Numerical and Analytical Method based Mathematical Approach to Estimate Water Pollution over 3-D Region

Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 50 Retrieval Number:100.1/ijam.B122205021025 DOI: 10.54105/ijam.B1222.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Numerical and Analytical Method based Mathematical Approach to Estimate Water Pollution over 3-D Region Tarjani Naik, Mukesh Patel, Rachna Patel Abstract: One of the major environmental issues is water pollution, which harms aquatic ecosystems, human health, and the sustainability of natural resources. To solve this problem, reliable mathematical models are required that can predict the behaviour of pollutants in three-dimensional (3-D) water systems. In this study, the temporal and spatial fluctuations of pollutant concentration in water are estimated using a 3-D diffusion model. The work's goals are to investigate the gradual rise in pollution levels in a 3-D region and assess the accuracy of analytical and numerical methods for addressing diffusion-based pollution issues. Two distinct mathematical techniques are employed to determine the level of pollution in water at a 3-D location: the Crank-Nicolson (C-N) method, a numerical method, and the Adomian Decomposition Method (ADM), an analytical method. The data collected from an experiment conducted in a 3-D cuboid tank, where water serves as the medium and iodised saltwater solution as the pollutant, are used to derive the initial and boundary conditions for the presented mathematical approaches. The dispersion of pollutants over time and space can be directly observed in this experiment, yielding crucial data for validating the mathematical model. The results of the experiments show that during all the time intervals, there is a rise in water pollution at all 3-D locations. Furthermore, the insignificant error in the form of parts per million (PPM) difference obtained when comparing the outcomes of the C-N with Experimental data (Exp. data) and ADM methodologies demonstrates the effectiveness of the proposed mathematical model. The benefit of this research lies in the use of mathematical approaches and experimental data to investigate water pollution in a 3-D region. The study demonstrates the validity of the proposed model and its ability to forecast the spread of pollution in water accurately. This method is also applicable to larger water systems than the experimental tank. The research enhances mathematical solutions to diffusion equations and provides valuable insights for developing pollution control measures, assessing water quality, and promoting sustainable water management. Keywords : Water Pollution · 3-D Diffusion Equation · Crank - Nicolson Method · Adomian Decomposition Method Mathematics Subject Classification : 35K57 · 65N06 Manuscript received on 26 September 2025 | First Revised Manuscript received on 02 October 2025 | Second Revised Manuscript received on 08 October 2025 | Manuscript Accepted on 15 October 2025 | Manuscript published on 30 October 2025. *Correspondence Author(s) Tarjani Naik *, Scholar, Department of Mathematics, Uka Tarsadia University, Bardoli (Gujarat), India. Email ID: [email protected], ORCID ID: 0009-0002-3392-1693 Dr. Mukesh Patel, Assistant Professor, Department of Mathematics, Uka Tarsadia University, Bardoli (Gujarat), India. Email ID: mukesh[email protected], ORCID ID: 0000-0001-5797-5288 Dr. Rachna Patel, Assistant Professor, Department of Computer Engineering, CGPIT, Uka Tarsadia University, Bardoli (Gujarat), India. Email ID: [email protected], ORCID ID: 0000-0002-5685-756X © The Authors. Published by Lattice Science Publication (LSP). This is an open-access article under the CC-BY-NC-ND license http://creativecommons.org/licenses/by-nc-nd/4.0/ Nomenclature: ADM: Adomian Decomposition Method PPM: Parts Per Million FTCS: Finite-Time Central Space ANN: artificial neural network TDS: Total Dissolved Solids MAE: Mean Absolute Error C-N: Crank-Nicolson I. INTRODUCTION One essential element of the earth is water; two-thirds of its surface is covered by it. Most people, particularly humans, rely on freshwater to ensure their growth [14]. In today's world, people want water that is sufficient in quantity and of superior quality [1]. Residential and industrial human activity-related water pollution is a serious issue in many nations [4]. An estimated 25 million people die from water pollution each year. Therefore, the issue of water quality is gaining huge attention worldwide [3]. There are multiple ways to characterize water pollution. Water pollution occurs when the physical, chemical, and biological aspects of water are altered in a manner that negatively impacts living organisms. Human activity is the leading cause of water contamination, harming both human health and the quality of the environment's water [16]. There are two approaches, numerical and analytical, that can be employed to address the issue of water pollution, and several techniques exist for solving mathematical equations related to this problem. In this research, a 3-D diffusion mathematical model is used to predict water pollution over a time interval with a constant diffusion rate. This model is demonstrated by considering water as a pollutant and iodised salt-water solution as another pollutant. A 3-D cuboid is prepared with an equidistant grid in all three directions, and the Experiment Is Conducted. data on water pollution in PPM is collected from each grid point of the cuboid over equal time intervals. The initial and boundary conditions are constructed from the experimental data, and two mathematical approaches, the C-N numerical method and the ADM analytical method, have been used to estimate the level of water pollution. Furthermore, their comparative analysis was performed to estimate errors. II. RELATED WORK Chen and Xu [6] developed a 1-D diffusion model to study water pollution in the Tuojiang River Basin. The model was implemented using the finite difference method, along with a Jacobian matrix. Zainab Yahya, Hanani Numerical and Analytical Method based Mathematical Approach to Estimate Water Pollution over 3-D Region 51 Retrieval Number:100.1/ijam.B122205021025 DOI: 10.54105/ijam.B1222.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Johari, and Nursalasawati Rusli [1] employed the onedimensional advection-diffusion model. She solved it using the Finite Difference Method (FTCS techniques and implicit C-N techniques) to forecast the transportation of water pollution concentration. Nigar Sultana and Laek Sazzad Andallah [7] solved the one-dimensional advection-diffusion equation using the second-order Lax-Wendroff method and Finite-Time Central Space (FTCS) to determine the concentration of water pollution in a river as well as the pollutant in the river at various times and locations. Abbas Parsaie, Amir Hamzeh Haghiabi [12], Omar Hireche, Abdelkader Saidane, Safia Meddah, and Mohamed Hadjel [13] developed a one-dimensional advection-diffusion model. In [12], this model was employed to estimate the longitudinal dispersion coefficient and simulate the transmission of pollution in rivers using the finite volume method and an artificial neural network (ANN). In [13], the Transmission Line Matrix Method was employed to estimate the longitudinal dispersion of pollutants and determine the maximum concentration over time. Delong Wan, Huiping Zeng [8], Tsegaye Simon, Purnachandra Rao Koya [10], R.V. Wagmare, and S.B. Kiwne [11] expanded their research work by adding some parameters to a one-dimensional advectiondiffusion model. In [8], the Pollution Index Method was employed to predict water quality based on specific parameters. In [10], the Runge-Kutta, splitting, and C-N methods were applied to calculate the numerical solution and study the dynamics of pollution in rivers. The splitting method separated the diffusion and reaction terms, and the CN and Runge-Kutta methods were used to find a numerical solution. In [11], the analytical method was used to find the system's solution. Kusuma, Ribal, Mahie, and Aris [5] extended a 1-D advection-diffusion model into a 2-D advection-diffusion model. She utilised a numerical compound finite difference method to investigate pollution levels in Unhas Lake, Indonesia. Saleh, Dimian, and Ibrahim [2] developed a 3-D advection-diffusion model. They employ both analytical methods, such as the Laplace transform, and numerical solutions through the finite difference method. They utilise dimensionless variables to forecast pollutant concentrations in rivers and examine the effectiveness of releasing clean water in reducing pollution levels. III. PROPOSED METHODOLOGY The issue of water pollution is addressed by the proposed mathematical method, as shown in Fig. 1. [Fig.1: Mathematical Method for 3-D Water Pollution Estimation] A. Water Pollution Mathematical Model The mathematical model for estimating water pollution is constructed based on the diffusion model in a 3-D region. The mathematical formulation of the rate of change in the concentration of the pollutant with respect to time 𝑡 at various 3-D locations in the 𝑥,𝑦 and 𝑧 directions is given in Eq. (1). 𝜕𝑤 𝜕𝑡=𝐷(𝜕2𝑤 𝜕𝑥2+𝜕2𝑤 𝜕𝑦2+𝜕2𝑤 𝜕𝑧2) … (1) where, 𝑡0 < t ≤ 𝑡𝑔, 𝑥0 < 𝑥 ≤ 𝑥ℎ, 𝑦0 < 𝑦 ≤ 𝑦𝑖, 𝑧0 < 𝑧 ≤ 𝑧𝑗 and the parameter 𝑤 is the concentration of the pollutant in 𝑥,𝑦,𝑧 directions, respectively, 𝑡 denotes the time, 𝑥,𝑦 and 𝑧 denotes directions, and 𝐷 The diffusion coefficient is constant and the same for all directions. B. Initial and Boundary Conditions for Mathematical Model The mathematical model of Eq. (1) has a numerical and analytical solution. 𝑤(𝑥,𝑦,𝑧,𝑡) that can be derived using CN and ADM, respectively. To solve the C-N and ADM methods, initial and boundary conditions are required for space and time. The initial condition for time 𝑡 is given in Eq. (2). 𝑤(𝑥,𝑦,𝑧,𝑡0)=∅0(𝑥,𝑦,𝑧) … (2) and boundary conditions for 𝑥,𝑦 and 𝑧 directions are as follows in Eq. (3) 𝑤(𝑥0,𝑦,𝑧,𝑡)=𝛹1(𝑦,𝑧,𝑡) 𝑤(𝑥𝑟,𝑦,𝑧,𝑡)=𝛹2(𝑦,𝑧,𝑡) 𝑤(𝑥,𝑦0,𝑧,𝑡)=𝛹3(𝑥,𝑧,𝑡) 𝑤(𝑥,𝑦𝑠,𝑧,𝑡)=𝛹4(𝑥,𝑧,𝑡) 𝑤(𝑥,𝑦,𝑧0,𝑡)=𝛹5(𝑥,𝑦,𝑡) 𝑤(𝑥,𝑦,𝑧𝑝,𝑡)=𝛹6(𝑥,𝑦,𝑡) } … (3) where, ∅0,𝛹1,𝛹2,𝛹3,𝛹4,𝛹5 𝑎𝑛𝑑 𝛹6 are known functions. C. Mathematical Method The diffusion equation can be solved using various analytical and numerical methods. The C-N method is used to obtain a numerical solution, while the ADM is utilized in this research to provide the analytical solution. i. Crank-Nicolson (C-N) Method [17] Analytic Method Adomian Decomposition Method (ADM) Mathematical Method 3-D Diffusion Model Initial and Boundary Conditions for Mathematical Model Water Pollution Mathematical Model Numerical Method Crank-Nicolson (C-N) Method Water Pollution Estimation Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 52 Retrieval Number:100.1/ijam.B122205021025 DOI: 10.54105/ijam.B1222.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. The C-N method is a finite difference method used to obtain the numerical solution of a mathematical diffusion model. The finite number of grid locations in the 𝑥,𝑦 and 𝑧 directions of the 3-D regions where water pollution is estimated in a finite time interval is stated as follows; The grid points (𝑥𝑎,𝑦𝑏,𝑧𝑐,𝑡𝑑) are given as 𝑥𝑎=𝑥0∶𝛿𝑥∶ 𝑥𝑟, 𝑎=0,1,2,…𝑟 𝑦𝑏=𝑦0∶𝛿𝑦∶ 𝑦𝑠, 𝑏=0,1,2,…𝑠 𝑧𝑐=𝑧0∶𝛿𝑧∶ 𝑧𝑝, 𝑐=0,1,2,…𝑝 … (4) 𝑡𝑑=𝑡0∶𝛿𝑡∶ 𝑡𝑙, 𝑑=0,1,2,…𝑙 Where 𝑟,𝑠,𝑝 and 𝑙 are integers and 𝛿𝑥,𝛿𝑦 , and 𝛿𝑧 are grid spacing of all three directions, respectively, and 𝛿𝑡 is a time step size and denotes 𝑤(𝑥𝑎,𝑦𝑏,𝑧𝑐,𝑡𝑑)=𝑤(𝑥,𝑦,𝑧,𝑡) in the finite difference approximation. 𝜕𝑤 𝜕𝑡 =𝑤(𝑥,𝑦,𝑧,𝑡+1)−𝑤(𝑥,𝑦,𝑧,𝑡) 𝛿𝑡 Forward difference for timespace derivative (5) 𝜕𝑤 𝜕𝑡 =𝑤(𝑥,𝑦,𝑧,𝑡)−𝑤(𝑥,𝑦,𝑧,𝑡−1) 𝛿𝑡 Backward difference for time-space derivative (6) 𝜕2𝑤 𝜕𝑥2=𝑤(𝑥+1,𝑦,𝑧,𝑡)−2𝑤(𝑥,𝑦,𝑧,𝑡)+𝑤(𝑥−1,𝑦,𝑧,𝑡) 𝛿𝑥2 The central difference for space derivative in 𝑥direction (7) 𝜕2𝑤 𝜕𝑦2=𝑤(𝑥,𝑦+1,𝑧,𝑡)−2𝑤(𝑥,𝑦,𝑧,𝑡)+𝑤(𝑥,𝑦−1,𝑧,𝑡) 𝛿𝑦2 The central difference for spatial derivative in 𝑦direction (8) 𝜕2𝑤 𝜕𝑧2=𝑤(𝑥,𝑦,𝑧+1,𝑡)−2𝑤(𝑥,𝑦,𝑧,𝑡)+𝑤(𝑥,𝑦,𝑧−1,𝑡) 𝛿𝑧2 The central difference for spatial derivative in 𝑧direction (9) Apply Eq. (5), (7) - (9) in Eq. (1), 𝑤(𝑥,𝑦,𝑧,𝑡+1)−𝑤(𝑥,𝑦,𝑧,𝑡) 𝛿𝑡= 𝐷[𝑤(𝑥+1,𝑦,𝑧,𝑡)−2𝑤(𝑥,𝑦,𝑧,𝑡)+𝑤(𝑥−1,𝑦,𝑧,𝑡) 𝛿𝑥2+ 𝑤(𝑥,𝑦+1,𝑧,𝑡)−2𝑤(𝑥,𝑦,𝑧,𝑡)+𝑤(𝑥,𝑦−1,𝑧,𝑡) 𝛿𝑦2+ 𝑤(𝑥,𝑦,𝑧+1,𝑡)−2𝑤(𝑥,𝑦,𝑧,𝑡)+𝑤(𝑥,𝑦,𝑧−1,𝑡) 𝛿𝑧2] (10) Further, apply Eq. (6), (7) - (9) in Eq. (1), 𝑤(𝑥,𝑦,𝑧,𝑡)−𝑤(𝑥,𝑦,𝑧,𝑡−1) 𝛿𝑡= 𝐷[𝑤(𝑥+1,𝑦,𝑧,𝑡)−2𝑤(𝑥,𝑦,𝑧,𝑡)+𝑤(𝑥−1,𝑦,𝑧,𝑡) 𝛿𝑥2+ 𝑤(𝑥,𝑦+1,𝑧,𝑡)−2𝑤(𝑥,𝑦,𝑧,𝑡)+𝑤(𝑥,𝑦−1,𝑧,𝑡) 𝛿𝑦2+ 𝑤(𝑥,𝑦,𝑧+1,𝑡)−2𝑤(𝑥,𝑦,𝑧,𝑡)+𝑤(𝑥,𝑦,𝑧−1,𝑡) 𝛿𝑧2] (11) Then, replace 𝑡 by 𝑡+1 in Eq. (11), 𝑤(𝑥,𝑦,𝑧,𝑡+1)−𝑤(𝑥,𝑦,𝑧,𝑡) 𝛿𝑡= 𝐷[𝑤(𝑥+1,𝑦,𝑧,𝑡+1)−2𝑤(𝑥,𝑦,𝑧,𝑡+1)+𝑤(𝑥−1,𝑦,𝑧,𝑡+1) 𝛿𝑥2+ 𝑤(𝑥,𝑦+1,𝑧,𝑡+𝛿𝑡)−2𝑤(𝑥,𝑦,𝑧,𝑡+1)+𝑤(𝑥,𝑦−1,𝑧,𝑡+1) 𝛿𝑦2+ 𝑤(𝑥,𝑦,𝑧+1,𝑡+1)−2𝑤(𝑥,𝑦,𝑧,𝑡+1)+𝑤(𝑥,𝑦,𝑧−1,𝑡+1) 𝛿𝑧2] (12) Now, add Eq. (10) & (12) and take the same grid spacing for all three directions that 𝛿𝑥=𝛿𝑦=𝛿𝑧=𝛿. 𝑤(𝑥,𝑦,𝑧,𝑡+1)−𝑤(𝑥,𝑦,𝑧,𝑡)=𝐷∗𝛿𝑡 2𝛿2[𝑤(𝑥+ 1,𝑦,𝑧,𝑡)+𝑤(𝑥−1,𝑦,𝑧,𝑡)+𝑤(𝑥,𝑦+1,𝑧,𝑡)+ 𝑤(𝑥,𝑦−1,𝑧,𝑡)+𝑤(𝑥,𝑦,𝑧+1,𝑡)+𝑤(𝑥,𝑦,𝑧− 1,𝑡)−6𝑤(𝑥,𝑦,𝑧,𝑡)+𝑤(𝑥+1,𝑦,𝑧,𝑡+1)+𝑤(𝑥− (13) 1,𝑦,𝑧,𝑡+1)+𝑤(𝑥,𝑦+1,𝑧,𝑡+1)+𝑤(𝑥,𝑦− 1,𝑧,𝑡+1)+𝑤(𝑥,𝑦,𝑧+1,𝑡+1)+𝑤(𝑥,𝑦,𝑧−1,𝑡+ 1)−6𝑤(𝑥,𝑦,𝑧,𝑡+1)] Taking 𝐷∗𝛿𝑡 𝛿2=𝜇 in Eq. (13) and simplifying that getting Eq. (14) 𝑤(𝑥,𝑦,𝑧,𝑡+1)=(1−3 𝜇) (1+3 𝜇)𝑤(𝑥,𝑦,𝑧,𝑡)+ 𝜇 2(1+3 𝜇)[𝑤(𝑥+1,𝑦,𝑧,𝑡)+𝑤(𝑥−1,𝑦,𝑧,𝑡)+ 𝑤(𝑥,𝑦+1,𝑧,𝑡)+𝑤(𝑥,𝑦−1,𝑧,𝑡)+𝑤(𝑥,𝑦,𝑧+ 1,𝑡)+𝑤(𝑥,𝑦,𝑧−1,𝑡)+𝑤(𝑥+1,𝑦,𝑧,𝑡+1)+ 𝑤(𝑥−1,𝑦,𝑧,𝑡+1)+𝑤(𝑥,𝑦+1,𝑧,𝑡+1)+ 𝑤(𝑥,𝑦−1,𝑧,𝑡+1)+𝑤(𝑥,𝑦,𝑧+1,𝑡+1)+ 𝑤(𝑥,𝑦,𝑧−1,𝑡+1)] (14) This Eq. (14) is the implicit formula for the C-N technique. To get the answer at the 𝑡+1 level, it is necessary to finding the solution at some locations of the 𝑡-level and 𝑡+1 levels. Therefore, this method requires the initial conditions shown in Eq. (2). D. Adomian Decomposition Method (ADM) [15] In ADM method, re-write Eq. (1) in the standard operator form as 𝐿𝑡𝑤=𝐷 (𝐿𝑥𝑥𝑤+𝐿𝑦𝑦𝑤+𝐿𝑧𝑧𝑤) … (15) Were, 𝐿𝑡=𝜕 𝜕𝑡 ,𝐿𝑥𝑥=𝜕2 𝜕𝑥2 ,𝐿𝑦𝑦=𝜕2 𝜕𝑦2 ,𝐿𝑧𝑧=𝜕2 𝜕𝑧2 . Taking the inverse operator of the operator 𝐿𝑡 exists and it defined as 𝐿𝑡−1(.)=∫(.)𝑑𝑡 𝑡 0 Thus, applying the inverse operator 𝐿𝑡−1 to Eq. (15) yields 𝐿𝑡−1 𝐿𝑡𝑤(𝑥,𝑦,𝑧,𝑡)=𝐷 (𝐿𝑡−1 𝐿𝑥𝑥𝑤+𝐿𝑡−1 𝐿𝑦𝑦𝑤 +𝐿𝑡−1 𝐿𝑧𝑧𝑤) Numerical and Analytical Method based Mathematical Approach to Estimate Water Pollution over 3-D Region 53 Retrieval Number:100.1/ijam.B122205021025 DOI: 10.54105/ijam.B1222.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. 𝑤(𝑥,𝑦,𝑧,𝑡)−𝑤(𝑥,𝑦,𝑧,0) =𝐷 (𝐿𝑡−1 𝐿𝑥𝑥𝑤+𝐿𝑡−1 𝐿𝑦𝑦𝑤 +𝐿𝑡−1 𝐿𝑧𝑧𝑤) 𝑤(𝑥,𝑦,𝑧,𝑡)=𝑤(𝑥,𝑦,𝑧,0)+𝐷 𝐿𝑡−1 (𝐿𝑥𝑥𝑤+ 𝐿𝑦𝑦𝑤 +𝐿𝑧𝑧𝑤) (16) In ADM, represent the solution suppose that 𝑤(𝑥,𝑦,𝑧,𝑡)=∑𝑤𝑛(𝑥,𝑦,𝑧,𝑡) ∞ 𝑛=0 … (17) Substituting Eq. (17) into (16), get ∑𝑤𝑛(𝑥,𝑦,𝑧,𝑡) ∞ 𝑛=0 =𝑤(𝑥,𝑦,𝑧,0)+ 𝐷 𝐿𝑡−1[ 𝐿𝑥𝑥∑𝑤𝑛(𝑥,𝑦,𝑧,𝑡) ∞ 𝑛=0 + 𝐿𝑦𝑦∑𝑤𝑛(𝑥,𝑦,𝑧,𝑡) ∞ 𝑛=0 +𝐿𝑧𝑧∑𝑤𝑛(𝑥,𝑦,𝑧,𝑡) ∞ 𝑛=0 ] 𝑤0(𝑥,𝑦,𝑧,𝑡)+𝑤1(𝑥,𝑦,𝑧,𝑡)+⋯= 𝑤(𝑥,𝑦,𝑧,0)+ 𝐷 𝐿𝑡−1[ 𝐿𝑥𝑥∑𝑤𝑛(𝑥,𝑦,𝑧,𝑡) ∞ 𝑛=0 + 𝐿𝑦𝑦∑𝑤(𝑥,𝑦,𝑧,𝑡) ∞ 𝑛=0 + 𝐿𝑧𝑧∑𝑤𝑛(𝑥,𝑦,𝑧,𝑡) ∞ 𝑛=0 ] … (18) Now, comparing the Eq. (18) on both sides getting the recurrent relation in the form of as follows 𝑤0(𝑥,𝑦,𝑧,𝑡)=𝑤(𝑥,𝑦,𝑧,𝑡0)=𝑤(𝑥,𝑦,𝑧,0)= ∅0(𝑥,𝑦,𝑧) (From Eq. (2)) and 𝑤𝑛+1(𝑥,𝑦,𝑧,𝑡)=𝐷 𝐿𝑡−1 (𝐿𝑥𝑥𝑤𝑛(𝑥,𝑦,𝑧,𝑡)+ 𝐿𝑦𝑦𝑤𝑛(𝑥,𝑦,𝑧,𝑡)+𝐿𝑧𝑧𝑤𝑛(𝑥,𝑦,𝑧,𝑡)) for 𝑛=0,1,2,… From which 𝑤1(𝑥,𝑦,𝑧,𝑡)=𝐷 𝐿𝑡−1 (𝐿𝑥𝑥𝑤0(𝑥,𝑦,𝑧,𝑡) + 𝐿𝑦𝑦𝑤0(𝑥,𝑦,𝑧,𝑡) +𝐿𝑧𝑧𝑤0(𝑥,𝑦,𝑧,𝑡)) 𝑤2(𝑥,𝑦,𝑧,𝑡)=𝐷 𝐿𝑡−1 (𝐿𝑥𝑥𝑤1(𝑥,𝑦,𝑧,𝑡) + 𝐿𝑦𝑦𝑤1(𝑥,𝑦,𝑧,𝑡) +𝐿𝑧𝑧𝑤1(𝑥,𝑦,𝑧,𝑡)) ⋮ 𝑤𝑛(𝑥,𝑦,𝑧,𝑡)=𝐷 𝐿𝑡−1 (𝐿𝑥𝑥𝑤𝑛−1(𝑥,𝑦,𝑧,𝑡) + 𝐿𝑦𝑦𝑤𝑛−1(𝑥,𝑦,𝑧,𝑡)+𝐿𝑧𝑧𝑤𝑛−1(𝑥,𝑦,𝑧,𝑡)) } … (19) Therefore, the estimation of the approximate solution ∅𝛾 by using 𝛾-term approximation. That is, ∅𝛾=∑𝑤𝑛(𝑥,𝑦,𝑧,𝑡) 𝛾−1 𝑛=0 … (20) Therefore, Eq. (20) is the approximate solution of the 3-D diffusion mathematical model. D. Water Pollution Estimation The level of water pollution at a 3-D grid location is predicted by the C-N mathematical approach over time intervals, and their results are validated through comparison with the ADM approach. Equation (14) employs the C-N approach to estimate the concentration of pollutants in water at various locations over different time periods. It is a 2-level implicit technique, which uses the values of the surrounding locations in the 𝑥,𝑦, and 𝑧 directions of the previous one-time (𝑡) level and the current time (𝑡+1) level to estimate the current time (𝑡+1) level water pollution at a particular location. This technique helps to understand pollution dispersion by simulating the spread of pollutants, such as an iodised salt-water solution, in a water body. Eq. (20) depicts the concentration of pollutants in water at a specific location (𝑥,𝑦,𝑧) and time 𝑡, based on a series solution obtained from the ADM. This formula indicates that the analytical solution is expressed as a sum of terms 𝑤0,𝑤1,𝑤2,…,𝑤𝛾−1.Here, 𝑤0 represents the initial condition, and Eq. (19) is utilized to determine the subsequent terms, 𝑤1,𝑤2,…,𝑤𝛾−1. Therefore, this equation provides a method for computing the pollutant concentration over both time and space using a series expansion technique. IV. EXPERIMENTED RESULTS AND DISCUSSION According to studies, the majority of previous research has focused on 1-D or 2-D diffusion mathematical models, which have been resolved using a variety of analytical and numerical methods. The proposed research extends the diffusion model into a 3D space to estimate water contamination. Additionally, the researchers used a 3-D dummy cuboid water tank with measurements of 4.25× 4.25×2.25 feet to demonstrate the 3-D water pollution diffusion model. To establish the 3-D grid location in a cuboid tank, a grid structure with a grid spacing of one foot is used in all directions. Each 3-D grid location represents approximately one cubic foot 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 enables the study of pollution spread in all three directions within the tank. Here, 960 Liters of water are used as a pollutant, and 40 Liters of an iodized salt-water solution are used as a pollutant. The contaminated water is measured by a Total Dissolved Solids (TDS) meter in PPM every 20 minutes. The various types of 3-D grid locations, categorised by their positions within cuboid water tanks, are illustrated in Fig. 2. [Fig.2: 3-D Grid Locations] Here, the range of 3-D grid positions in the 𝑥,𝑦 and 𝑧 space directions is set as 𝑟=4,𝑠=4,𝑝=2 and 𝑙=5 respectively in Eq. (4). Therefore, the time interval would be 𝑡0≤ 𝑡 ≤ 𝑡5 and the space intervals may be defined as 𝑥0≤ 𝑥 ≤ 𝑥4,𝑦0≤ 𝑦 ≤ 𝑦4 and 𝑧0≤ 𝑧≤𝑧2. Moreover, taking into consideration 𝑥0=𝑦0=𝑧0=𝑡0=0,𝑥4=𝑦4=4,𝑧2= 2 and 𝑡5=5The range of various 3-D grid locations can be classified as shown in Table I. Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 54 Retrieval Number:100.1/ijam.B122205021025 DOI: 10.54105/ijam.B1222.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Table I: Types of 3-D grid locations Types 3-D Grid Locations Left Boundary 𝑥=0 ,0 ≤ 𝑦 ≤ 4,0<𝑧<2 Right Boundary 𝑥=4 ,0 ≤ 𝑦 ≤ 4,0<𝑧<2 Rear Boundary 0< 𝑥< 4,𝑦=0,0<𝑧<2 Front Boundary 0< 𝑥< 4,𝑦=4,0<𝑧<2 Top Boundary 0 ≤ 𝑥 ≤ 4,0 ≤ 𝑦 ≤4,𝑧=0 Bottom Boundary 0 ≤ 𝑥 ≤ 4,0 ≤ 𝑦 ≤4,𝑧=2 Unknown (𝑥1=1)< 𝑥<(𝑥3=3), (𝑦1=1) ≤ 𝑦 ≤ (𝑦3=3), 𝑧=(𝑧1=1) The mathematical model is simulated in MATLAB with spatial increments of 1 foot in the 𝑥,𝑦,𝑎𝑛𝑑 𝑧 directions and a time step of 1 unit, which corresponds to 20 minutes. The graphical representation of the Exp. data at different time intervals, as shown in Fig. 3. Time 𝑡=0 𝑡=1 Range 495−2680 𝑃𝑃𝑀 635−2720 𝑃𝑃𝑀 Time 𝑡=2 𝑡=3 Range 695−2780 𝑃𝑃𝑀 763−2840 𝑃𝑃𝑀 Time 𝑡=4 𝑡=5 Range 817−2920 𝑃𝑃𝑀 858−3000 𝑃𝑃𝑀 [Fig.3: Water Pollution at 3-D grid Locations of Exp. Data] Figure 3 illustrates that the water pollution levels at each 3D grid point increase over time. Initially, at time t=0, the pollution levels range from 495−2680 𝑃𝑃𝑀. After 120 minutes, the range has increased to 858−3000 𝑃𝑃𝑀. Now, using the Multi-Poly Regression model on the observed Exp. Based on the data, the necessary initial and boundary conditions for space and time are derived. It can be stated as in Eq. (21) – (27), with their corresponding Mean Absolute Error (MAE) and Standard Deviation of Mean Absolute Error (𝑆𝐷𝑀𝐴𝐸). Numerical and Analytical Method based Mathematical Approach to Estimate Water Pollution over 3-D Region 55 Retrieval Number:100.1/ijam.B122205021025 DOI: 10.54105/ijam.B1222.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Initial and Boundary Conditions MAE 𝑺𝑫𝑴𝑨𝑬 Initial Condition for time 𝑡=(𝑡0=0) 𝑤(𝑥,𝑦,𝑧,0)=903.55∗𝑧−71.87∗𝑧2−73.67∗𝑦−78.60∗𝑦∗𝑧+24.22∗𝑦∗𝑧2+6.24∗𝑦2+5.34∗ 𝑦2∗𝑧−2.86∗𝑦2∗𝑧2−325.08∗𝑥−117.94∗𝑥∗𝑧+49.86∗𝑥∗𝑧2+44.60∗𝑥∗𝑦+5.69∗𝑥∗𝑦∗𝑧− 3.57∗𝑥∗𝑦∗𝑧2−3.98∗𝑥∗𝑦2+0.45∗𝑥∗𝑦2∗𝑧+213.34∗𝑥2−21.77∗𝑥2∗𝑧+2.06∗𝑥2∗𝑧2− 19.12∗𝑥2∗𝑦+1.87∗𝑥2∗𝑦∗𝑧+0.23∗𝑥2∗𝑦2−73.13∗𝑥3+0.05∗𝑥3∗𝑧+2.69∗𝑥3∗𝑦+1159.17+ 8.26∗𝑥4; 0.0014 0.0011 (21) Left Boundary 𝑤(0,𝑦,𝑧,𝑡)=50.15∗𝑡+1989.34∗𝑧+4.43∗𝑧∗𝑡2+2.22∗𝑦∗𝑡+0.04∗𝑦∗𝑡2−128.40∗𝑦∗𝑧+ 9.22∗𝑦2−1.04∗𝑦2∗𝑡−0.093∗𝑦3; 0.0012 0.00076 (22) Right Boundary 𝑤(4,𝑦,𝑧,𝑡)=−0.12∗𝑦∗𝑡2−1.22∗𝑦2∗𝑡+958+106.32∗𝑡+4.04∗𝑡2−47.04∗𝑦+11.88∗𝑦∗𝑡− 1.44∗𝑦2−1.47𝑒−14∗𝑦3; 0.0011 0.00085 (23) Rear Boundary 𝑤(𝑥,0,𝑧,𝑡)=42.23∗𝑡+1812.02∗𝑧+5.076∗𝑧∗𝑡2−69.87∗𝑥∗𝑡−0.43∗𝑥∗𝑡2−91.36∗𝑥2+ 21.83∗𝑥2∗𝑡+9.56∗𝑥3; 0.0012 0.0008 (24) Front Boundary 𝑤(𝑥,4,𝑧,𝑡)=40.08∗𝑡+4.8∗𝑡2+1504.56∗𝑧−0.27∗𝑥∗𝑡2−65.76∗𝑥∗𝑡+22.35∗𝑥2∗𝑡−76.76∗ 𝑥2+5.88∗𝑥3; 0.0019 0.0012 (25) Top Boundary 𝑤(𝑥,𝑦,0,𝑡)=55.20∗𝑡+4.54∗𝑡2−72.73∗𝑦+1.76∗𝑦∗𝑡+0.048∗𝑦∗𝑡2+5.86∗𝑦2−0.99∗𝑦2∗ 𝑡−0.007∗𝑦2∗𝑡2−325.096∗𝑥−74.92∗𝑥∗𝑡−0.17∗𝑥∗𝑡2+42.35∗𝑥∗𝑦+2.14∗𝑥∗𝑦∗𝑡− 0.0007∗𝑥∗𝑦∗𝑡2−3.51∗𝑥∗𝑦2−0.046∗𝑥∗𝑦2∗𝑡+214.80∗𝑥2+22.88∗𝑥2∗𝑡−0.011∗𝑥2∗𝑡2− 18.21∗𝑥2∗𝑦−0.022∗𝑥2∗𝑦∗𝑡+0.13∗𝑥2∗𝑦2−74.013∗𝑥3−0.087∗𝑥3∗𝑡+2.59∗𝑥3∗𝑦+ 1159.17+8.39∗𝑥4; 0.0015 0.0012 (26) Bottom Boundary 𝑤(𝑥,𝑦,2,𝑡)=2675.5+42.94∗𝑡+4.38∗𝑡2−130.74∗𝑦+3.02∗𝑦∗𝑡+0.002∗𝑦∗𝑡2+0.025∗𝑦2∗ 𝑡2+4.8∗𝑦2−1.18∗𝑦2∗𝑡−353.56∗𝑥−75.64∗𝑥∗𝑡−0.34∗𝑥∗𝑡2−0.023∗𝑥∗𝑦∗𝑡2+39.9∗𝑥∗ 𝑦+2.38∗𝑥∗𝑦∗𝑡−0.043∗𝑥∗𝑦2∗𝑡−2.58∗𝑥∗𝑦2+0.059∗𝑥2∗𝑡2+176.82∗𝑥2+23.54∗𝑥2∗𝑡− 0.029∗𝑥2∗𝑦∗𝑡−15.7∗𝑥2∗𝑦+0.17∗𝑥2∗𝑦2−0.27∗𝑥3∗𝑡−73.82∗𝑥3+2.82∗𝑥3∗𝑦+8.43∗𝑥4; 0.0012 0.0009 (27) In this experiment, it is essential to decide the value of the diffusion rate of the iodized salt-water solution into the water tank volume for the proposed mathematical model. It is based on Fick's first law phenomenon, described in Eq. (28) [18]. 𝐷=−𝐷𝑣 𝑑𝐶 𝑑𝑓 … (28) Where, 𝐷 is the diffusion rate, 𝐷𝑣 is the diffusivity rate of iodized salt-water solution (cm2 s) and 𝑑𝐶 𝑑𝑓 is the average concentration gradient. The negative sign in Eq. (28) indicates that the flow moves from areas of high concentration to areas of low concentration. The diffusivity rate of the iodised salt-water solution is obtained from Eq. (29). 𝐷𝑣=4 𝑉𝑥𝑐 𝜋𝑑𝑐2𝑁𝑀𝐶𝑀 𝑑𝑘 𝑑𝑡 … (29) Where, 𝑉 is the volume of water in diffusion vessel (Liter (L) or cm3), 𝑥𝑐 is the capillaries’ length (cm), 𝑑𝑐 is the diameter of capillaries (cm), 𝑁 is the number of capillaries, 𝑀 is the molar concentration of iodized salt solution (mol/L), 𝐶𝑀 is the slope of conductivity change per unit molar concentration change (µS*L/mol) and 𝑑𝑘 𝑑𝑡 is the slope of conductivity change per unit time (µS/s). In this experiment, 𝑉= 2500 𝑐𝑚3,𝑥𝑐= 0.4 𝑐𝑚,𝑑𝑐= 0.1 𝑐𝑚 and 𝑁 = 50 in this experiment. The iodized salt's molar concentration is 𝑀=𝑤𝑠 𝑀𝑤×1 (𝐿) 𝑉𝑠 … (30) Where, 𝑤𝑠 is the weight of solute (gm), 𝑀𝑤 is the molecular weight of solute (gm/mol) and 𝑉𝑠 is the volume of solvent (L). Here, 𝑤𝑠= 2500 gm iodized salt 𝑀𝑤 = 𝑁𝑎++𝐶𝑙−+𝐼−+𝑀𝑔+(impurity) = 23+35.5+127+24 = 209.5 gm/mol 𝑉𝑠= 40 L Applying these values into Eq. (30), 𝑀=2500 gm 209.5gm mol×1 L 40 L 𝑀 = 0.29 mol/L The graphs below, in Figs. 4 and 5, show the slopes of conductivity changes per unit of molar concentration change and conductivity changes per unit of time obtained. [Fig.4: Conductivity Vs. Molar] Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 56 Retrieval Number:100.1/ijam.B122205021025 DOI: 10.54105/ijam.B1222.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. [Fig.5: Conductivity Vs. Time] Fig. 4 and 5 demonstrate that the molar coefficient reflects the slope of conductivity variation for each unit of molar concentration, with a value of 𝐶𝑀= 2.1311×10^4 (µ𝑆∗ 𝐿)/𝑚𝑜𝑙, while the time coefficient indicates the slope of conductivity change relative to time, represented by 𝑑𝑘/𝑑𝑡= 2.1789 µ𝑆/𝑠. Now, substitute all the values in Eq. (29), obtain that 𝐷𝑣=4×2500 cm3×0.4 𝑐𝑚×2.1789 µSs 3.14×(0.1)2cm2×50×0.29(𝑚𝑜𝑙 𝐿)×2.1311×104µS∗L mol 𝐷𝑣=8715.6 0.97028×104 cm2 𝑠 𝐷𝑣=8982.56×10−4cm2 𝑠 In the experiment, the z-axis consists of three distinct layers within the range. 0≤𝑧≤2, Namely the 1st layer (Top), 2nd layer (Middle), and 3rd layer (Bottom) of the cuboid tank. Now, the average concentration gradient for the 1st and 2nd layers is (𝑑𝐶 𝑑𝑓)1=𝑐2−𝑐1 𝑓2−𝑓1 … (31) the 2nd and 3rd layers are (𝑑𝐶 𝑑𝑓)2=𝑐3−𝑐2 𝑓3−𝑓2 … (32) where, 𝑐1is the average of the first layer (𝑧=0) pollution = 1387.219𝑃𝑃𝑀 𝑐𝑚3,𝑐2 is the average of the second layer (𝑧=1) pollution = 1381.575𝑃𝑃𝑀 𝑐𝑚3, 𝑐3 is the average of the third layer (𝑧=3) pollution = 1372.26𝑃𝑃𝑀 𝑐𝑚3. 𝑓1,𝑓2 and 𝑓3 are the distance (cm) between the layers. Here, 𝑓1= 0 feet = 0 cm, 𝑓2= 1 feet = 30.48 cm, 𝑓3= 2 feet = 60.96 cm. After all these values are replaced in Eq. (31) and (32), it is obtained that (𝑑𝐶 𝑑𝑓)1=−0.185𝑃𝑃𝑀 𝑐𝑚4 and (𝑑𝐶 𝑑𝑓)2=−0.3056 𝑃𝑃𝑀/ 𝑐𝑚4. Consequently, the concentration gradient's average value is 𝑑𝐶 𝑑𝑓=−0.24PPM cm4 . Now, the diffusion coefficient of the iodised salt solution is obtained by substituting each of the necessary values into Eq. (28), which yields Eq. (33). 𝐷=− 8982.56×10−4cm2 𝑠×−0.24 PPM cm4 𝐷=2155.8×10−4 PPM cm2 𝑠 𝐷=0.21558PPM cm2 𝑠 (33) The suggested mathematical model is implemented in MATLAB, taking into account the following considerations. 𝛿𝑥 =𝛿𝑦=𝛿𝑧=1 𝑓𝑜𝑜𝑡,𝛿𝑡=1 unit (20 minutes), and a diffusion coefficient 𝐷=0.21558 𝑃𝑃𝑀/(𝑐𝑚² 𝑠) Which is assumed to be uniform in the 𝑥,𝑦,𝑎𝑛𝑑 𝑧 directions. To assess water pollution levels across every point in a 3-D grid over a period of time, apply Eq. (21) – (27) to solve Eq. (1) numerically via the C-N method. It is graphically plotted in Fig. 6. Time 𝑡=0 𝑡=1 Range 495.52−2678.80 𝑃𝑃𝑀 635.39−2722.82 𝑃𝑃𝑀 Time 𝑡=2 𝑡=3 Numerical and Analytical Method based Mathematical Approach to Estimate Water Pollution over 3-D Region 57 Retrieval Number:100.1/ijam.B122205021025 DOI: 10.54105/ijam.B1222.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Range 696.07−2778.91 𝑃𝑃𝑀 763.88−2843.78 𝑃𝑃𝑀 Time 𝑡=4 𝑡=5 Range 817.91−2917.41 𝑃𝑃𝑀 858.61−2999.82 𝑃𝑃𝑀 [Fig.6: Water Pollution at 3-D Grid Locations by C-N] Fig. 6 illustrates that the concentration of pollutant iodised salt-water solution in water tanks at various 3-D grid locations progressively rises over time. The resultant water pollution level in the C-N technique ranges from 495.52− 2678.80 𝑃𝑃𝑀 at the beginning time (𝑡=0) and increases to 858.61−2999.82 𝑃𝑃𝑀 at time 𝑡=5 (120 minutes). There has been a significant increase in water contamination, which is expected to continue rising until it reaches its saturation point. However, the C-N method gives a numerical solution for the 3-D diffusion model for water pollution, accurately reflecting the phenomena that occurred in real-time during the experiment. Still, another mathematical solution technique should be used to validate the results. Therefore, to obtain the solution of the proposed diffusion model for predicting the amount of water pollution at the exact 3-D location over the same time duration, ADM has been employed as an analytical solution approach. The analytical solution of Eq. (1), obtained by applying the ADM and using the initial condition given by Eq. (21) at a time ( 𝑡 = 0 ) is expressed in Eq. (34). 𝑤(𝑥,𝑦,𝑧,𝑡)= 96.91∗𝐷2∗𝑡2+𝐷∗𝑡∗(103.8∗ 𝑥2+9.02∗𝑥∗𝑦+1.207∗𝑥∗𝑧−347∗𝑥− 5.255∗𝑦2+3.736∗𝑦∗𝑧+10.19∗𝑦−1.586∗ 𝑧2−32.87∗𝑧+295.4)+8.264∗𝑥4+2.692∗ 𝑥3∗𝑦+0.05∗𝑥3∗𝑧−73.13∗𝑥3+0.2296∗𝑥2∗ 𝑦2+1.868∗𝑥2∗𝑦∗𝑧−19.112∗𝑥2∗𝑦+2.064∗ 𝑥2∗𝑧2−21.77∗𝑥2∗𝑧+213.3∗𝑥2+0.4536∗ 𝑥∗𝑦2∗𝑧−3.979∗𝑥∗𝑦2−3.565∗𝑥∗𝑦∗𝑧2+ 5.689∗𝑥∗𝑦∗𝑧+44.6∗𝑥∗𝑦+49.86∗𝑥∗𝑧2− 117.9∗𝑥∗𝑧−325.1∗𝑥−2.857∗𝑦2∗𝑧2+ 5.336∗𝑦2∗𝑧+6.238∗𝑦2+24.22∗𝑦∗𝑧2− 78.6∗𝑦∗𝑧−73.67∗𝑦−71.87∗𝑧2+903.5∗𝑧+ 1159; (34) By substituting the values of 𝑥,𝑦,𝑧,𝑎𝑛𝑑 𝑡 in an alternating manner, Eq. (34) is solved, which indicates the water pollution levels at every location within the 3-D grid. Fig. 7 provides a 4-D graphical depiction of the results from the ADM concerning water pollution levels for each 3-D grid location throughout the given time interval. Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 58 Retrieval Number:100.1/ijam.B122205021025 DOI: 10.54105/ijam.B1222.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Time 𝑡=0 𝑡=1 Range 495.50−2678.52 𝑃𝑃𝑀 641.86−2730.56 𝑃𝑃𝑀 Time 𝑡=2 𝑡=3 Range 709.16−2791.42 𝑃𝑃𝑀 785.29−2861.10 𝑃𝑃𝑀 Time 𝑡=4 𝑡=5 Range 848.15−2939.60 𝑃𝑃𝑀 897.52−3026.91 𝑃𝑃𝑀 [Fig.7: Water Pollution at 3-D Grid Locations by ADM] Fig. 7 demonstrates that, following 120 minutes of testing (𝑡=5), the water pollution level at 3-D grid locations is in the range of 897.52−3026.91 𝑃𝑃𝑀, having started (𝑡=0) in the range of 495.50−2678.52 𝑃𝑃𝑀. Additionally, it has been observed that water contamination in various areas increases monotonically with time. Now, two different approaches, C-N and ADM, have provided numerical and analytical solutions to Eq. (1), respectively. Comparing the C-N results with Experimental Data. Data and ADM results can validate the accuracy of predicting water pollution levels at various 3-D grid locations. The comparison is based on error estimation, calculated as the absolute difference between the water pollution levels at each 3-D grid location over time. The estimated error is displayed graphically in Fig. 8. 𝑻𝒊𝒎𝒆 𝑪−𝑵 𝒗𝒔 𝑬𝒙𝒑. Data 𝑪−𝑵 𝒗𝒔 𝑨𝑫𝑴 𝑡=0 0 10 1 7 1319253137434955616773 Error Location t=0 0 10 1 7 1319253137434955616773 Error Location t=0