BIBECHANA Vol. 22, No. 1, April 2025, 41-51 ISSN 2091-0762 (Print), 2382-5340 (Online) Journal homepage: http://nepjol.info/index.php/BIBECHANA Publisher:Dept. of Phys., Mahendra Morang A. M. Campus (Tribhuvan University)Biratnagar Dynamics of wter pollution concentration with uniform and exponential increment of pollutants Jeevan Kafle1, Ranjan Kumar Chaudhary1, Bekha Ratna Dangol2,∗ 1Centarl Department of Mathematics, Tribhuvan University, Kathmandu, Nepal 2Department of Mathematics,Patan Multiple Campus, Tribhuvan University Lalitpur, Nepal ∗Corresponding author. Email: bdangol37@gmail.com Abstract The study focuses on the one dimensional Advection-Dispersion Equation (ADE) to study the dynamics of concentration of water pollution when the pollutants at the source is increasing uniformly and exponentially. Analytical solutions are obtained by using Laplace transform and numerical solutions are by Finite Difference Method (FDM). The steady state case is studied. The dynamics of pollution concentration along the length of river channel are shown through two dimensional plots by varying the rate of added pollutants, cross sectional area of river and water flow velocity. The pollution concentration decreases along the length of river (downstream) for each analysis. The analytical and numerical solutions are shown in three dimensional plots. The analytical and numerical solutions are compared with the help of relative error. The relative errors calculated for uniform and exponential increment of pollutants at the source are compared while studying the dynamics of the concentration. The environmental and chemical engineering may substantially benefit from such studies. Keywords Advection, Dispersion, Pollutant Concentration, Laplace transform, Finite Difference, Relative error. Article information Manuscript received: May 13, 2024; Revised: January 14, 2025; Accepted: January 16, 2025 DOI https://doi.org/10.3126/bibechana.v22i1.65799 This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons. org/licenses/by-nc/4.0/ 1 Introduction The rapid and unplanned development of urbaniza- tion, unregulated industries and highly dense pop- ulation in the urban areas are becoming a seri- ous problem in the natural environment [1]. More specifically, the degradation of the natural environ- ment are due to poor waste management, public transports, excessive use of pesticides in agricul- ture, deforestation, release of toxic materials from industries, unmanaged drainage and many more [1]. In some Eastern philosophies, water is often metaphorically linked to human civilization. How- ever, contamination in these water resources like pond, river and lake at the urban areas poses a sig- nificant challenge to the communities, vegetation and other species that rely on them [1]. The ma- jor causes for these water pollution are unmanaged growth of urbanization, industries, and lack of pub- lic awareness. Indeed, rivers play a crucial role in aquatic ecosystems by transporting water and nu- trients to various areas [2]. However, when contam- 41 http://nepjol.info/index.php/BIBECHANA bdangol37@gmail.com https://doi.org/10.3126/bibechana.v22i1.65799 https://creativecommons.org/licenses/by-nc/4.0/ https://creativecommons.org/licenses/by-nc/4.0/ Jeevan Kafle et al./ BIBECHANA 22 (2025) 41-51 42 inated with pollutants, these waterways can become hazardous to human health and other species. Con- sumption of contaminated water or exposure to pol- lutants can lead to sickness and even death among both humans as well as wildlife. These highlight the critical importance of protecting water sources from pollution [3]. As per a recent World Health Organi- zation (WHO) report, inadequate water quality is responsible for 3.1% of fatalities and contributes to 80% of illnesses [4]. In the study of water pollution, biochemical oxygen demand (BOD) serves as the gauge for assessing water pollution levels [5]. This analysis employed most of the parameters outlined by Pimpunchat et al. (2007), under the assump- tion that the predominant pollutants are biochemi- cal wastes [6]. For instance, Bagmati river is widely recognized as the most polluted river in Nepal with its biochemical oxygen demand (BOD) consistently on the rise [7] (See, Fig. 1). The extreme BOD is achieved due to the consequence of high levels of organic matter, high iron, and low dissolved oxy- gen [8]. Industrial waste and drainage from urban dwellers are responsible for the main source of BOD loads, leading to the increased concentration of the pollutants along the river. Prior studies on the wa- ter quality of Bagmati river revealed the degrada- tion in its lower portion with several significant pa- rameters that exceeded the limits set by the Na- tional Surface Water Quality Standards and Clas- sification [7]. The ongoing deterioration of water quality in the Bagmati river, revered as a holy river, serves as a compelling motivation for researchers to conduct systematic studies on water pollution. Pol- lution in river channel takes place due to disposal of waste materials from point and non-point pollution sources [5]. Understanding the transport and dispersion of pol- lutants from the source to body across the medium can substantially contribute for identifying sources of pollution, assessing the impact of contaminants on aquatic ecosystems, guiding and implementing pollution control measures and facilitating public awareness [1]. In overall, understanding, manag- ing, and mitigating the impacts of pollution on wa- ter resources are the important aspects in the study of water pollution [6]. To describe the pollution dis- persion in the water body, physical and mathemati- cal models are developed. Physical models are small scale representations of the water body. However, numerical experiments using mathematical models can cover the larger domain. These mathematical models are divided into statistical and deterministic models [1]. Statistical models rely on the analysis of historical monitoring data, while deterministic models are constructed based on mathematical de- scription of physical processes. Figure 1: Polluted Bagmati river (Left) and BOD Status in Bagmati river at major locations (cities) at different years (Right) [8]. BOD is in rise over time (t) and location (x), where the river flows downstream. More specifically, BOD contents con- sistently exceeded the limit for moderately con- taminated water (2-8 mg L−1) and even crossed the threshold for treated municipal sewage (20 mg L−1). For the mathematical models to study water pollution, various conservation equations have been developed to describe the movement of substances subject to both advection and diffusion [9]. Advec- tion refers as the transport of substances by the flow of water and dispersion refers to the spreading of substances due to concentration gradients, typically from areas of higher concentration to areas of lower concentration [6]. With the rise of environmental engineering and the need to model the transport of pollutants in underground and surface water, the advection-dispersion equation gained prominence and is developed during the latter half of the 20th century [9]. Due to the reliability, the advection and dispersion equations have been emerged in sev- eral fields, including environmental sciences, chem- istry, and physics. These equations are incredibly employed for the spreading of pollution which may take the form of any substance (gas, liquid or solid) or energy ( radioactivity, heat, sound, or light) [10]. Marusic [11] developed a model to describe the dis- Jeevan Kafle et al./ BIBECHANA 22 (2025) 41-51 43 persion of pollutants in systems resembling rivers over the temporal changes in pollutant concentra- tions. Pochai et al. [12] proposed a convection- diffusion model with constant coefficients to opti- mize contaminant levels in waste water treatment and successfully solved it by employing the finite element method. The primary objective was to re- duce the initial cost of water treatment, thereby achieving more cost-effective treatment solutions. Later on, it was extended to two dimensions by Tabuenca et al. [13]. The advection-diffusion equa- tions are employed for various prospective: predic- tion of the transport concentration of pollutants (Johari et al. [14]); contamination of rivers (Pim- punchat et al. [6]); deriving pollutant concentra- tion field (Miller et al. [15]). Poudel et al. [?, 10], solving one dimensional ADE for pollutant concen- tration with zero dispersion and steady state cases are analyzed for oxygen concentration with zero as well as non zero dispersion coefficients. Find- ing the analytical and numerical solutions for the developed models are another important aspects of this field. Lots of efforts have been made in finding analytical solutions in some states and re- duced situations using different approaches. The example includes the use of Laplace transform and with Greens functions for pollutant concentration (Carslaw and Jaeger [17]; Poudel et al. [10]); fi- nite difference approach in semi-infinite mediums (Savovic and Djordjevich [18]); semi infinite do- main with zero initial concentration (Van and Alves [9]); Laplace transform approach for the temporally and spatially dependent solute dispersion in a one- dimensional semi-infinite porous medium (Kumar et al. [19]). The purpose of the study is to obtain the analyt- ical and numerical solutions of advection disper- sion equation modeled for uniform and exponen- tial increment of pollution concentration at point source and hence observe the dynamics of pollution concentration at different spatial points along the length of the river. Laplace transform technique is used for analytical solution and Finite Differ- ence Scheme, Forward Time Central Space Scheme (FTCSS) is used for numerical solution. The ob- tained solutions are compared with the help of rel- ative error for more reliable result. 2 Mathematical Model and Numerical Method 2.1 The Governing Equation In this model developed by Manitcharoen and Pim- punchat (2020) [5], the involved parameters are based on the assumption that the contaminants are only due to the biochemical wastes [6]. The rate of change of the pollution concentration with position (x) and time (t) is expressed as ∂(AP ) ∂t = Dp ∂2(AP ) ∂x2 − ∂(vAP ) ∂x −K1 X X + k (AP )+qH(x), 0 ≤ x < L, t > 0 (1) where, H(x) is the Heaviside function as suggested by Chapra [20] (1997), defined as H(x) = { 1, 0 < x < L, 0, otherwise. (2) Here, v is the flow velocity of water along the river, P is the pollution concentration, Dp is the dis- persion coefficient, K1 is the pollutant degradation rate coefficient, q is added pollutant rate, k is half- saturated oxygen demand concentration, X repre- sents concentration of dissolved oxygen, and A is the cross sectional area of the river [5]. Pollutant of oxygen concentration quantities are considered homogeneous throughout the cross-section of river and can fluctuate over the its length. This pre- sumption meets the requirement as proposed by Dobbin [21]. For convenience, we assume that the parameters A, q, Dp, and K1 remain constant both spatially and temporally [22]. The pollution con- centration at the source may increase uniformly or sometimes, it may increase exponentially. The case of insignificant k (k≈0) is taken into consideration for analysis as it isnot possible to use Laplace trans- form technique to solve advection dispersion equa- tion [5]. From equation (1), the rate of change of the pollution concentration P1(x, t) resulting from a uniform increase in the pollution concentration at source is ∂(AP1) ∂t = Dp ∂2(AP1) ∂x2 − ∂(vAP1) ∂x −K1AP1 + qH(x). (3) Similarly, the pollution concentration P2(x, t) re- sulting from exponential rise in the pollution con- centration at source is ∂(AP2) ∂t = Dp ∂2(AP2) ∂x2 −∂(vAP2) ∂x −K1AP2+q(1−exp(−λx))H(x), (4) where the arbitrary constant λ represents the expo- nential pollution source term. Initially, the domain is free of solutes. So, the initial condition is P (x, t) = 0, x ≥ 0, t = 0. (5) Jeevan Kafle et al./ BIBECHANA 22 (2025) 41-51 44 The concentration at the origin is P0 and the con- centration gradient is supposed to be zero at infinite length of course. So, boundary conditions are P (x, t) = P0, x = 0, t > 0, (6) ∂P (x, t) ∂x = 0, x → ∞, t > 0. (7) 2.2 Analytical Solution Let the function P (x, t) is assumed to be bounded and defined for t > 0. Let P̄ (x, s) be Laplace transform of the function P (x, t), where s is transformed variable of t [5]. Using Laplace transform for equations (3) and (4), we get respectively, DpA d2P̄1(x, s) dx2 − vA dP̄1(x, s) dx −K1AP̄1(x, s) +A(sP̄1(x, s)− P1(x, 0)) + q s = 0, (8) DpA d2P̄2(x, s) dx2 − vA dP̄2(x, s) dx −K1AP̄2(x, s) +A(sP̄2(x, s)− P2(x, 0)) + q(1− exp(−λx)) s = 0, (9) where s > 0, x ≥ 0. The initial (5) and boundary conditions (6) and (7) are transformed respectively as P̄ (x, s) = 0, x ≥ 0, s = 0; P̄ (x, s) = P0 s , x = 0, s > 0; dP̄ dx (x, s) = 0, x → ∞, s > 0. (10) Using (10) in (8), we get d2 dx2 (P̄1(x, s))− v Dp d dx (P̄1(x, s))− K1 + s Dp P̄1(x, s) = − q sADp . (11) Auxiliary equation of (11) is m2 − v Dp m− K1 + s Dp = 0. (12) This is quadratic in m whose roots are m1 = v + √ v2 + 4Dp(K1 + s) 2Dp , and m2 = v − √ v2 + 4Dp(K1 + s) 2Dp . Thus, complimentary function and particular integral respectively are, c1e m1x + c2e m2x and q As(K1 + s) . The solution of (8) is P̄1(x, s) = c1e m1x + c2e m2x + q As(K1 + s) . (13) Using boundary condition (10), c1 + c2 = P0 s − q sA(K1 + s) . (14) Differentiating equation (13) with respect to x, we get dP̄1(x, s) dx = c1m1e m1x + c2m2e m2x. Again, by using boundary condition (10) in above equation, we get c1 = 0. Thus from (13), required solution is P̄1(x, s) = ( P0 s − q sA(K1 + s) ) em2x + q As(K1 + s) . Jeevan Kafle et al./ BIBECHANA 22 (2025) 41-51 45 Also, we can write P̄1(x, s) = ( P0 s − q sA(K1 + s) ) exp (( γ − √ s+ β2 Dp ) x ) + q As(K1 + s) , (15) where, γ = v 2Dp , β = √ v2 4Dp +K1. Now, we find the solution of equation (9) by the similar process as above. Using (10) in (9), we get d2 dx2 (P̄2(x, s))− v Dp d dx (P̄2(x, s))− K1 + s Dp P̄2(x, s) = −q(1− exp(−λx)) sADp . (16) Auxiliary equation of (16) is m2 − v Dp m− K1 + s Dp = 0. (17) So, roots of equation (17) are m1 = v + √ v2 + 4Dp(K1 + s) 2Dp , and m2 = v − √ v2 + 4Dp(K1 + s) 2Dp . Thus complimentary function and particular integral respectively are, c1e m1x + c2e m2x and q As(K1 + s) − qe−λx As(K3 + s) where, K3 = K1 − λv − λ2Dp. Thus, the solution of (16) is P̄2(x, s) = ( P0 s − q sA(K1 + s) + qe−λx As(K3 + s) ) em2x + q As(K1 + s) − qe−λx As(K3 + s) . (18) Using boundary conditions (10) and calculating same as above, we get the solution of equation (16) as P̄2(x, s) = ( P0 s − q sA(K1 + s) + qe−λx As(K3 + s) ) em2x + q As(K1 + s) − qe−λx As(K3 + s) . It can be written as P̄2(x, s) = ( P0 s − q sA(K1 + s) + qe−λx As(K3 + s) ) exp (( γ − √ s+ β2 Dp ) x ) + q As(K1 + s) − qe−λx As(K3 + s) . (19) Taking inverse Laplace transform to the equations (15) and (19), the respective analytical solutions of equations (3) and (4) are P1(x, t) = q AK1 (1− exp(−K1t)) + 1 2 ( P0 − q AK1 ) exp (( β√ Dp + γ ) x ) erfc ( x 2 √ Dpt + β √ t ) + exp (( −β√ Dp + γ ) x ) erfc ( x 2 √ Dpt − β √ t ) + q 2AK1 exp ( v Dp x−K1t ) .erfc ( x 2 √ Dpt + γ √ Dpt ) + exp(−K1t)erfc ( x 2 √ Dpt − γ √ Dpt ) , (20) Jeevan Kafle et al./ BIBECHANA 22 (2025) 41-51 46 P2(x, t) = q AK1 (1− exp(−K1t))− q AK3 exp(−λx)(1− exp(−K3t)) + 1 2 ( P0 − q AK1 + q AK3 ) . exp (( β√ Dp + γ ) x ) erfc ( x 2 √ Dpt + β √ t ) + exp (( −β√ Dp + γ ) x ) .erfc ( x 2 √ Dpt − β √ t ) + q 2AK1 exp ( v Dp x−K1t ) erfc ( x 2 √ Dpt + γ √ Dpt ) + exp(−K1t)erfc ( x 2 √ Dpt − γ √ Dpt ) − q 2AK3 exp (( θ√ Dp + γ ) x−K3t ) .erfc ( x 2 √ Dpt + θ √ t ) + exp (( −θ√ Dp + γ ) x−K3t ) erfc ( x 2 √ Dpt − θ √ t ) , (21) where, θ = √ v2/4Dp + vλ+Dpλ2. 2.3 The steady state case Taking t→∞ to the equations (20) and (21) gives the steady state case as follows P1(x, t → ∞) = q AK1 + ( P0 − q AK1 ) exp (( γ − β√ DP ) x ) , (22) and P2(x, t → ∞) = q AK1 − q AK3 exp(−λx) + ( P0 − q AK1 + q AK3 ) exp (( γ − β√ Dp ) x ) . (23) Now, taking x→∞ to calculate downstream pollutant concentration, we get P1(x → ∞, t → ∞) = P2(x → ∞, t → ∞) = q AK1 . (24) 2.4 Numerical technique Here, we use Forward Time and Central Space Scheme (FTCSS) to find numerical solution of equations (3) and (4). These equations in finite difference form can be written as [5], Pn+1 1m − Pn 1m k = Dp h2 (P1m+1 − 2Pn 1m + Pn 1m−1)− v 2h (Pn 1m+1 − Pn 1m−1)−K1P n 1m + q A , (25) Pn+1 2m − Pn 2m k = Dp h2 (P2m+1 − 2Pn 2m + Pn 2m−1)− v 2h (Pn 2m+1 − Pn 2m−1)−K1P n 2m + q A (1− exp(−λxn m)) (26) where, m and n refer to the discrete step size h and k respectively. The initial and boundary conditions (10) can be written respectively as [5] Pm,0 = 0, x ≥ 0; P0,n = P0, t > 0; PM,n = PM−1,n, x → ∞, t ≥ 0. (27) The stability condition is Dpk/h 2≤1/2 for the above schemes. 3 Results and Discussion We comparatively study the dynamics of the wa- ter pollution concentration over the river channel when the pollutants at the source increases uni- formly and exponentially along with the variation of the rate of added pollutant, cross section area of the water channel, and flow velocity over time. The parametric values are extracted from the dif- ferent existing literatures. For various plots to study the concentration dynamics, the values for involved parameters are taken as: Dp=10 m2 hr−1, Jeevan Kafle et al./ BIBECHANA 22 (2025) 41-51 47 v=4 mhr−1, A=10 m2, K1=1 hr−1, P0=1 kgm−3, q=1.02 kgm−1 hr−1 and λ=0.06289 [5]. 3.1 Comparison of dynamics of con- centrations P1 and P2 Keeping the same parameter values, the dynamics of the pollution concentrations P1 along the river for the uniformly increasing pollutants and P2 for exponentially increasing pollutants are revealed in the Fig. 2. Figure 2: Comparison of pollution concentrations P1 and P2 at different spatial regions when the pol- lutant at source increases uniformly and exponen- tially respectively. The concentration P2 is higher than P1 at the vicinity of the source of pollutant at x = 0 and then both saturates to the almost same level as the water flows substantially downstream. Initially, at the source x = 0, pollution concen- tration P2 is relatively larger than that of P1 as pol- lutants at the source is increasing exponentially for P2 and uniformly for P1. Both the concentrations decrease non-linearly over the distance and time un- til the flow travels 18 km. In this moment, the rate of decrement of P2 is substantially faster than that of P1. After traveling about 18 km from the source, both the concentrations decreases asymptotically. As a result, both graphs look like the same. How- ever, they vary by the incredibly smaller margin. The applied model enables to study the dynamics for the significantly longer travel distance. How- ever, we study only upto 30 km of travel distance. The concentration decreases rapidly for exponen- tial increment case rather than uniformly increment form along the length of the river channel. 3.2 Dynamics of concentrations P1 and P2 as rate of added pollutant varies The concentration dynamics of pollution P1 and P2 are shown in Fig. 3 as the rate of added pollutants q varies for q = 0.04, 1.02 and 2.00. The dispersion rate of the pollution is the greatest when the rate of added pollutants q is the least. On the contrary, the rate is the least when q is the greatest. Figure 3: Evolution of pollution concentration A: P1, B: P2 at different spatial regions along the river as rate of added pollutants (q) varies. The pollution concentration decreases at slow rate as the rate of added pollutants increases. As a result, the effect of rate of added pollutants q near the upstream is very less and that near the downstream is dominant in both the cases. But in case of Fig. 3B, the effect of rate of added pollu- tants q is very less near the origin as it is affected by exponential source term λ. For the case Fig. 3A, the concentration decreases non-linearly till x=19 km and then after it goes asymptotically whereas for the case Fig. 3B, it decreases non-linearly till x=21 km and then after goes asymptotically. The concentration of pollutants decreases at slow rate at any cross section of the river as the rate of added pollutants q increases. 3.3 Dynamics of concentration P1 and P2 as cross section area of the river channel varies The variation of pollution concentrations P1 and P2 with different cross sectional area A of the river Jeevan Kafle et al./ BIBECHANA 22 (2025) 41-51 48 channel, along the length of the river is shown in Fig. 4 as the cross sectional area of the river varies for A=5, 10, 80. Figure 4: Evolution of pollution concentration A: P1, B: P2 at different spatial regions along the river as cross sectional area A of the river channel varies. The pollution concentration decreases as the cross sectional area of the river channel increases. When the cross sectional area of the river chan- nel is less, then there is not enough space for pol- lutants to be dispersed more rapidly. Thus, the concentration of pollutants remains condensed and is more for least area. Thus, the pollution concen- tration decreases at any cross section of the river as the cross sectional area increases. The rate of decrement of concentration is more when the cross sectional area is more and is less for less area. The effect of cross sectional area is less for case Fig. 4A and very less for case Fig. 4B near the upstream and dominant near the downstream. The increment of pollutants is exponential in case of Fig. 4B, so the concentration profile near the origin seems to be very close. But it is not so close in case of uniform increment. For the case Fig. 4A, the concentration decreases non-linearly till x=21 km and then after it goes asymptotically whereas for the case Fig. 4B, it decreases non-linearly till x=23 km and then after goes asymptotically. In overall, the concentration decreases along the length of river channel as cross sectional area increases. 3.4 Dynamics of concentration P1 and P2 as flow velocity of the river varies The variation of pollution concentrations P1 and P2 with different water flow velocity v along Figure 5: Evolution of pollution concentration A: P1, B: P2 at different spatial regions along the river as the flow velocity of water v varies. The pollution concentration increases as the flow velocity of water increases. the length of the river is shown in Fig. 5. With the increase of velocity, advection becomes more dominant than dispersion and hence the rate of decrement of concentration of the pollution de- creases with increase in water flow velocity along the length of the river. If the water flow velocity is least, then pollutants get dispersed very near to the origin and concentration decreases so rapidly. But when water flow velocity is high then the concen- tration does not get chance to disperse so rapidly at the same position and hence rate of decrement of concentration becomes slower. For v=2, 4 & 6, concentration decreases non-linearly till the posi- tion x=16, 20 & 23 km in case of Fig. 5A, and x=17, 21 & 25 km in case of Fig. 5B, respectively and then goes asymptotically. Jeevan Kafle et al./ BIBECHANA 22 (2025) 41-51 49 3.5 Dynamics of the concentration as travel distance and travel time both varies We obtained the analytical and numerical solutions of Advection Dispersion Equation (ADE) with uni- form and exponential increment of pollutants at ori- gin by employing Laplace transform technique and Finite difference technique respectively. Figure 6: Analytical: A and numerical: B solution of equation (3). The both solutions do not provide exactly same values but provide nearly comparable values. Fig. 6 shows the pollution concentration P1(x, t) at each grid point of time and space. In both cases Fig. 6A and Fig. 6B, concentration at origin is the highest and goes down as distance in- creases. The concentration decreases rapidly along the distance whereas it decreases slowly along time. The concentration at peak point i.e at x=30, t=3, is very less approximate to zero but not actually zero. Though values of concentration obtained from analytical and numerical solutions are not exactly same, they are close to each other. Thus nature of both the three dimensional plots are same but not exact. Similarly, Fig. 7 shows the nearly same thing as described for Fig. 6 i.e. concentration at origin is the highest and decreases rapidly along the dis- tance and slowly along the time. But concentra- tion at origin for Fig. 7 is higher than that of Fig. 6. This clears that Forward Time Central Space Scheme (FTCSS) is somehow good scheme to solve these modeled equations but not perfect. Figure 7: Analytical: A and numerical: B solution of equation (4). The both solutions do not provide exactly same values but provide nearly comparable values 3.6 Error Analysis The solutions obtained analytically and numerically can be compared with relative error which is defined by [5], Relative error = ∣∣∣∣∣Panalytical − Pnumerical Panalytical ∣∣∣∣∣ (28) The relative errors of pollution concentrations for both the cases by h=1 and k=0.03 at t=3 hours are tabulated below. The domain for distance is taken up to 20 km from the origin to analyze the rela- tive error between analytical and numerical solu- tion of Advection-Dispersion Equation (ADE). Ta- ble 1 shows the relative error for P1 and Table 2 shows that for P2. From both the tables, Table 1 and Table 2, we can see that concentrations goes on decreasing with increase in distance. The relative error is less near the origin and more at far from the origin within the domain. The error for both pollution concentrations P1 and P2 is shown with the help of histogram below in Fig. 8, where red colored pillar shows the analytical solution whereas green colored pillar shows the numerical solution. As the analytical and numerical solutions for P1 and P2 are compared, the relative error is higher in computing P2 rather than P1, which can be seen clearly from the tables given below. Table 1: Relative Error of pollution concentration P1 [Kg m−3] at t = 3 hours. Distance (km) Analytical solution Numerical solution Error Error % 0 2.301356 2.283520 0.007750 0.770 5 0.945251 0.930611 0.015400 1.548 10 0.409821 0.399460 0.025281 2.530 15 0.193824 0.183838 0.05152 5.152 20 0.104777 0.098491 0.05999 5.999 Jeevan Kafle et al./ BIBECHANA 22 (2025) 41-51 50 Table 2: Relative Error of pollution concentration P2 [Kg m−3] at t = 3 hours. Distance (km) Analytical solution Numerical solution Error Error % 0 2.405330 2.427610 0.009260 0.926 5 1.089690 1.121450 0.029140 2.914 10 0.507550 0.477210 0.059770 5.977 15 0.182539 0.169762 0.069990 6.990 20 0.085641 0.092935 0.085160 8.516 Figure 8: Comparison of analytical and numeri- cal solutions for water pollution concentration with uniform (Left) and exponential (Right) increment of pollutants at origin. In both the cases analytical as well as numerical solutions are obtained close to each other. 4 Conclusion In this study, one dimensional Advection- Dispersion Equation (ADE) is applied to study the dynamics of water pollution concentration at dif- ferent spatial region along the length of river as the pollutants in the source increasing uniformly and exponentially. The analytical solutions of the em- ployed model are presented by employing Laplace transform and steady state state case is studied. It has been found that concentration becomes a fixed positive constant when time and space both tends to infinity. Dispersion coefficient of pollution concentration is taken to be non zero. The results show that the pollution concentration at the source is relatively larger in the case of exponentially incre- ment than that for uniformly increment. However, as the water flows substantially, these concentra- tions vary with small margin due to their advection and dispersion through the medium of water. The variations of concentrations P1 and P2 are pre- sented along the length of the river with respect to the variation in rate of added pollutants, cross- sectional area of the river, water flow velocity. The results reveals that as the water flows downstream, the concentration decreases. However, their rate of decrement varies. As the rate of added pollu- tants increases, the concentration decreases slowly. Similarly, as velocity of the water increases, the water pollution advects rather than the dispersion that slows the decrease in concentration variation. When the area of cross section of the river channel increases, there will high discharge of water through the channel. Consequently, pollution concentration quickly decreases. The three dimensional plots reveal that pollution concentration is the highest at origin (source) at the initial time and it decreases over time and space. However, it quickly decreases with traveled distance rather than traveled time due to the increment of pollutants either uniformly or exponentially. The numerical solutions of the model equations are ob- tained by Finite Difference Method (FDM) using Forward Time Central Space Scheme (FTCSS). As the analytical and numerical solutions for P1 and P2 are compared, the relative error is higher in com- puting P2 rather than P1. These results may be useful in better understanding the water pollution that may contribute to environmental engineer- ing, hydrology, contaminant transport modeling and chemical engineering where the advection and dispersion of contaminants is widely applied. References [1] P. Goyal and A. Kumar. Mathematical model- ing of air pollutants: An application to Indian urban city. Air Quality-Models and Applica- tions. doi:10.5772/16840, 2011. Jeevan Kafle et al./ BIBECHANA 22 (2025) 41-51 51 [2] A. S. Wadi, M. F. Dimian, and F. N. Ibrahim. Analytical solution of one-dimensional advection-dispersion equation of the pollutant concentration. Journal of Earth System Science, 123(6):1317–1324, 2014. [3] S. A. Alruman, A. F. El-kott, and M. A. Keshk. Water pollution: Source and treatment. Amer- ican Journal of Environmental Engineering, 6(3):88–98, 2016. [4] M. Haseena, M. F. Malik, A. Javed, S. Ar- shad, N. Adif, S. Zulqar, and J. Hanif. Water pollution and human health. Environmental Risk Assessment and Remediation, 1(3):16–19, 2017. [5] N. Manitcharoen and B. Pimpunchat. Analyt- ical and numerical solutions of pollution con- centration with uniformly and exponentially increasing forms of sources. Journal of Applied Mathematics, 2020:1–9, 2020. [6] B. Pimpunchat, W. L. Sweatman, W. Tri- ampo, G. C. Wake, and A. Parshotam. Mod- elling river pollution and removal by aeration. Modelling and Simulation Society of Australia and New Zealand, pages 2431–2437, 2007. [7] B. K. Mishra, R. K. Regmi, Y. Masago, K. Fukushi, P. Kumar, and C. Saraswat. As- sessment of Bagmati river pollution in Kath- mandu Valley: Scenario-based modeling and analysis for sustainable urban development. Sustainability of Water Quality and Ecology, 9:67–77, 2017. [8] K. Pokhrel. Water quality monitoring of Bag- mati river basin, Kathmandu Valley. Doctoral dissertation, Amrit Campus, 2023. [9] M. T. Van Genuchten and W. J. Alves. Analyt- ical solution of the one-dimensional convective- dispersive solute transport equation volume bulletin no.1661. US Department of Agricul- ture, Agricultural Research Service, 1982. [10] K. Poudel, J. Kafle, and P. S. Bhandari. Analytical solution for advection-dispersion equation of the pollutant concentration us- ing Laplace transformation. Journal of Nepal Mathematical Society, 4(1):33–40, 2021. [11] G. A. L. Marusic. A study on the mathemat- ical modeling of water quality in river type aquatic systems. Journal WSEAS Transac- tions on Fluid Mechanics, 8(2):80–89, 2013. [12] N. Pochai, S. Tangmanee, L. J. Crane, and J. J. Miller. A mathematical model of water pollu- tion control using the finite element method. Proceedings in Applied Mathematics and Me- chanics, 6(1):755–756, 2006. [13] P. Tabuenca, J. Vila, J. Cardona, and A. Samartin. Finite element simulation of dis- persion in the Bay of Santander. Advances in Engineering Software, 28(5):313–332, 1997. [14] H. Johari, N. Rusli, and Z. Yahya. Finite difference formulation for prediction of water pollution. Materials Science and Engineering, 318(012005):1–10, 2018. [15] J. J. Miller, N. Pochai, S. Tangmanee, and L. J. Crane. A water quality computation in the uniform channel. Journal of Interdisciplinary Mathematics, 11(6):803–814, 2008. [16] G. E. Roberts and H. Kaufman. Table of Laplace transforms. W. E. Saunders Co., Philadelphia and London, 1969. [17] H. S. Carslaw and J. C. Jaeger. Conduction of heat in solids. Clarendon Press, Oxford, UK, 1959. [18] S. Savović and A. Djordjevich. Finite difference solution of the one-dimensional advection– diffusion equation with variable coefficients in semi-infinite media. International Journal of Heat and Mass Transfer, 55(15-16):4291–4294, 2012. [19] A. Kumar, D. K. Jaiswal, and N. Kumar. Ana- lytical solutions to one-dimensional advection– diffusion equation with variable coefficients in semi-infinite media. Journal of Hydrology, 380(3-4):330–337, 2010. [20] C. S. Chapra. Surface water-quality modeling. The McGraw-Hill Companies, 1997. [21] W. E. Dobbins. BOD and oxygen relationships in streams. Journal of the Sanitary Engineer- ing Division, 90(3):53–78, 1964. [22] G. Li. Stream temperature and dissolved oxy- gen modeling in the lower Flint River basin, GA. Doctoral dissertation, University of Geor- gia, 2006. [23] G. E. Roberts and H. Kaufman. Table of Laplace transforms. W. E. Saunders Co., Philadelphia and London, 1969. [24] D. K. Salkuyeh. On the finite difference ap- proximation to the convection-diffusion equa- tion. Applied Mathematics and Computation, 179(1):79–86, 2006. [25] R. K. Singh, H. Yabar, N. Nozaki, and R. Rakwal. Analyzing waste problems in de- veloping countries: Lessons for Kathmandu, Nepal through analysis of the waste system in Tsukuba city, Japan. Journal of Scientific Re- search and Reports, 8(6):1–13, 2015. Introduction Mathematical Model and Numerical Method The Governing Equation Analytical Solution The steady state case Numerical technique Results and Discussion Comparison of dynamics of blackconcentrations P1 and P2 Dynamics of blackconcentrations P1 and P2 as rate of added pollutant varies Dynamics of concentration P1 and P2 as cross section area of the river channel varies Dynamics of concentration P1 and P2 as flow velocity of the river varies Dynamics of the concentration as travel distance and travel time both varies Error Analysis Conclusion