Ratio Mathematica Volume 47, 2023 Mathematical modelling and application of reduced differential transform method for river pollution Manan A. Maisuria* Priti V. Tandel† Abstract This paper presents the mathematical model of pollutant transport in a river. To effectively find the analytical solution of the advection- diffusion equation under various forms of suitable initial conditions, the reduced differential transform method (RDTM) is used. Three different initial concentration function cases, including rational, ex- ponential, and power, are analyzed for the present model. A 2D and 3D visual comparison of the solutions obtained for each case is also shown. This article discusses the sufficient condition for convergence of the reduced differential transform approach to solving non-linear differential equations.The convergence results for the concentration functions in each case are briefly described. The present method is highly effective and more efficient in solving real-world problems. For all cases, the amount of phosphate pollutant concentration at var- ious distances and time levels has been examined using numerical and graphical representations. While analyzing actual world prob- lems, the current study demonstrates its effectiveness. Keywords: Pollutant transport equation; Reduced differential trans- form method; Convergence; River pollution 2020 AMS subject classifications: 35A22 ,35C10, 35G05 1 *Veer Narmad South Gujarat University, Surat, Gujarat, India; manan- maisuria.maths21@vnsgu.ac.in. †Veer Narmad South Gujarat University, Surat, Gujarat, India; pvtandel@vnsgu.ac.in. 1Received on November 17, 2022. Accepted on March 21, 2023. Published online on April 10, 2023. DOI: 10.23755/rm.v41i0.955. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 342 Manan A. Maisuria, Priti V. Tandel 1 Introduction Accidents involving environmental contamination are common in the process plant, particularly in sectors like the chemical sector, manufacture of agricul- tural chemicals, natural gas extraction, etc [Li et al., 2009]. Organic materials and heavy metals are frequent and harmful contaminants in water pollution in- cidents, and examples of organic materials include benzene, naphthalene, phe- nol, anthracene, alcohol, etc. As a typical hydrocarbon, benzene is poisonous, and cancer-causing [Nomura et al., 2019] It may be inhaled or absorbed via the skin. Because of its low vapor pressure, benzene may be easily spread through the air. Exposure to benzene in the past is a common factor in developing leukemia [Jiang et al., 2018, Tsuji et al., 2018, Meszaros et al., 2017]. In many places, industrial or household human activity-related water contamination is a serious issue [Tchobanoglous et al., 1991]. The contamination of water sources is re- sponsible for the deaths of over 25 million people annually. Models to manage and forecast water quality are vital. When analyzing river water quality, numer- ous aspects must be addressed, including dissolved oxygen, nitrates, chlorides, phosphates, suspended particles, environmental hormones, and chemical oxygen demand, such as heavy metals and bacteria. Agricultural pollution may degrade surface, and groundwater [Knight et al., 2000]. To satisfy its many needs, society relies heavily on river water, one of the few abundant sources of freshwater [Shi et al., 2019]. To ensure an undisturbed freshwater supply, specific water quality requirements along the rivers must be maintained [Chen et al., 2016]. Agricultural non-point source pollution (ANPSP), caused by the use of agrochemicals in farming, significantly impacts water quality and aquatic ecosystems [Bryan and Kandulu, 2011, Borges et al., 2017]. In 1925, the well-known model of Streeter and Phelps characterized the equilibrium of dis- solved oxygen in rivers, marking the beginning of the era of mathematical water quality models. Since then, there have been many updates to this model [Streeter, 1925, James, 1978]. Weighted discretizations and the two-dimensional modified equation method solved the linear, constant coefficient advection-diffusion equa- tion. The modified equivalent equation determines one- and multi-dimensional fi- nite difference method accuracy [Noye and Tan, 1989]. The Eulerian-Lagrangian localized adjoint method (ELLAM) solves the nonlinear Buckley-Leverea equa- tion, which has degenerate diffusion and sharpening near-shock solutions [Dahle et al., 1995]. It is thought that a suitable strategy for identifying and evaluat- ing the production of nutrients produced by management scenarios, which may aid in project prioritization and improve water quality, is extensive modeling of the surface water using tried-and-true techniques. They should be simulated as management scenarios before implementing plans to evaluate their effectiveness 343 Mathematical modelling and application of reduced differential transform method for river pollution [Fakouri et al., 2019]. Numerous research on the impact of various water management strategies on water quality and quantity have been carried out using multiple models and ex- perimental techniques in diverse places of different sizes with varying objectives. They used a MIKE11 pattern on the Pasikhan River and simulated nitrate and phosphate contaminant concentration. The effects of dumping waste water and draining water into rivers are significant and impact the river’s water quality. In addition, Kerich assessed the chemicals used in the Ahero Irrigation Scheme and provided many recommendations to enhance the quality of water retrieved from the drainage canals. For this reason, the most efficient means of purifying water for human consumption in the area were biodegradable chemicals for pest and herbicide management and bio-sand filters [Kerich, 2020]. Groundwater quality in the Blinaja River basin was also investigated by Çadraku using irrigation water quality criteria. According to the findings, the groundwater in the research region is of sufficient quality for irrigating the crops. As well as addressing surface water issues, specific recommendations were made for preserving groundwater qual- ity [Çadraku, 2021]. To solve the advection-dispersion equation (ADE) in rivers backward in time and a one-dimensional domain for different pollution loading patterns, an unique analytical approach was devised using the quasi-reversibility (QR) technique and the Fourier transform tool. To avoid the issue being ill-posed during the inverse solution process, a stability factor is added to the initial trans- port equation in this approach [Permanoon et al., 2022]. Mass movement is reg- ulated by the molecular diffusion of solutes between mobile and static water in aquifers like the Chalk, which have long diffusion path lengths [Bibby, 1981]. In this paper, we formulate a one-dimensional mathematical model of pollu- tant transport. The governing equation is a 1D advection-diffusion equation solved by the reduced differential transform method (RDTM). This method requires an initial condition. To generate the initial condition, we have used the concentration of the river Khobistskali for PO4 pollutant component [Tsuji et al., 2018]. Also, we have discussed the convergence of analytic solutions obtained by RDTM. Section 2 covers the mathematical formulation of this problem. Section 3 contains the fundamental ideas behind the reduced differential transform method. In section 4, The process for achieving the convergence of the analytic series solution given by RDTM has been discussed. Section 5 includes the numerical outcomes and the convergence of the method for its effectiveness. 2D and 3D plots show a visual representation of the obtained solutions. Section 6 provides a summary of the conclusion. 344 Manan A. Maisuria, Priti V. Tandel 2 Mathematical Formulation of the problem Reaction, diffusion, advection, absorption, and sedimentation all have a role in transporting the pollutant material farther downstream. Variables such as the kind of pollution, its physicochemical characteristics, flow characteristics, and the surrounding environment all have a role. Thus, parameters linked to the flow of pollution are prioritized above those relating to the pollutant’s nature [Kim and Chapra, 1997]. A linear partial differential equation, the mass transfer equation, is often uti- lized in research on water, soil, petroleum, the living environment, and several engineering subjects. A linear parabolic partial differential equation, the afore- mentioned equation is of the first and second orders in terms of time and space, respectively. In the one-dimensional domain (along the river length), the general form of this equation under unstable and non-uniform flow regimes is as follows [Amiri et al., 2021]. A ∂c ∂t = ∂ ∂x ( ADx ∂c ∂x ) − Av ∂c ∂x − Akc + Af (1) where c =Pollutant concentration, Dx =Diffusion coefficient along the x- direction, v = Mean flow velocity, k = Coefficient of non-conservation, A = Flow area, f = Source term, x = Distance from starting point of domain, t = Time dimension. For the entire study, the area of the cross-section of the river is considered a constant. To analyze this problem, we have used the concentration of pollutant substance PO4 of river Khobistskali. Here Khobistskali river’s length is 44800 m. Using past collected data, at time t = 0 (any fixed time), the concentration is assumed as a rational, exponential, and power form, and the following values of parameters are used to obtain the solutions. Dx = 0.55 m2 sec , v = 0.534930 m sec , k = 0 1 sec and f = 0 [Kachiashvili et al., 2007]. Goodness of fit Curve fitting SSE R-square Adjusted R-square RMSE Rational 3.89E-06 0.9884 0.9876 0.0003662 Exponential 1.62E-05 0.9515 0.9499 0.0007354 Power 5.55E-06 0.9834 0.9823 0.0004376 Table 1: Statistical indices of initial function Graphical representations of curve fitting for the initial condition are shown in Figures 1, 2, and 3. Table 1 lists the goodness of fit values for various initial conditions. Hence this problem is studied for three following different cases: 345 Mathematical modelling and application of reduced differential transform method for river pollution 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x(m) - distance 104 0.004 0.006 0.008 0.01 0.012 0.014 0.016 c (m g /L ) -c o n c e n tr a ti o n Rational curve fitting of c(x,0) c vs. x untitled fit 1 Goodness of fit: SSE: 3.889e-06 R-square: 0.9884 Adjusted R-square: 0.9876 RMSE: 0.0003662 Figure 1: Rational curve fitting of c(x, 0). 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x(m) - distance 104 0.004 0.006 0.008 0.01 0.012 0.014 0.016 c (m g /L ) - c o n c e n tr a ti o n Exponential curve fitting of c(x,0) c vs. x untitled fit 1 Goodness of fit: SSE: 1.623e-05 R-square: 0.9515 Adjusted R-square: 0.9499 RMSE:0.0007354 Figure 2: Exponential curve fitting of c(x, 0). 346 Manan A. Maisuria, Priti V. Tandel 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x(m) - distance 104 0.004 0.006 0.008 0.01 0.012 0.014 0.016 c (m g /L ) - c o n c e n tr a ti o n Power curve fitting of c(x,0) c vs. x untitled fit 1 Goodness of fit: SSE: 5.552e-06 R-square: 0.9834 Adjusted R-square: 0.9823 RMSE: 0.0004376 Figure 3: Power curve fitting of c(x, 0). Case-1 Rational initial function For this case, c(x, 0) = 0.0004008x + 251.4 x + 1.39e + 04 (2) with c(44800, t) = 5.846e − 13t2 + 3.463e − 08t + 0.004597 (3) Case-2 Exponential initial function For this case, c(x, 0) = 0.01533e−3e−05x − 0.0001573 (4) with c(44800, t) = 6.887e − 13t2 + 6.167e − 08t + 0.003846 (5) Case-3 Power initial function For this case, c(x, 0) = −0.007562x0.1384 + 0.03762 (6) with c(44800, t) = 6.846e − 08t + 0.004247 (7) 347 Mathematical modelling and application of reduced differential transform method for river pollution 3 Reduced Differential Transform Method Let b(ω, τ) be a two-variable function. Suppose b(ω, τ) is written as b(ω, τ) = f(ω)g(τ).b(ω, τ) can be represented as the following using the features of the differential transform: b(ω, τ) = ∞∑ i=0 F(i)ω i ∞∑ j=0 G(j)τ j = ∞∑ k=0 Bk(ω)τ k (8) where Bk(ω) is referred to as the t-dimensional spectrum function of b(ω, τ). Bm(ω) = 1 m! [ ∂m ∂τm b(ω, τ) ] τ=0 (9) The original function is denoted by the lowercase [b(ω, τ)], whereas the altered function is denoted by the capital [B(ω, τ)]. The way to define the differential inverse transform of Bk(ω) is as follows: b(ω, τ) = ∞∑ m=0 Bm(ω)τ m (10) From Equations (9) and (10), we get b(ω, τ) = ∞∑ m=0 1 m! [ ∂m ∂τm b(ω, τ) ] τ=0 τm (11) Let us consider the following nonlinear PDE, to understand the basic concept of RDTM. Tb(ω, τ) + Pb(ω, τ) + Ob(ω, τ) = f(ω, τ) (12) with initial condition b(ω, 0) = η(ω) , where T = ∂ ∂τ , Pb(ω, τ) is a linear term that has partial derivatives, while Ob(ω, τ) is a non-linear term, and f(ω, τ) is a source term [Al-Amr, 2014]. By applying the transform on equation (12), we get (m + 1)Bm+1(ω) = Fm(ω) − PBm(ω) − OBm(ω) (13) where Bm(ω),Fm(ω),PBm(ω) and OBm(ω) are transform of b(ω, τ),f(ω, τ),Pb(ω, τ) 348 Manan A. Maisuria, Priti V. Tandel and Ob(ω, τ) respectively. We are able to write this down based on the initial con- dition. B0(ω) = η(ω) (14) From equations (13) and (14), we get the values of Bm(ω). After that, an approx- imation solution is produced by carrying out an inverse transformation on the set of values {Bm(ω)} n m=0. This transformation yields an approximation solution as b̃n(ω, τ) = n∑ m=0 Bm(ω)τ m (15) where n is the order of approximation answer. Consequently, the exact solution is given by [Al-Amr, 2014], b(ω, τ) = lim n→∞ b̃n(ω, τ) (16) Function Transformation b(ω, τ) Bm(ω) = 1 m! [ ∂m ∂τm b(ω, τ) ] τ=0 αf(ω, τ) ± βg(ω, τ) αFm(ω) + βGm(ω) ωkτn ωkδ(m − n) ωkτnb(ω, τ) ωkBm−n(ω) l(ω, τ) = f(ω, τ)g(ω, τ) Lm(ω) = ∑m r=0 Fr(ω)Gm−r(ω) ∂r ∂τr b(ω, τ) (m+r)! m! Bm+r(ω) ∂ ∂ω b(ω, τ) ∂ ∂ω Bm(ω) Table 2: Transform Table[Al-Amr, 2014, Keskin and Oturanc, 2010, Srivastava et al., 2014] 4 Convergence of RDTM To understand the convergence, Let us consider the solution of equation (13) in power series form as follow b(ω, τ) = ∞∑ n=0 Bn(ω)τ n = ∞∑ n=0 Bnτ n (17) 349 Mathematical modelling and application of reduced differential transform method for river pollution which is obtained by equation (16) [Moosavi Noori and Taghizadeh, 2021, Saeed and Mustafa, 2017]. Theorem 4.1. If ∑∞ n=0 Bnτ n is given series, [1] ∃ 0 < β < 1 ∋ ∥Bn+1∥∥Bn∥ ≤ β ⇒ series is convergent. [2] ∃ β > 1 ∋ ∥Bn+1∥∥Bn∥ ≥ β ⇒ series is divergent. Proof. Let (C[l], ∥.∥) represent the Banach space that contains all continuous functions on l that satisfy the norm ∥.∥. Also, let’s suppose that ∥B0(ω)∥ ≤ M, where M is an integer in the positive range. Let {δn}∞n=0 be a partial sum δn = B0 + B1 + B2 + ... + Bn If we can prove that {δn}∞n=0 is a Cauchy sequence in Banach Space, then we can conclude that the sequence of partial sum is convergent in Banach Space. As a result, at this point, we shall demonstrate that the series of partial sums follows the Cauchy sequence. We take ∥δn+1 − δn∥ = ∥Bn+1∥ ≤ β∥Bn∥ ≤ ... ≤ βn+1∥B0∥ ≤ βn+1M Therefore,∀n, m ∈ N, n ≥ m ,We have ∥δn − δm∥ = ∥(δn − δn−1) + (δn−1 − δn−2) + ... + (δm+1 − δm)∥ ≤ ∥δn − δn−1∥ + ∥δn−1 − δn−2∥ + ... + ∥δm+1 − δm∥ ≤ 1−β n−m 1−β β m+1∥B0∥ Now here 0 < β < 1, We obtain lim n,m→∞ ∥δn − δm∥ = 0 Hence {δn}∞n=0 is Cauchy sequence in Banach Space. Therefore, given series is convergent. 350 Manan A. Maisuria, Priti V. Tandel 5 Result and discussion We take three cases of initial function in rational, exponential and power form and obtain three solutions using reduced differential transform method. Case-1 Rational initial function Solving equation (1) by RDTM with initial condition (2), we get c(x, t) = M0 + M1t + M2t 2 + M3t 3 + M4t 4 + ... where M0 = 0.0004008x+251.4 x+1.39e+04 M1 = 0.0270 ( 4863 x + 67605700 ) (x+13900)3 M2 = 8.9236e−06   x2 + 219193889400 x +1523735609830000   (x+13900)5 M3 = 9.8159e−10   x3 + 1599028070073300 x2 +22233066963300870000 x +103043693991250631000000   (x+13900)7 M4 = 1.0798e−13   186421425071787 x4 +10368864699446257200 x3 +216270792211342627620000 x2 +2004850662624966111612000000 x +6969433288195869001126700000000   (x+13900)9 Here, ∥M1∥ ∥M0∥ = 3.4116405e − 05 < 1, ∥M2∥ ∥M1∥ = 3.4972143e − 05 < 1, ∥M3∥ ∥M2∥ = 3.4976844e − 05 < 1, ∥M4∥ ∥M3∥ = 3.4981545e − 05 < 1 Therefore, the solution function c(x, t) is convergent. 351 Mathematical modelling and application of reduced differential transform method for river pollution 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 10 4 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 Figure 4: c(x, t) at t = 1000 sec 352 Manan A. Maisuria, Priti V. Tandel 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 10 4 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 Figure 5: c(x, t) at x = 7000 m 353 Mathematical modelling and application of reduced differential transform method for river pollution Two-dimensional representations of the concentration function for case-1 are provided in figures 4, and 5, respectively, for the fixed values of (t = 1000sec),and (x = 7000m).As shown in figure 4, we can see that the value of concentration decreases as the length (x) variable increases. In figure 5, we can see that the value of concentration rises as the passage of time (t) increases. x(m)\t(sec) 4500 9000 13500 18000 22500 27000 31500 36000 1400 0.019467 0.023775 0.030123 0.039479 0.053046 0.072264 0.098812 0.134601 5740 0.014666 0.016968 0.020065 0.024264 0.029941 0.03754 0.047573 0.060618 10080 0.011796 0.013224 0.015034 0.017353 0.020329 0.024138 0.028979 0.035078 14420 0.009888 0.010858 0.01204 0.013494 0.015289 0.017506 0.020238 0.023587 18760 0.008527 0.009229 0.01006 0.011051 0.01224 0.013668 0.015383 0.017437 23100 0.007507 0.008038 0.008654 0.009372 0.010213 0.011201 0.012363 0.013728 27440 0.006715 0.007131 0.007605 0.008148 0.008772 0.009493 0.010326 0.01129 31780 0.006082 0.006416 0.006792 0.007217 0.007698 0.008246 0.00887 0.009581 36120 0.005564 0.005839 0.006144 0.006485 0.006868 0.007297 0.00778 0.008325 40460 0.005133 0.005362 0.005616 0.005896 0.006206 0.006551 0.006936 0.007365 44800 0.004768 0.004963 0.005176 0.00541 0.005667 0.00595 0.006264 0.00661 Table 3: c(x, t) for Case-1 The values of the concentration function for case-1 are shown in table 3. Case-2 Exponential initial function Solving equation (1) by RDTM with initial condition (4), we get c(x, t) = A0 + A1t + A2t 2 + A3t 3 + A4t 4 + ... where A0 = 0.01533e −3e−05x − 0.0001573 A1 = 2.4602e − 07 e− 3 x 100000 A2 = 1.9741e − 12 e− 3 x 100000 A3 = 1.0561e − 17 e− 3 x 100000 A4 = 4.2370e − 23 e− 3 x 100000 Here, ∥A1∥ ∥A0∥ = 1.6221987e − 05 < 1, ∥A2∥ ∥A1∥ = 8.0241975e − 06 < 1, 354 Manan A. Maisuria, Priti V. Tandel ∥A3∥ ∥A2∥ = 5.3494650e − 06 < 1, ∥A4∥ ∥A3∥ = 4.0120987e − 06 < 1 Therefore, the solution function c(x, t) is convergent. Figures 6 and 7 illustrate, 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 10 4 2 4 6 8 10 12 14 16 10 -3 Figure 6: c(x, t) at t = 1000 sec respectively, two-dimensional representations of the concentration function for case-2 with the fixed values of (t = 1000 sec) and (x = 7000 m). As shown in figure 6, the concentration value falls as the distance (x) variable rises. It is clear from graph 7 that when time (t) grows, so does the value of concentration. The values of a concentration function for case-2 are presented in table 4. Case-3 Power initial function Solving equation (1) by RDTM with initial condition (6), we get c(x, t) = P0 + P1t + P2t 2 + P3t 3 + P4t 4 + ... 355 Mathematical modelling and application of reduced differential transform method for river pollution 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 10 4 0.012 0.013 0.014 0.015 0.016 0.017 0.018 0.019 0.02 0.021 0.022 Figure 7: c(x, t) at x = 7000 m x(m)\t(sec) 4500 9000 13500 18000 22500 27000 31500 36000 1400 0.015643 0.016826 0.018098 0.019465 0.020934 0.022512 0.024208 0.026029 5740 0.013714 0.014753 0.015869 0.01707 0.018359 0.019745 0.021233 0.022832 10080 0.012021 0.012933 0.013913 0.014966 0.016099 0.017315 0.018622 0.020025 14420 0.010534 0.011335 0.012195 0.01312 0.014114 0.015182 0.016329 0.017561 18760 0.009229 0.009932 0.010687 0.011499 0.012372 0.01331 0.014317 0.015398 23100 0.008083 0.0087 0.009363 0.010076 0.010842 0.011665 0.01255 0.013499 27440 0.007077 0.007619 0.008201 0.008827 0.0095 0.010222 0.010998 0.011832 31780 0.006194 0.006669 0.007181 0.00773 0.008321 0.008955 0.009637 0.010368 36120 0.005418 0.005836 0.006285 0.006767 0.007286 0.007843 0.008441 0.009083 40460 0.004738 0.005104 0.005498 0.005922 0.006377 0.006866 0.007391 0.007955 44800 0.00414 0.004462 0.004808 0.00518 0.005579 0.006009 0.00647 0.006965 Table 4: c(x, t) for Case-2 356 Manan A. Maisuria, Priti V. Tandel where P0 = −0.007562x0.1384 + 0.03762 P1 = 3.4537e−07 (1621 x+1436) x2327/1250 P2 = 1.2902e−04 x2+4.9388e−04 x+7.2656e−04 x4827/1250 P3 = 4.2826e−05 x3+3.7801e−04 x2+0.0015 x+0.0025 x7327/1250 P4 = 1.6389e−05 x4+2.6028e−04 x3+0.0020 x2+0.0078 x+0.0138 x9827/1250 Here, ∥P1∥ ∥P0∥ = 6.4108463e − 05 < 1, ∥P2∥ ∥P1∥ = 0.0001649 < 1, ∥P3∥ ∥P2∥ = 0.0002379 < 1, ∥P4∥ ∥P3∥ = 0.0002747 < 1 Therefore, the solution function c(x, t) is convergent. Figures 8 and 9 are two- dimensional representations of the concentration function for case-3 with fixed parameters ( t = 1000sec) and (x = 7000m), respectively. As shown in figure 8, as the distance (x) variable increases, the concentration value decreases. Graph 9 demonstrates that as time (t) increases, so does the value of concentration. The x(m)\t(sec) 4500 9000 13500 18000 22500 27000 31500 36000 1400 0.034284 0.149754 0.560422 1.579016 3.633997 7.269557 13.14562 22.03783 5740 0.014373 0.017404 0.02281 0.032238 0.047824 0.072202 0.108497 0.160329 10080 0.011539 0.012841 0.014609 0.017063 0.020482 0.025201 0.031609 0.040152 14420 0.00987 0.010713 0.011738 0.013011 0.014609 0.016626 0.019168 0.022356 18760 0.00866 0.009291 0.010017 0.010867 0.011871 0.013066 0.014494 0.016203 23100 0.007701 0.008209 0.008775 0.009415 0.010143 0.010977 0.011937 0.013046 27440 0.006902 0.007329 0.007796 0.008311 0.008883 0.009521 0.010237 0.011043 31780 0.006215 0.006584 0.006983 0.007416 0.007888 0.008405 0.008974 0.009603 36120 0.00561 0.005937 0.006285 0.006659 0.007062 0.007498 0.007971 0.008486 40460 0.005069 0.005362 0.005673 0.006003 0.006355 0.006732 0.007137 0.007573 44800 0.004579 0.004845 0.005126 0.005422 0.005735 0.006068 0.006423 0.006801 Table 5: c(x, t) for Case-3 values of a concentration function for case-3 are shown in table 5. Figure 10 give the 3D graphical comparison of solution obtained for all cases. Also figures 11 and 12 give the 2D graphical comparison of solution obtained for all cases for t = 3000 sec and x = 40000 m respectively. 357 Mathematical modelling and application of reduced differential transform method for river pollution 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 10 4 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Figure 8: c(x, t) at t = 1000 sec 358 Manan A. Maisuria, Priti V. Tandel 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 10 4 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Figure 9: c(x, t) at x = 7000 m 359 Mathematical modelling and application of reduced differential transform method for river pollution Figure 10: 3D Comparison of c(x, t) 360 Manan A. Maisuria, Priti V. Tandel 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 10 4 0 0.005 0.01 0.015 0.02 0.025 Figure 11: 2D Comparison of c(x, 3000) 361 Mathematical modelling and application of reduced differential transform method for river pollution 0 0.5 1 1.5 2 2.5 3 3.5 4 10 4 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 10 -3 Figure 12: 2D Comparison of c(40000, t) 362 Manan A. Maisuria, Priti V. Tandel 6 Conclusion In this paper, we outlined the main components of a mathematical model that various ways to predict chemical concentrations in rivers due to pollutant dis- charges. We get initial condition from old collected data in different form like rational, exponential and power. We attain three different solutions from different form of initial conditions. We conclude that concentration rises as time(t) rises. Concentration decreases as length(x) increases. The fundamental benefit of the RDTM is that it offers the user a quick converging power series form with neatly calculated terms that contains an analytical approximation, and in many situa- tions, an exact solution. There is no discretization or unavoidable presumptions while using RDTM. Sometimes RDTM is superior to other techniques (DTM). Furthermore, if highly polluted regions can be located physically, the research framework may provide a useful strategy for more economical watershed man- agement.According to the findings, phosphate levels along the river fall when fer- tiliser usage is reduced. Among the management options, the use of less fertiliser greatly reduces river pollution. The current work provides a helpful tool for a scholar to compare and contrast the performance of different models and yields intriguing and practical outcomes. It may be possible to extend this research to consider the two-dimensional advection-diffusion equation. References M. O. Al-Amr. New applications of reduced differential transform method. Alexandria Engineering Journal, 53(1):243–247, 2014. doi: 10.1016/j.aej.2014.01.003. S. Amiri, M. Mazaheri, and N. Bavandpouri Gilan. Introducing a new method for calculating the spatial and temporal distribution of pollutants in rivers. Interna- tional Journal of Environmental Science and Technology, 18(12):3777–3794, 2021. doi: 10.1007/s13762-020-03096-y. R. Bibby. Mass transport of solutes in dual-porosity media. Water Resources Research, 17(4):1075–1081, 1981. doi: 10.1029/WR017i004p01075. L. G. A. Borges, A. Savi, C. Teixeira, R. P. de Oliveira, M. L. F. De Camil- lis, R. Wickert, S. F. M. Brodt, T. F. Tonietto, R. Cremonese, L. S. da Silva, et al. Mechanical ventilation weaning protocol improves medi- cal adherence and results. Journal of critical care, 41:296–302, 2017. doi: 10.1016/j.jcrc.2017.07.014. 363 Mathematical modelling and application of reduced differential transform method for river pollution B. A. Bryan and J. M. Kandulu. Designing a policy mix and sequence for mitigat- ing agricultural non-point source pollution in a water supply catchment. Water resources management, 25(3):875–892, 2011. doi: 10.1007/s11269-010-9731- 8. H. S. Çadraku. Groundwater quality assessment for irrigation: case study in the blinaja river basin, kosovo. Civil Engineering Journal, 7(9):1515–1528, 2021. J. Chen, H. Shi, B. Sivakumar, and M. R. Peart. Population, water, food, energy and dams. Renewable and Sustainable Energy Reviews, 56:18–28, 2016. doi: 10.1016/j.rser.2015.11.043. H. K. Dahle, R. E. Ewing, and T. F. Russell. Eulerian-lagrangian localized adjoint methods for a nonlinear advection-diffusion equation. Computer methods in ap- plied mechanics and engineering, 122(3-4):223–250, 1995. doi: 10.1016/0045- 7825(94)00733-4. B. Fakouri, M. Mazaheri, and J. M. Samani. Management scenarios method- ology for salinity control in rivers (case study: Karoon river, iran). Journal of Water Supply: Research and Technology-Aqua, 68(1):74–86, 2019. doi: 10.2166/aqua.2018.056. A. James. Mathematical models in water pollution control. Wiley, 1978. B. Jiang, Y. Wen, Z. Li, D. Xia, and X. Liu. Theoretical analysis on the removal of cyclic volatile organic compounds by non-thermal plasma. Water, Air, & Soil Pollution, 229(2):1–12, 2018. doi: 10.1007/s11270-018-3687-3. K. Kachiashvili, D. Gordeziani, R. Lazarov, and D. Melikdzhanian. Modeling and simulation of pollutants transport in rivers. Applied mathematical modelling, 31 (7):1371–1396, 2007. doi: 10.1016/j.apm.2006.02.015. E. C. Kerich. Households drinking water sources and treatment methods options in a regional irrigation scheme. Journal of Human, Earth, and Future, 1(1): 10–19, 2020. doi: 10.28991/HEF-2020-01-01-02. Y. Keskin and G. Oturanc. Reduced differential transform method for generalized kdv equations. Mathematical and Computational applications, 15(3):382–393, 2010. doi: 10.3390/mca15030382. K. S. Kim and S. C. Chapra. Temperature model for highly transient shal- low streams. Journal of Hydraulic Engineering, 123(1):30–40, 1997. doi: 10.1061/(ASCE)0733-9429(1997)123:1(30). 364 Manan A. Maisuria, Priti V. Tandel R. L. Knight, V. W. Payne Jr, R. E. Borer, R. A. Clarke Jr, and J. H. Pries. Con- structed wetlands for livestock wastewater management. Ecological engineer- ing, 15(1-2):41–55, 2000. doi: 10.1016/S0925-8574(99)00034-8. J. Li, B. Zhang, M. Liu, and Y. Wang. Numerical simulation of the large-scale malignant environmental pollution incident. Process Safety and Environmental Protection, 87(4):232–244, 2009. doi: 10.1016/j.psep.2009.03.001. N. Meszaros, B. Subedi, T. Stamets, and N. Shifa. Assessment of surface water contamination from coalbed methane fracturing-derived volatile contaminants in sullivan county, indiana, usa. Bulletin of Environmental Contamination and Toxicology, 99(3):385–390, 2017. doi: 10.1007/s00128-017-2139-x. S. R. Moosavi Noori and N. Taghizadeh. Study of convergence of re- duced differential transform method for different classes of differential equa- tions. International Journal of Differential Equations, 2021, 2021. doi: 10.1155/2021/6696414. S. Nomura, Y. Ito, S. Takegami, and T. Kitade. Development and validation of an assay method for benzene in the delgocitinib drug substance using conventional hplc. Chemical Papers, 73(3):673–681, 2019. doi: 10.1007/s11696-018-0608- 2. B. Noye and H. Tan. Finite difference methods for solving the two-dimensional advection–diffusion equation. International Journal for Numerical Methods in Fluids, 9(1):75–98, 1989. doi: 10.1002/fld.1650090107. E. Permanoon, M. Mazaheri, and S. Amiri. An analytical solution for the advection-dispersion equation inversely in time for pollution source identifi- cation. Physics and Chemistry of the Earth, Parts A/B/C, 128:103255, 2022. doi: 10.1016/j.pce.2022.103255. R. K. Saeed and A. A. Mustafa. Numerical solution of fisher–kpp equation by us- ing reduced differential transform method. In AIP Conference Proceedings, vol- ume 1888, page 020045. AIP Publishing LLC, 2017. doi: 10.1063/1.5004322. H. Shi, J. Chen, S. Liu, and B. Sivakumar. The role of large dams in promoting economic development under the pressure of population growth. Sustainability, 11(10):2965, 2019. doi: 10.3390/su11102965. V. K. Srivastava, M. K. Awasthi, and S. Kumar. Analytical approximations of two and three dimensional time-fractional telegraphic equation by reduced differen- tial transform method. Egyptian Journal of Basic and Applied Sciences, 1(1): 60–66, 2014. doi: 10.1016/j.ejbas.2014.01.002. 365 Mathematical modelling and application of reduced differential transform method for river pollution H. W. Streeter. Studies of the pollution and natural purification of the ohio river, part iii, factors concerned in the phenomena of oxi- dation and reareation. Public health bulletin, (146), 1925. URL http://udspace.udel.edu/handle/19716/1590. G. Tchobanoglous, F. L. Burton, and H. D. Stensel. Wastewater engineering. Management, 7(1):4, 1991. M. Tsuji, T. Kawahara, K. Uto, N. Kamo, M. Miyano, J.-i. Hayashi, and T. Tsuji. Efficient removal of benzene in air at atmospheric pressure using a side-on type 172 nm xe2 excimer lamp. Environmental Science and Pollution Research, 25 (19):18980–18989, 2018. doi: 10.1007/s11356-018-2103-2. 366