Research ArticleAnalytical and Numerical Solutions of PollutionConcentration with Uniformly and Exponentially IncreasingForms of Sources

N. Manitcharoen and B. Pimpunchat

Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand

Correspondence should be addressed to B. Pimpunchat; busayamas.pi@kmitl.ac.th

Received 8 August 2019; Revised 24 December 2019; Accepted 13 March 2020; Published 1 April 2020

Academic Editor: Mariano Torrisi

Copyright © 2020 N. Manitcharoen and B. Pimpunchat. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original workis properly cited.

The study of pollution movement is an important basis for solving water quality problems, which is of vital importance in almostevery country. This research proposes the motion of flowing pollution by using a mathematical model in one-dimensionaladvection-dispersion equation which includes terms of decay and enlargement process. We are assuming an added pollutantsources along the river in two cases: uniformly and exponentially increasing terms. The unsteady state analytical solutions areobtained by using the Laplace transformation, and the finite difference technique is utilized for numerical solutions. Solutionsare compared by relative error values. The result appears acceptable between the analytical and numerical solutions. Varying thevalue of the rate of pollutant addition along the river (q) and the arbitrary constant of exponential pollution source term (λ) isdisplayed to explain the behavior of the incremental concentration. It is shown that the concentration increases as q and λincrease, and the exponentially increasing pollution source is a suitable model for the behavior of incremental pollution alongthe river. The results are presented and discussed graphically. This work can be applied to other physical situations described byadvection-dispersion phenomena which are affected by the increase of those source concentrations.

1. Introduction

One of themajor problems for area-connected water resourcesis water pollution. The source of this problem stems from theexpansion of the community, the increase of industrial plants,and the increase of agricultural farms. Rivers carry water andnutrients to areas all around them, which is why contaminatedwater is having an impact on human health and is the leadingcause of disease worldwide. Sources of water pollution occurdue to the disposal of untreated point and nonpoint sourcepollutants. Point-source pollution refers to the pollutants dis-charged into a river from one discrete location or point, suchas industrial or residential wastewater. The indicator usedto measure the amount of pollution in water is biochemicaloxygen demand (BOD). In this study, it is assumed that pol-lutants are largely biochemical waste and use the parametersfrom the Tha Chin river, a particular river in Thailand, as perPimpunchat et al. (2007) [1, 2].

In Thailand, the Tha Chin river is one of the country’smost polluted river. It has a continuously enlarged BOD.Agriculture is, apparently, the most important source ofBOD loads affecting BOD concentrations along the river,followed by aquaculture, swine farms, rice cultivation, andagro-industry. Previous papers on the water quality of theTha Chin river found that the lower part of the river wasdegraded, and that several major parameters exceeded theNational Surface Water Quality Standards and Classificationlimits. The major water quality problems were low dissolvedoxygen (DO), high ammonia nitrogen, high fecal coliformbacteria, high turbidity, and high organic matter (BOD).The degradation of water quality in rivers has affected thewater quality and natural resources of the Gulf of Thailand.[3–5]. Figure 1 shows the monitoring of BOD for the thirdquarter of the year (April-June) between 2011 and 2018; itplotted from raw data that obtained from the mobile applica-tion developed by Regional Environmental Office 5 [6]. It

HindawiJournal of Applied MathematicsVolume 2020, Article ID 9504835, 9 pageshttps://doi.org/10.1155/2020/9504835

indicates that BOD concentrations repeatedly exceededthe Class 4 BOD standards (2-4mg/L) throughout the river[7, 8]. Understanding the movement of pollution is usefulfor controlling and predicting water quality in rivers. Forthe assessment of the transport of contaminant changes inwater quality (e.g., BOD), an advection-dispersion equation(ADE) is commonly used. ADE is a partial differential equa-tion (PDE) derived from the mass balance applied to a massvolume unit in a river. The solution includes both analyticalsolutions and numerical solutions, based on the complica-tions of the model. The Laplace transform technique hasbeen widely applied to develop analytical solutions toADE. Previous works in which ADE has been used in anal-ysis include M.Th. Van Genuchten and W.J. Alves (1982),who presented analytical solutions of a one-dimensionalconvective-dispersive solute transport equation under a vari-ety of conditions [9]; Kumar et al. (2010), who obtained ana-lytical solutions for temporally and spatially dependentsolute dispersion in a one-dimensional semi-infinite porousmedium by using Laplace transform technique [10]; S. Savo-vic’ and A. Djordjevich (2012), who described numericalsolution by using the finite difference method for the 1-Dadvection diffusion partial differential equation with variablecoefficients in semi-infinite media [11]; and Pimpunchatet al., who studied a mathematical model of pollution in river,presenting a simple model and providing some analyticalsolutions at steady state flows [1, 2].

In this study, the objective is to propose unsteady statesolutions of the pollutant concentration by consideringadvection-dispersion equations in one dimension whichincludes terms of decay and increasing sources. Increasingsources were analysed by assuming the rate of pollutant addi-tion along the river: q in two cases and uniformly increasing

and exponentially increasing forms. We approximate thearbitrary constants: λ of exponentially increasing pollutionsource terms for predicting the behavior of concentrationmovement. Analytical solutions are obtained by using theLaplace transform technique and comparing the results byrelative error with the numerical solution by using theexplicit finite difference technique.

2. Materials and Methods

2.1. The Governing Equation. The unsteady flow in the riveris modeled as 1-D using a single spatial xðmÞ to describethe distance down the river from its source. Quantities, suchas pollutants or oxygen concentrations, are only allowed tovary along the length of the river and are treated as homoge-neous across the river cross-section. This assumption is justi-fied by fulfilling Dobbin’s criterion [12]. We use a singlequantity to measure water pollution; the concentration ofthe pollutant Pðx, tÞ (kgm-3) is assumed to vary with time t(day). This is in the same form as a water pollution equation,2.1, presented by Pimpunchat et al. [2]. The rate of change ofthe concentration with position x and time t are expressed as

∂ APð Þ∂t

=Dp∂2 APð Þ∂x2

−∂ vAPð Þ∂x

− K1X

X + kAP

+ qH xð Þ, 0 ≤ x < L ≤∞, t > 0ð Þ,ð1Þ

where HðxÞ is the Heaviside function

H xð Þ =1, 0 < x < L,0, otherwise,

(ð2Þ

30

25

20

15

BOD

(mg/

L)

10

5

0318 290 275 255 237 202 190 140 118

Km from river mouth82 60 53 45 34 16 0

201620172018Standard

20112012201320142015

Figure 1: BOD status in the Tha Chin river (April-June) during 2011-2018. (Modified from raw data which is available from [6]).

2 Journal of Applied Mathematics

this equation is standard and was develop in Chapra [13]. Weconsider a river where pollutants are discharged in the formof wastes. It is assumed that these pollutants use dissolvedoxygen Xðx, tÞ for various biochemical and biodegradationprocesses. The discharge of pollutants into the river is atthe constant rate q, A is the cross-section area of the river,Dp is the dispersion coefficient of pollutant in the x direction,v is the water velocity in the x direction, and K1 is the degra-dation rate coefficient for pollutants. For convenience, allparameters, such as A, Dp, v, K1, and q, hold constant valuesover time and space. Analysis is considered by the case ofnegligible k ðk ≈ 0Þ, and pollution sources are considered intwo cases, uniformly increasing P1ðx, tÞ and exponentiallyincreasing pollution sources P2ðx, tÞ, demonstrated by Equa-tions (3) and (4), respectively

∂ AP1ð Þ∂t

=Dp∂2 AP1ð Þ∂x2

−∂ vAP1ð Þ

∂x− K1AP1 + qH xð Þ, ð3Þ

∂ AP2ð Þ∂t

=Dp∂2 AP2ð Þ∂x2

−∂ vAP2ð Þ

∂x− K1AP2

+ q 1 − exp −λxð Þð ÞH xð Þ,ð4Þ

where λ is an arbitrary constant of exponential pollutionsource terms. The exponentially increasing form of pollutionsources Equation (4) assumed by, the referred papers wasfound that the downstream pollution of the Tha Chin riveris higher than the upstream, which is caused by geography,including various contributions from branch river to riverand the increasing wastewater that comes from swine andrice farming [4–6]. Figure 2 sketches the physical system ofthe pollution sources described in Equations (3) and (7).

Let the domain be supposedly initially solute free, thus,initial conditions are

P x, tð Þ = 0, x ≥ 0, t = 0: ð5Þ

Boundary conditions at the origin of domain are consid-ered by the uniformly increasing source concentration. Theconcentration gradient at the infinity is assumed to be zero.Then, boundary conditions associated with Equation (3) ina semi-infinite domain are as follows;

P x, tð Þ = P0, x = 0, t > 0, ð6Þ

∂P∂x

x, tð Þ = 0, x⟶∞, t > 0, ð7Þ

where P0 is source concentration at the origin.

2.2. Analytical Technique. Given a function Pðx, tÞ definedfor all t > 0 and assumed to be bounded, the Laplace trans-form in t considering x as parameter can be applied to Equa-tion (8) [14, 15];

�P x, sð Þ = L P x, sð Þ ; t⟶ sf g =ð∞0e−stP x, tð Þdt, ð8Þ

where s is called the transform variable. Applying Laplacetransformation to Equations (3) and (4) then, we get

DpAd2P1 x, sð Þ

dx2− vA

dP1 x, sð Þdx

− K1A�P x, sð Þ

− A sP1 x, sð Þ − P1 x, 0ð Þ� �+ q

s= 0, x ≥ 0, s > 0,

ð9Þ

7 × 10−8

6

Rate

of p

ollu

tant

addi

tion

: q−(

kg/m

) day

5

4

3

2

1

00 100 200

Flow

Distance: x (km)300

UniformlyExponential form

Figure 2: Physical system of the rate of pollutant addition q.

3Journal of Applied Mathematics

DpAd2P2 x, sð Þ

dx2− vA

dP2 x, sð Þdx

− K1AP2 x, sð Þ

− A sP2 x, sð Þ − P2 x, 0ð Þ� �+ q

s1 − exp −λxð Þð Þ

= 0, x ≥ 0, s > 0:

ð10Þ

Transformed initial and boundary conditions on Equa-tions (5), (6), and (7) give

�P x, 0ð Þ = 0, ð11Þ

�P 0, sð Þ = P0s, ð12Þ

and

d�Pdx

x, sð Þ = 0 as x⟶∞: ð13Þ

Evaluating the solutions of Equations (9) and (10) byusing initial and boundary conditions. Equations (11), (12),and (13) provide their solutions in the Laplace domain,which may be written as

�P1 x, sð Þ = P0s

−q

As s + K1ð Þ� �

exp γ −

ffiffiffiffiffiffiffiffiffiffiffis + β2

Dp

s !x

!

+ qAs s + K1ð Þ ,

ð14Þ

�P2 x, sð Þ = P0s

−q

As s + K1ð Þ + qAs s + K3ð Þ

� �exp

� γ −

ffiffiffiffiffiffiffiffiffiffiffis + β2

Dp

s !x

!+ qAs s + K1ð Þ

−q exp −λxð ÞAs s + K3ð Þ ,

ð15Þ

where γ = v/2Dp, β = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2/4Dp + K1

p, and K3 = K1 − vλ −

Dpλ2.

2.3. Numerical Technique. In this section, we present thenumerical technique, in order to use the explicit finite differ-ence technique by using the forward differences scheme forthe time and the central derivatives for the space. Thus,Equations (3) and (7) in the finite difference form can bewritten as

P1n+1m − P1

nm

Δt=

Dp

Δx2Pn1m+1 − 2P1

nm + P1

nm−1ð Þ

−v

2Δx P1nm+1 − P1

nm−1ð Þ − K1P1

nm

+ qA

+ 0 Δx2, Δt� �

,

ð16Þ

P2n+1m − P2

nm

Δt=

Dp

Δx2P2

nm+1 − 2P2

nm + P2

nm−1ð Þ

−v

2Δx P2nm+1 − P2

nm−1ð Þ − K1P2

nm

+ qA

1 − exp −λxnmð Þð Þ + 0

Δx2, Δt� �

,

ð17Þ

where indexes m and n refer to the discrete step size Δx andthe time step size Δt, respectively. The initial condition (4)and boundary conditions (5) and (6) for Equations (3) and(7) can be expressed in the finite difference form as;

Pm,0 = 0, x ≥ 0,P0,n = P0, t > 0,PM,n = PM−1,n, x⟶∞,t ≥ 0:

ð18Þ

3. Results and Discussions

3.1. Analytical Solutions. The solutions of the Equations (14)and (15) are found by inverting the Laplace transformdenoted by the complex variable Equation (16).

P x, tð Þ = L−1 �P x, sð Þ ; s⟶ t� �

= 12πi

ðc+i∞c−i∞

�P x, sð Þe−stds, s > 0:ð19Þ

Applying the shift theorem and the convolution theorem[16], then the analytical solution with uniformly increasingpollution source P1ðx, tÞ is

P1 x, tð Þ = qAK1

1 − exp −K1tð Þð Þ + 12 P0 −

qAK1

� �exp

� βffiffiffiffiffiffiDp

p + γ

!x

!erfc x

2 ffiffiffiffiffiffiffiDpt

p + βffiffit

p !

+ exp −βffiffiffiffiffiffiDp

p + γ

!x

!erfc x

2 ffiffiffiffiffiffiffiDpt

p − βffiffit

p !

+ q2AK1

exp vDp

x − K1t

!erfc

� x

2 ffiffiffiffiffiffiffiDpt

p + γffiffiffiffiffiffiffiDpt

q !+ exp −K1tð Þ erfc

� x

2 ffiffiffiffiffiffiffiDpt

p − γffiffiffiffiffiffiffiDpt

q !,

ð20Þ

4 Journal of Applied Mathematics

and for the analytical solution with exponentially increasingpollution source P2ðx, tÞ is

P2 x, tð Þ = qAK1

1 − exp −K1tð Þð Þ

−q

AK3exp −λxð Þ 1 − exp −K3tð Þð Þ

+ 12 Po −

qAK1

+ qAK3

� �exp

� βffiffiffiffiffiffiDp

p + γ

!x

!erfc x

2 ffiffiffiffiffiffiffiDpt

p + βffiffit

p !

+ exp −βffiffiffiffiffiffiDp

p + γ

!x

!erfc x

2 ffiffiffiffiffiffiffiDpt

p − βffiffit

p !

+ q2AK1

exp vDp

x − K1t

!erfc

� x

2 ffiffiffiffiffiffiffiDpt

p + γffiffiffiffiffiffiffiDpt

q !+ exp −K1tð Þ erfc

� x

2 ffiffiffiffiffiffiffiDpt

p − γffiffiffiffiffiffiffiDpt

q !−

q2AK3

exp

� θffiffiffiffiffiffiDp

p + γ

!x − K3t

!erfc x

2 ffiffiffiffiffiffiffiDpt

p + θffiffit

p !

+ exp −θffiffiffiffiffiffiDp

p + γ

!x − K3t

!erfc

� x

2 ffiffiffiffiffiffiffiDpt

p − θffiffit

p !,

ð21Þ

where θ =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2/4Dp + vλ +Dpλ

2q

.

3.2. The Steady State Solution. The steady solution is derivedfrom Equations (20) and (21) by taking limit t⟶∞.Hence, pollutant concentrations in this state give

P1 xð Þ = qAK1

+ P0 −q

AK1

� �exp γ −

βffiffiffiffiffiffiDp

p !

x

!,

P2 xð Þ = qAK1

−q

AK3exp −λxð Þ +

�P0 −

qAK1

+ qAK3

�exp γ −

βffiffiffiffiffiffiDp

p !

x

!:

ð22Þ

The downstream pollutant concentration limit werecalculated by x⟶∞ therefore gives

P1 x⟶∞, t⟶∞ð Þ = P2 x⟶∞, t⟶∞ð Þ = qAK1

:

ð23Þ

This limit is the same result which was obtained byPimpunchat et al. [1].

3.3. Analytical and Numerical Simulation. Numerical solu-tions are obtained by rearranging Equations (16) and (17);then, the numerical solutions must satisfy

Pn+1m = BPn

m+1 + CPnm + EPn

m−1 +Q, ð24Þ

where B =DPΔt/Δx2 − vΔt/2Δx, C = 1 − 2DPΔt/Δx2 − K1Δt,E =DPΔt/Δx2 + vΔt/2Δx, Q1 = qΔt/A for uniformly increas-ing pollution source, and Q2 = qð1 − exp ð−λxnmÞÞΔt/A for

0

0.5

1

1.5

2

2.5

3

0 10 20 30 40 50Distance from source (km)

q = 0.06 kg m−1 day−1

q = 0.48 kg m−1 day−1

Uniformly increasing form of source at t = 1 day× 10−5

Conc

entr

atio

n−P

1 (kg

m−3

)

0

0.5

1

1.5

2

2.5

3

0 10 20 30 40 50Distance from source (km)

q = 0.48 kg m−1 day−1

q = 0.06 kg m−1 day−1

Exponentially increasing form of source at t = 1 day× 10−5

Conc

entr

atio

n−P

2 (kg

m−3

)

q = 0.06 kg m−1 day−1

q = 0.48 kg m−1 day−1

0

0.5

1

1.5

2

2.5qqqqq = 0.48 kg m−1 da

q = 0.06 kg m−1 day

Conc

entr

atio

n−P

2PP

(kg

m−3

)Figure 3: Analysis of pollutant concentrations P1 and P2 of various values of the rate of pollutant addition (q) at t = 1 day and 0:06 ≤ q ≤ 0:48:

5Journal of Applied Mathematics

exponentially increasing pollution source. The x and t meshesmust be chosen DpΔt/ðΔxÞ2 ≤ 1/2 in order to ensure stability.

Relative error =Panalytical − Pnumerical

Panalytical

: ð25Þ

The parameter values used in this study are the same asper Pimpunchat et al. [1]: Dp = 3:456 × 106 (m2day-1), v =4:32 × 104 (mday-1), A = 2:1 × 103 (m2), K1 = 8:27 (day-1),and P0 = 1 × 10−9 (kgm-3). Figure 3 shows pollutant concen-trations with various rates of pollutant addition along theriver by various distance steps x at time t = 1 day. It startingfrom q = 0:06 to q = 0:48 (kgm-1day-1), the subfigure on theleft-hand side shows q in the form of uniform increase byEquation (20), and the right-hand side shows q in the formof exponential increase from Equation (21). The arbitraryconstant of the exponential pollution source term (λ =0:06289day-1) is assumed by the total rate of pollutant addi-tion q being reduced by 5% along the river (L = 318 km),approximately. At the same position x, the result of addingpollutants by different terms makes it clear that the concen-tration in the right subfigure is less than the concentrationin the left subfigure. The concentrations increasingly varyobviously with q; P increases as q increases. The rate of bothconcentrations increases rapidly near the origin pollutionsource and slowly far away. By Equation (6), the concentra-tion gradient at infinity is assumed to be zero; when the dis-tance is long enough, the pollutant concentrations convergeto a positive constant. The range of distance at which the con-centration is unchanged happens downstream; addition byuniformly increasing pollution source is shorter than additionby exponentially increasing pollution source.

The comparison between the analytical and numericalsolutions obtained by the Laplace transform technique andfinite different technique is illustrated in Figure 4 by step size

Δx = 0:1 and Δt = 0:001. The concentration values are shownin the longitudinal region 0 ≤ x ≤ 50 km at different timet = 0:01 (blue line), t = 0:1 (green line), and t = 1 (red line),respectively. The relative errors which are calculated byEquation (25) are provided in Table 1. The relative error ishigh near the origin and then decreases into a positive value.The maximum percentage relative error is excellent agree-ment between the analytical and the numerical solutions; itis less than 0.07% when x ≤ 1 km and less than 0.05% when1 < x ≤ 5 km, approximately. When the distance is longenough, the concentration values converge into a positiveconstant, causing the relative errors to gradually decrease

0

0.5

1

1.5

2

2.5

3

3.5

0 2 4 6 8 10 12 18 20Distance from source (km)

Uniformly increasing form of source× 10−6

Conc

entr

atio

n−P

1 (kg

m−3

)

14 16

t = 1 day

t = 0.1 day

t = 0.01 day

0

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25 30 45 50Distance from source (km)

Exponentially increasing form of source× 10−6

Conc

entr

atio

n−P

2 (kg

m−3

)

35 40

t = 1 day

t = 0.1 day

t = 0.01 day

NumericalAnalytical

Figure 4: Analytical and numerical solutions of pollutant concentrations P1 and P2 at t = 0:01, 0.1, and 1 day.

Table 1: Relative error of pollutant concentrations by Δx = 0:1 andΔt = 0:001 at t = 1 day.

Distance (km)Analyticalsolutions

Numericalsolutions

Relative errors

Uniformly increasing pollution source P1

5 2:10964 × 10−6 2:10957 × 10−6 3:48535 × 10−5

10 2:93091 × 10−6 2:93085 × 10−6 1:95423 × 10−5

15 3:25077 × 10−6 3:25074 × 10−6 1:02938 × 10−5

20 3:37535 × 10−6 3:37534 × 10−6 5:14846 × 10−6

25 3:42387 × 10−6 3:42387 × 10−6 2:47106 × 10−6

30 3:44277 × 10−6 3:442277 × 10−6 1:14859 × 10−6

Exponentially increasing pollution source P2

5 3:52318 × 10−7 3:52349 × 10−7 8:67962 × 10−5

10 9:63291 × 10−7 9:63313 × 10−7 2:27143 × 10−5

15 1:54747 × 10−6 1:54748 × 10−6 7:10092 × 10−6

20 2:02779 × 10−6 2:02779 × 10−6 2:07408 × 10−6

25 2:39946 × 10−6 2:39946 × 10−6 3:55824 × 10−7

30 2:67901 × 10−6 2:67900 × 10−6 1:95815 × 10−7

6 Journal of Applied Mathematics

into a positive constant value. In addition, pollutant concen-tration with 100 mesh points is depicted in Figure 5. Thegraph shows the different levels of increasing concentrationsby surface shading.

3.4. Capacity Approximation. The daily load of contamina-tion depends on the term of added pollutant q and thelength of river L. Certainly, that in case q increases theconcentration increases, as depicted by Figure 3. Figure 6illustrates the behavior of pollutant concentrations (P2/P0)by vary λ (0:25 × 10−5 ≤ λ ≤ 0:46667 × 10−5) compared withthe observed data between water quality sampling stationsTC 15 and TC 13. These stations were chosen because theywere separated between the middle and lower parts of riversand the BOD standard esteem multiplied (2-4mg/L). In thismanner, λ is simulated by the concentration at the end toextend by one time. After that, we utilize these values for esti-mated assessing the capacity load of pollutant per daybetween the two stations by L = 36 km, q = 60, load is about2,160 kg per day by uniformly incremental but the daily loadof the exponentially is less than up to 90% which appeared

approximation computation these values by Table 2. By thissimulation, we can offer at the same rate q the analytical solu-tion with exponentially increasing pollution source as a suit-able model for illustrating the behavior of incrementalpollution along the river.

3.5. Variable Coefficients. In our model, the medium of trans-port has a uniform dispersion and velocity. To apply ourmodel to the case of spatially temporally dependent coeffi-cients, it can therefore be modeled with variable coefficients as

∂ APð Þ∂t

= ∂∂x

Dp x, tð Þ ∂ APð Þ∂x

− v x, tð ÞAP �

− K1AP + qH xð Þ,

ð26Þ

let Dpðx, tÞ =Dpf1ðx, tÞ, vðx, tÞ = v0 f2ðx, tÞ, f1ðx, tÞ = fðmtÞ, and f2ðx, tÞ = 1 [10]. This model applies to temporallydependent dispersion along a uniform flow. By these trans-formations, dX/dx = −1/f1ðx, tÞ or X =

Ðdx/f1ðx, tÞ and T =

1

0.8

0.6

Pollutant concentration P1 (x,t) of uniformly increasing form of pollution source

0.4

Distance (km)

Distance (km)

0.2

0 0 1 2 3 4 5 6 7 8 9 10

× 10−6

0.5

1

1.5

2

2.5

1

0.8

0.6

Pollutant concentration P2 (x,t) of exponentially increasing form of pollution source

0.4Time (day)

Time (day)

0.2

0 0 1 2 3 4 5 6 7 89 10

× 10−7

54321

0

6789

Figure 5: Pollutant concentrations P1ðx, tÞ and P2ðx, tÞ with 100 mesh points by Δx = 0:1 and Δt = 0:01.

7Journal of Applied Mathematics

Ð t0dt/f ðmtÞ (Crank (1975)); then, Equation (26) reduces to

our model Equation (3) ∂ðAPÞ/∂T =Dpð∂2ðAPÞ/∂X2Þ + v0ð∂ðAPÞ/∂XÞ − K1AP + q, where Dp and v0 are constants inwhich the analytical solution can be obtained by using theLaplace transform technique the same with this work.

In addition, the model can apply to spatially dependentdispersion along a nonuniform flow by Dpðx, tÞ is consid-ered proportional to the square of vðx, tÞ instead of propor-tion to vðx, tÞ. Both terms in Equation (26) are obtained asfollows: f1ðx, tÞ = ð1 + axÞ2 and f2ðx, tÞ = ð1 + axÞ, where axis nondimensional and a is nonzero real constant accountingfor the variations in velocity and dispersion due to inhomo-geneity [10]. From previous transformation, we will haveanother spatial variable X = 1/að1 + axÞ then Equation (26)reduces to

∂ APð Þ∂t

= a2DpX2 ∂

2 APð Þ∂X2 + av0X

∂ APð Þ∂X

− av0X + K1ð ÞAP + q:

ð27Þ

To evaluate the analytical solution of Equation (27),additional studies can be conducted in this work and theprevious work [10]. Another case for the cross-section areain Equation (3) vary with position x, and we may get thedesired equation as

∂P∂t

=Dp∂2P∂x2

− v∂P∂x

− K1P + qA xð Þ , ð28Þ

in which the analytical solutions can be obtained by usingLaplace transformation techniques in this work as well.

4. Concluding Remarks

The unsteady state solutions of pollutant concentration byconsidering advection-dispersion equations in 1-D are pro-posed by using the Laplace transform technique and theexplicit finite difference technique, for analytical and numer-ical solutions, respectively. The model is presented by consid-ering the rate of pollutant addition along the river byuniformly increasing and exponentially increasing forms.The trend of concentrations is expanded with time and space,which is affected by the rate of pollutant addition along theriver q. It is obviously found that the limits of the down-stream concentrations that are increased q by the exponen-tially increasing form are less than the increased q by theuniformly increasing form, which is the result of the behaviorof added pollutant sources. In reality, this model is suitablefor the river which has pollution source that varies with

2

1.5

1P2

/P0

0.5

0118 Distance from river mount (km) 82

20162015

2013, 20172014

𝜆 = 4.16667 × 10−5

𝜆 = 2.5 × 10−5

𝜆 = 4.16667 × 10−5

𝜆 = 2.5 × 10−5

Figure 6: Behaviors of pollutant concentration (P2/P0) at t = 1 day by various arbitrary constants of exponentially increasing pollution sourceterms (0:25 × 10−5 ≤ λ ≤ 0:46667 × 10−5 day-1) with q = 60 kg/km day.

Table 2: Various λ of exponentially increasing pollution source forP2/P0 concentration and daily load difference.

Exponentially increasing pollution source L = 36 km and q = 60λ

P2/P0 at the endnumerical solutions

Daily load (kg/day)difference (%)

2:5 × 10−5 1.32439 99.95501%

2:63889 × 10−5 1.39788 99.52516%

2:77778 × 10−5 1.47136 99.95002%

4:16667 × 10−5 2.20598 99.92504%

8 Journal of Applied Mathematics

position by the pollution sources in the downstream which ishigher than the downstream, as appeared within the ThaChin river and the Chao Praya River in Thailand. The sourceof contamination of the case study river will be spread fromthe center to the end, causing the concentration of contami-nation will continuously increment. The arbitrary constant λwas presented and illustrated to significantly compare thebehavior of exponential increasing form of sources. The solu-tions of this study can be used to predict the effect of addedpollutants on the maximum allowable loads of pollutants thata river can have from all sources by changing the value of qand λ. The results can be applied in many physical situationsdescribed by advection-dispersion phenomena and in mak-ing decisions to support the planning and managing of riverwater quality issues.

Data Availability

The result of this study was from Analytical and Numericaltechniques. We did not use the raw data.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first author is grateful to King Mongkut’s Universityof Technology North Bangkok for the financial supportduring her study doctor of philosophy. Finally, the authorswould like to express their gratitude to the anonymous ref-erees for very helpful suggestions and comments which ledto improvements of the original manuscript.

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