826	3.452954	6.32677
0.5	1.324093	2.807882	5.052338
0.6	1.119741	2.328956	4.11013
0.7	0.965058	1.968107	3.404136
0.8	0.846285	1.692361	2.868241
0.9	0.753864	1.478794	2.456363
1	0.681054	1.311259	2.136

Table 21. $q(x,0.07,0.4)$ for distinct values of k.

x	$\mathit{k}=1$	$\mathit{k}=3$	$\mathit{k}=5$
0.1	3.913954	5.732002	8.794366
0.2	3.445156	4.805046	7.069187
0.3	3.085737	4.123853	5.825489
0.4	2.802767	3.611715	4.911947
0.5	2.574741	3.218179	4.228409
0.6	2.387303	2.90952	3.707685
0.7	2.230644	2.662828	3.304071
0.8	2.097897	2.462277	2.986054
0.9	1.984141	2.29675	2.731601
1	1.885759	2.158305	2.525098

Figure 13. Comparison of pollutant concentration for fixed k.

Figure 14. Comparison of DO concentration for fixed k.

5. Conclusions

In the present study, we have presented the two-dimensional time fractional mathematical model for river pollution and its remediation for an unsteady state and investigated the effect of aeration. We successfully applied the fractional reduced differential transform method to solve the coupled nonlinear two-dimensional time fractional advection-diffusion equations. The research area is supposed to be initially polluted in the exponentially decayed form. We have observed that the concentration of DO and the value of pollutants decrease as the length and width of the river increase. The concentration value of DO and pollutants increases as time increases. When fractional order values rise from 0 to 1, the concentration of pollutant and DO decreases. We also checked the effect of different parameters on concentrations. We observed that the concentration of pollutant and DO increases as the longitudinal and transverse diffusion coefficients of the pollutant and DO increase. Also, the concentration of contaminants and DO will rise as the river’s longitudinal and transverse seepage velocity increases, and it will increase if the half-saturated oxygen demand concentration for pollutant degradation increases.