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A Numerical and Analytical Framework for Estimating Water
Pollution in A 3-D Aquatic Region Using Diffusion Model with Du
Fort Frankel and Adomian Decomposition Methods
1
Tarjani Naik,
2
Mukesh Patel,
3
Rachna Patel
1
Ph.D. Scholar, Department of Mathematics, Uka Tarsadia University, Bardoli, Gujarat, India,
2
Assistant Professor, Department of Mathematics, Uka Tarsadia University, Bardoli, Gujarat, India
3
Assistant Professor, Department of Computer Engineering, CGPIT, Uka Tarsadia University, Bardoli, Gujarat, India
DOI:
https://doi.org/10.51583/IJLTEMAS.2025.1410000040
Received: 09 October 2025; Accepted: 15 October 2025; Published: 07 November 2025
Abstract: Water pollution is an important environmental issue that affects human health, aquatic habitats, and the sustainability
of natural resources. Reliable mathematical models that can forecast the behaviour of pollutants in three-dimension (3-D) water
systems are needed to address this issue. In this study, a 3-D diffusion model is used to determine the periodic and location-based
fluctuation of the pollutant concentration in water. Investigating the slow increase in pollution levels in a 3-D area and evaluating
the precision of analytical and numerical approaches to diffusion-based pollution problems are the objectives of this work. Two
approaches are used to accomplish this: the Adomian Decomposition Method (ADM), which is an analytical approach, and the
Du Fort Frankel (DF) scheme, which is a numerical approach. Initial and boundary conditions required for the modelling are
provided by experimented data (Exp. data) from a 3-D cuboid tank filled with water and introduced with an iodized salt water
solution as the pollutant. Direct monitoring of pollution dispersion across time and space is made possible by this configuration,
producing useful data for confirming the mathematical models. The results of the experiment verify that the levels of pollutants
rise with time at each location in the 3-D region. When comparing the outcomes of the DF approach with Exp. data and ADM, an
insignificant difference, measured in parts per million (PPM), is observed, demonstrating the reliability and strength of the
suggested model. This study is important because it combines mathematical models and realistic observations to examine
pollution of water in 3-D. The results show that the model is accurate and applicable to both controlled experiments and larger-
scale water systems. Additionally, this work advances the mathematical solutions for diffusion equations and offers useful
information for sustainable resource use, pollution control, and water quality evaluation.
Keywords Water pollution
·
3-D Diffusion equation
·
Du Fort Frankel method
·
Adomian Decomposition
method
Mathematics Subject Classification 35K57
·
65N06
I. Introduction
One of the essential components of the earth is water. Approximately two-thirds of the surface of the earth is covered with it.
Most of the world's population, especially humans, rely on freshwater for survival [21]. People today look for water that is both
sufficient and of high quality [1, 2]. Residential and industrial water contamination caused by human activities is a significant
concern in a lot of countries [4]. An estimated 25 million people each year pass away due to the severe effects of water
contamination. As a result, the water quality issue is grabbing massive attention across the world [3]. Water pollution can be
categorized in multiple ways, arising from changes in water's physical, chemical, and biological characteristics that harm living
organisms. Mainly caused by human actions, this pollution negatively affects both human health and the quality of the
environment's water [1, 22]. Two methods, numerical and analytical, can address water pollution issues. Various numerical and
analytical techniques are available for solving mathematical equations related to water pollution problems.
This research employs a 3-D diffusion mathematical model to predict water pollution levels over a specified time with assuming a
constant diffusion rate. The model considers water as the pollute and an iodized salt-water solution as the pollutant. A 3-D cuboid
is created with uniformly spaced grid points in all directions, and experimental data on water pollution, measured in parts per
million (PPM), is gathered from each grid point at regular time intervals. The initial and boundary conditions for the
mathematical model are derived from the collected data. Two mathematical methods, namely the DF(DF) numerical method and
the Adomian Decomposition analytical method, are utilized to estimate water pollution levels. A comparative analysis between
DF Vs. ADM and Exp. data is conducted to evaluate their accuracy and determine any potential errors.
Related Work
Through the use of mathematical models of differential equations, A.K. Misra, J.B. Shukla, and Peeyush Chandra [6]
examined the simultaneous impacts of saturation and water pollution on the concentration of the dissolved oxygen (DO) in a body
of water. The transportation of water pollution concentration was predicted by Zainab Yahya, Hanani Johari, and
Nursalasawati Rusli [1] using 1-D advection-diffusion model and resolved by the Finite Difference Method (FTCS techniques
and an Implicit Crank Nicolson techniques). The second-order Lax-Wendorff method and Finite Time Central Space (FTCS)
were used by Nigar Sultana and Laek Sazzad Andallah [7] to solve the 1-D advection-diffusion equation for determining the
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concentration of water pollution in rivers as well as the pollutant in rivers at different times and locations. The 1-D advection-
diffusion model was derived by Abbas Parsaie, Amir Hamzeh Haghiabi [12], Safia Meddah, Omar Hireche, Mohamed
Hadjel, and Abdelkader Saidane [13]. This model was utilized in [12] to simulate the spread of pollution in rivers using the
Finite Volume Method and Artificial Neural Network (ANN) and to estimate the longitudinal dispersion coefficient. The
maximum concentration for a certain length of time and the estimation of the longitudinal dispersion of pollutants were
determined in [13] using the Transmission Line Matrix Method. A research work was enlarged by adding some parameters to the
1-D Advection diffusion model by Delong Wan, Huiping Zeng [8], Nonparent Pochai, J. J. H. Miller, L. J. Crane, and
Suwon Tangmanee, [9], Tsegaye Simon, Purnachandra Rao Koya [10], R. V. Waghmare and S. B. Kiwne [11]. The
Pollution Index Method was employed in [8] to forecast the water quality using several parameters. This model was used in [9] to
assess the expense of purifying water and calculate the concentration of pollutants using the finite element method. The dynamics
of river pollution were examined in [10], and the numerical solution was found utilizing the Splitting Method, the Crank-Nicolson
Method, and the Runge-Kutta Method. In this case, the diffusion and reaction terms were separated using the splitting approach,
and the numerical solution was found using the Crank-Nicolson and Runge-Kutta methods. The analytical method was employed
in [11] to determine the system's solution. C. A. Poffal, J. R. Zabadal, and S. B. Leite [5] expanded on a study utilizing a 2-
dimensional Advection-Diffusion model and addressed the dispersion of substances and microorganisms in rivers and lakes
through the application of a new analytical technique (iterative method).
II. Proposed Methodology
The proposed mathematical approach addresses the problem of water pollution which is described in Fig. 1.
Fig. 1 Mathematical method for 3-D water pollution estimation
Water Pollution Mathematical Model
The mathematical model for estimating water pollution is constructed based on the diffusion model in 3-D region. The rate at
which pollution concentration varies concerning time
at different 3-D location in direction is mathematically formulated
as in Eq. (1).
(1)
where, and the parameter indicate the pollutant concentration along
the
and direction, while refers to time, and and represent directions. The diffusion rate remains constant across all
directions.
Initial and Boundary Conditions for Mathematical Model
The numerical and analytical solution
of the mathematical model described in Eq. (1) is obtained by DF and ADM,
respectively, which requires initial and boundary conditions concerning space and time. The initial conditions for time are as in
Eq. (2).
(2)
And boundary conditions for and directions are as in Eq. (3)
(3)
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Where, are known functions.
Mathematical Method
The diffusion equation can be solved using various analytical and numerical techniques. This research utilizes the DF method to
obtain numerical solutions and the ADM for the analytical approach.
Du Fort Frankel Method (DF) [17, 18, 19]
The DF Method is a finite difference method used to get a numerical solution for the mathematical diffusion model. The finite
number of grid locations in and directions of 3-D regions at which the water pollution is estimated over a finite time
interval is expressed as follows [20];
The grid points (
) are given as
(4)
in which and are integers and and are grid spacing of all three directions respectively and is a time step size.
We denote
in the finite difference approximation.
The central difference for time-space
derivative
(5)
The spatial derivative's central difference in
the direction
(6)
The spatial derivative's central difference in
the direction
(7)
The spatial derivative's central difference in
the direction
(8)
Applying Eq. (5) - (8) in Eq. (1),
(9)
Consider the same grid spacing in Eq. (9) for all three directions that
(10)
Letting in Eq. (10),
(11)
Now, replace by the mean of the values and i.e. in
Eq. (11),
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Therefore, the following finite difference formula is used by the DF method to solve the 3-D diffusion Eq. (1):
(12
)
This Eq. (12) is the explicit formula of the DF method, and here, finding the solution at the level, requires the solution at
some location of level and level. Therefore, this method requires two initial conditions, shown in Eq. (2).
Adomian Decomposition Method (ADM) [14, 15, 16, 24]
In ADM method, re-write Eq. (1) in the standard operator form as
(13)
where,
Taking the inverse operator of the operator
exists and it defined as
Thus, applying the inverse operator to Eq. (13) yields
(14)
In ADM, represent the solution suppose that
(15)
Substituting Eq. (15) into (14), getting that
(16)
Now, comparing the Eq. (16) on both sides getting the recurrent relation in the form of as follows
(From Eq. (2))
and
for
From which
(17)
Therefore, the estimation of the approximate solution by using -term approximation. That is,
(18)
Therefore, Eq. (18) is the approximate solution of the 3-D diffusion mathematical model.
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Water Pollution Estimation
The DF mathematical approach estimates the water pollution level at a 3-D grid location over a time interval, whose result is
validated by comparing it with the ADM approach. Eq. (12) estimates the pollutant concentration in water at different locations
over time by the DF method. It is a 3-level explicit method, in which the current time
water pollution level at a particular
location is estimated by using the surrounding locations’ value in
and directions of the previous one-time level and the
respective location value of the last two-time level . This method simulates the spread of pollutants, such as an iodized
salt-water solution, in a water body, helping us understand how contamination disperses.
Eq. (18) represents the pollutant concentration in water at any given point
and time , using a series solution derived
from the ADM. The formula states that the analytical solution is a sum of the terms
. Here, is an initial
condition, and Eq. (17) is used to calculate the remaining terms,
.Thus, this equation provides a way to calculate
the concentration of pollutants over time and space using a series expansion approach.
III. Experimented Result and Discussion
It is examined that the existing research mainly focused on either 1-D or 2-D diffusion mathematical models solved by different
numerical and analytical approaches. The proposed work extends the diffusion model into a 3-D region to estimate water
pollution. Furthermore, researchers have demonstrated the 3-D water pollution diffusion model in a 3-D dummy cuboid water
tank having dimensions
feet. A 3-D grid structure with a 1 feet grid distance apart is used in the cuboid tank
to define 3-D grid locations. Each 3-D grid location represents approximately one cubic feet of water volume area, so that the
total volume of tank is covered within 75 grid locations. As a result, it is assumed that the amount of pollutant present at a
particular location will be consider as the average pollution of one cubic foot water volume area. Thus, this grid setup helps to
study the pollution spreads in all three directions inside the tank. Here, 960 litres (L) of water are used in a tank as a pollute,
whereas 40-litres iodized salt-water solution is used as a pollutant, and the polluted water is measured in Parts Per Million (PPM)
by Total Dissolved solids (TDS) meter over every 20 minutes of the time interval. The different types of 3-D grid locations based
on their respective position in the cuboid water tank are represented in Fig. 2.
Fig. 2 3-D Grid Locations
Here, the 3-D grid locations in
and space direction are set as and respectively in Eq. (4) so that
the space intervals can be defined as;
and the time interval would be
Furthermore, considering and the range of different types of 3-D grid
locations can be classified as in the Table 1.
Table 1 Types of 3-D grid locations
Types
3-D grid locations
Left Boundary
Right Boundary
Rear Boundary
Front Boundary
Top Boundary
Bottom Boundary
Unknown
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The mathematical model is executed with spatial increments of 1 foot in the directions and a time step of 1 unit,
equivalent to 20 minutes. The visual depiction of the Exp. data at various time intervals is illustrated in Fig. 3.
Time
Range
Time
Range
Fig. 3 Water pollution at 3-D grid locations of Exp. data
Fig. 3 shows that the levels of water pollution at each 3-D grid point rise over time. At time , the levels of pollution vary
between . Following 120 minutes, the range has risen to
The required initial and boundary conditions for time and space are derived using a Multi-Poly Regression model on the obtained
Exp. data, which can be expressed as in Eq. (19) – (26) with their respective Mean Absolute Error (MAE) and Standard Deviation
of Mean Absolute Error
Initial and Boundary Conditions
MAE
Initial Condition for time
0.0014
0.0011
(19)
Initial Condition for time
0.0012
0.0009
(20)
Left Boundary
0.0012
0.00076
(21)
Right Boundary
0.0011
0.00085
(22)
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Rear Boundary
0.0012
0.0008
(23)
Front Boundary
0.0019
0.0012
(24)
Top Boundary
0.0015
0.0012
(25)
Bottom Boundary
0.0012
0.0009
(26)
In the experiment of a proposed mathematical model, it is necessary to fix the value of the diffusion rate of the iodized salt-water
solution into the volume of the water tank. It is derived based on the phenomena of Fick’s first law, which is defined in Eq. (27)
[23].
(27)
Where, is the diffusion rate, is the diffusivity value of iodized salt-water solution , and is the average
concentration gradient. In Eq. (27), the negative sign denotes that the flow proceeds from the area of high concentration to the
area of low concentration.
The diffusivity value of iodized salt-water solution can be derived from Eq. (28).
(28)
Here, is the volume of water in diffusion vessel (L or ), is the capillaries’ length (cm), is the diameter of capillaries
(cm),
is the number of capillaries, is the molar concentration of iodized salt water solution (mol/L), is the slope of
conductivity change per unit molar concentration change (µS*L/mol) and
is the slope of conductivity change per unit time
(µS/s).
In this experiment,
= 2500 , = 0.4 cm, = 0.1 cm, = 50. The molar concentration of iodized salt is
(29)
Where, is the weight of solute (gm), is the molecular weight of solute (gm/mol) and is the volume of solvent (L).
Here, = 2500 gm iodized salt
= (impurity)
= 23+35.5+127+24 = 209.5 gm/mol
= 40 L
Substituting these values into Eq. (29), that
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= 0.29 mol/L
The experiment shows that the conductivity of iodized salt water solution is changed as its molar concentration level is vary.
Based on the results obtained during experiment the slope of conductivity per unit molar concentration
is measured from
curve fitting of conductivity Vs. molar concentration which is graphically presented in Fig. 4. Furthermore, in the experiment it is
observed that the conductivity is varying as the time changed. Based on the experimented results the conductivity changes per
unit time is obtained by curve fitting between conductivity Vs. time as graphically presented in Fig. 5.
Fig. 4 Conductivity Vs. Molar
Fig. 5 Conductivity Vs. Time
Fig. 4 and 5 shows that, the coefficient of molar indicates slope of the conductivity change per unit molar concentration
and the coefficient of time indicated the slope of conductivity change per unit time
Now, substituting all the required values in Eq. (28), getting that
In the experiment, the has three different layers as such as 1
st
layer (Top), 2
nd
layer (Middle) and 3
rd
layer
(Bottom) of cuboid tank.
Now, the average value of the concentration gradient for 1
st
and 2
nd
layer is
(30)
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and for 2
nd
and 3
rd
layer is
(31)
where, is the average of 1
st
layer pollution = , is the average of 2
nd
layer pollution =
, is the average of 3
rd
layer pollution = Moreover are the distance (cm)
between layers. Here,
= 0 feet = 0 cm, = 1 feet = 30.48 cm, = 2 feet = 60.96 cm. Now, substituting all these values in Eq.
(30) and (31), getting that
and whose average value is of
. By applying all the required values in Eq. (27), gets Eq. (32)
(32)
The proposed mathematical model is simulated in MATLAB by considering feet, unit (20 minutes),
and the diffusion rate
is considered similar in all the directions and
To estimate the water pollution level at each 3-D grid location over a time interval by applying Eq. (19 - 26) as the initial and
boundary conditions for the numerical solution of Eq. (1) by DF method, which is graphically represented in Fig. 6.
Time
Range
Time
Range
Fig. 6 Water pollution at 3-D grid locations by DF
Fig. 6 shows that the concentration level of pollutant iodized salt-water solution in water tank at different 3-D grid locations
gradually increases over time. In the DF method, the resultant water pollution level has the range of at
the initial time that is increased to the range of at the time (120 minutes). A significant
increment in water pollution to be noted can also be increased as the time passed up to its saturation limit.
Although the DF method has given a numerical solution of the given 3-D diffusion model for water pollution that reflects the
real-time phenomena that actually happened during the experiment, another mathematical solution approach should be adopted to
validate the obtained results. Thus, ADM has been used as an analytic approach to find the solution of the proposed diffusion
model for predicting the level of water pollution at the same 3-D locations at the same time interval.
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The required analytical solution of Eq. (1) by applying the ADM approach using the initial condition ( ) given in Eq. (19) is
expressed as in Eq. (33).
(33)
Now, solving Eq. (33) by substituting the value of the diffusion rate derived in Eq. (32) and alternating that gives
the level of water pollution at all 3-D grid locations. Fig. 7 displays a graphical representation of the ADM's outcomes for the
water pollution level at each 3-D grid location over the time period.
Time
Range
Time
Range
Fig. 7 Water pollution at 3-D grid locations by ADM
Fig. 7 shows that the water pollution level at 3-D grid locations is initially
in the range of , and
after 120 minutes , it is in the range of Also, there has been a monotonically increment in water
pollution at different locations with respect to the time interval.
Now, two different approaches, DF and ADM, have given numerical and analytical solutions of Eq. (1), respectively. The
comparison between numerical DF outcomes with ADM outcomes and Exp. data can validate the accuracy level of prediction of
water pollution levels at different 3-D grid locations. The error estimation criteria are used for the comparison, which can be
obtained by taking the absolute difference between the water pollution levels at each 3-D grid location across the time interval.
The estimated error is graphically plotted in Fig. 8.
Data
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Fig. 8 Comparative analysis between DF Vs. Exp. data and ADM Results
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Fig. 8 shows that, the comparative analysis between DF Vs. Exp. data has the error range in between 0.0113 PPM to 17.585 PPM
and 0.019 to 55.369 PPM in DF Vs. ADM. Also, the analysis of the comparative study between all 75 locations and six-
time intervals during the whole duration, it has been observed that the DF methods error against the Exp. data and ADM method
is visually shown in Fig. 9 for both PPM difference (error) and
Fig. 9 Error Estimation of DF Vs. Exp. data and ADM
Fig. 9 shows that, in both DF Vs. Exp. data and DF Vs. ADM comparison, 100% of the PPM difference lies under 60 PPM which
is highly negligible in the form of PPM because a change of up to 60 PPM is considered a negligible change while above 60 PPM
represents a significant change in the level of water pollution [25,26]. As a result, it can be said that 100% of the PPM difference
does not affect the water pollution level.
Thus, DF methods results are validated with the Exp. data, the analytical solution obtained by the ADM method and implemented
boundary conditions. It concludes that the result obtained by the DF method can be reliable with the Exp. data and the analytical
ADM results for predicting water pollution levels over a time interval.
As described in Table 1, there are several types of 3-D grid locations in cuboid water tank. In which, at every time interval, the
water pollution level at the top, bottom, left, right, rear and front boundary locations is estimated based on the boundary
conditions given in Eq. (21) – (26) while the water pollution level at nine unknown locations is calculated by DF method. These
nine locations can be labelled as . Here, it is essential to compare the
water pollution level numerically obtained by DF with the Exp. data and the analytical values derived by the ADM approach at
each time interval, graphically represented in Fig. 10.
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Fig. 10 DF, Exp. data and ADM Result at Nine Unknown Locations
Fig. 10 shows that the water pollution level at these nine unknown locations is gradually rising over a time interval in the cases of
DF, Exp. data and ADM approaches. Furthermore, the DF method result is very similar to Exp. data and ADM at all locations.
Also, Fig. 11 graphically represents each location-wise progression of water pollution levels over time intervals by DF, Exp. data
and ADM approaches.
Fig. 11 Water pollution level at unknown Location
Fig.11 shows that, the resultant graphs seem identical in all the cases of unknown locations with respect to time, which leads to
negligible errors in the estimation of water pollution levels.
IV. Conclusion
The present research effectively illustrates a mathematical model based on 3-D diffusion model for forecasting the levels of water
pollution in a cuboid water tank. A more accurate estimation of the spread of pollutant is made possible by this research's
extension of the analysis into three spatial directions. Iodized salt water has been used as the pollutant in the experiment, which
produced accurate experimental data for validation. Pollutant concentrations increased gradually over time, according to both
analytical outcomes from the ADM and numerical results from the DF approach. During DF approach, in the given time period
the water pollution is incremented at minimum range as 363.09 PPM while at maximum range of 321.02 PPM, while in ADM
approach the increment was found to be at minimum range 401.67 PPM and at maximum range of 348.39 PPM across the cuboid
water tank. According to comparative analysis, it is found that the variations in DF, ADM, and Exp. data are almost identical.
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Moreover, the error analysis revealed that the variations in the pollution level in both DF and ADM with Exp. data is less than 60
PPM, which is considered to be insignificant changed in the water pollution level in the form of PPM. This confirms that the DF
technique is a trustworthy numerical scheme that aligns with both analytical predictions and experimental results. Overall, a
trustworthy framework for predicting water pollution in 3-D regions is offered by the validated 3-D diffusion model. The
reliability is confirmed by the high convergence of approaches, and the strategy can be expanded to larger natural water systems
for monitoring, pollution prevention, and water quality management.
Declaration of Competing Interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Research Funding
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
Authors contribution
Tarjani Naik: Conceptualization, Methodology / Study design, Validation, Formal analysis, Resources, Investigation, Data
curation, Writing – original draft, Writing – review and editing, Visualization. Mukesh Patel: Conceptualization,
Conceptualization, Validation, Formal analysis, Writing – original draft, Supervision. Rachna Patel: Software, Formal analysis,
Resources, Data curation.
Data Availability
The data that support the findings of this study are available on request from the corresponding author, [Mukesh Petel,
mukesh.mt@gmail.com]. The data are not publicly available due to [restrictions e.g. their containing information that could
compromise the privacy of research participants]
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