Forecasting Pollution Using Numerical Simulation Implementing Artificial Bee Colony Optimization

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Introduction
Optimization is the theory of methods that make a mathematical function or system maximally useful or minimize its disadvantages. Optimization methods are used in various disciplines of research to identify solutions that maximize or decrease some parameters of the subject. Te purpose of optimization is to discover an optimal or near-optimal solution with little computing efort. Tere has been a steady increase since 1960s in the pursuit of developing robust algorithms for challenging optimization problems by modelling them after biological systems [1]. Nowadays, while discussing optimization methods, researchers talk about evolutionary algorithms. Goat's genetic programming [2] and Fogel et al.'s evolutionary algorithm [3] are among the most well-known algorithms in this category; other notable contributors are Holland's evolutionary methods [4], Rechenberg's, and Schwefel's work [5]. Hybrid versions combining several paradigms are also very prevalent. Among the many reports, the work by Back and Schwefel [6], Michalewicz et al. [7], and Fogel [8] has provided a comprehensive overview of the current state of evolutionary algorithms. Perhaps the most well-known evolutionary computing approaches today are genetic algorithms, which are efective stochastic search and optimization strategies with broad applicability. Recently, the genetic algorithm community has focused much of its efort on industrial engineering optimization issues, resulting in a growing amount of research and practical implementations.
Swarm intelligence algorithms [9] have recently caught the attention of numerous research experts in related domains. Swarm intelligence (SI) is the collective behaviour of self-organized systems. Typically, SI systems consist of a population of simple agents interacting locally with one another and their environment. Te source of inspiration is frequently nature, particularly biological systems [10]. Fleeces of birds, schools of fsh, and colonies of social insects like termites, ants, and bees are all well-known instances of such swarms. Despite the fact that honey bee colonies exhibit the self-organizing features necessary for SI, researchers have been interested in the behaviour of these swarm systems to characterize novel intelligent techniques.
Te artifcial bee colony (ABC) [11] is one of the metaheuristic optimization algorithms that has received the greatest attention and has been successfully implemented in various applications such as the forecasting of transportation energy and energy demand [12,13].
Due to the applicability of the ABC algorithm in optimizing the objective function efectively in terms of time and computational complexity, the method is applied in the numerical technique to obtain the unknown parameter. Te present work is intended to optimize (minimize) the maximum numerical errors (L ∞ ) for the solutions of Burgers' equation using the diferential quadrature method involving an exponential B-spline basis function using artifcial bee colony optimization (ABC-EDQ) for the unknown parameter involved in exponential B-spline. Te application of the method reduces the considered partial diferential equation to an ordinary diferential equation (ODE) by approximating the space derivatives through the usage of a diferential quadrature method. Te subsequent step includes computing the solution to the converted equations through the use of MATLAB solvers.
Burgers' equation came into existence in 1915 by Harry Bateman [14] followed by Burgers in 1948 [15] and hence was named "Burgers' equation." Burgers' equation makes one think of the Navier-Stokes equation for one dimension. Burgers' equation is an elementary partial diferential equation which is widely practiced for applications in various felds of applied mathematics such as fuid mechanics, nonlinear acoustics, gas dynamics, and trafc fow.
Recently in 2022, researchers established an energy equation combined with the viscous Burgers' equation as a quantitative model for estimating river water thermal pollution [16]. Termal pollution is a pollution that causes a decrease in the oxygen level of water, making aquatic life harder. It occurs due to various human activities that includes electric generating plants, steel melting facilities, and industrial boilers. Termal pollution in rivers, lakes, and waterways can be studied using the energy equation and the viscous Burgers' equation. Te proposed model is a nonlinear system of partial diferential equations (PDEs), which may be thought of as an initial and boundary value problem (IBVP). Te authors have examined an explicit second-order Lax-Wendrof type technique to obtain the numerical solution of the equation and visually represented the numerical solutions as a temperature profle, which exhibits good qualitative agreement with the real heat transport occurrence.
In the context of pollution, this equation plays an important role. In a work reported by researchers in 2019, Burgers' equation has also been used as a model to forecast pollution, emphasizing hydro and water pollution [17].
In this paper, the authors propose a correction strategy based on Burgers' equations to improve the estimation and prediction of pollution.
Burgers' equation is given by Here, the coefcient of kinematic viscosity is given as ], a parameter which is greater than zero with a small positive value while α represents a positive constant which behaves as a driving force.
Due to various applications of Burgers' equation, there are various well-known numerical methods reported in the literature to solve this equation.
Various well-known numerical methods have been applied to fnd the numerical solution of Burgers' equation in the last few years, such as the diferential quadrature method (DQM) with modifed cubic B-spline [18], cubic Bspline basis functions in standard form [19], modifed trigonometric cubic B-spline [20], and exponential modifed cubic B-spline [21]. Te fnite element method has also been applied to the equation with the Hopf-Cole transformation by transforming the equation to the linear heat equation [22] and with the Galerkin fnite element approach [23]. Te equation has also been experimented with the collocation approach with quadratic B-spline basis functions [24], quartic B-spline [25], cubic B-spline [26], and modifed cubic B-splines [27]. Many researchers have also implemented the fnite diference approach in diferent forms, such as the fnite diference method, which has been used with the parameter-uniform implicit diference scheme [28], the fourth-order fnite diference method [29], and the implicit fourth-order compact fnite diference scheme [30].
Tis is how the research is presented. In Section 1, an introduction to the scheme and its signifcant contributions in several areas are presented. Following this is an explanation of the numerical scheme in Section 2, which includes a discussion of ABC, exponential B-spline, and their implementation in the diferential quadrature method. In Section 3, numerical examples of the equation are provided with various aspects of parameters and for diferent time levels. Te thoughts on the usefulness of the system are included in Section 4.

Scheme Description (ABC-EDQ)
Te proposed method uses artifcial bee colony optimization to establish the parameter for the execution of the exponential B-spline using the diferential quadrature method.

Exponential B-Spline Diferential Quadrature Method (EDQ).
Te diferential quadrature technique is a wellknown method for solving partial diferential equations that have been used with a variety of basic functions [31]. Many problems, including Fisher's equation [32], the Telegraph equation [33], the Korteweg-De Vries equation [34], the nonlinear Schrodinger equation [35], and many more, have been successfully solved numerically using this method. Tis approach has also been used for fractional diferential equations [36], indicating its applicability beyond partial diferential equations.
To make this method work, frst consider the domain xϵ is a smooth function over the solution domain, approximated for its r th derivatives with respect to x as a linear sum of all functional values in terms of weighting coefcients a i,j as follows: (2) To determine the requisite weighting factors, the suggested method employs exponential B-splines, a version of ordinary B-splines. Te purpose is to represent the solution using the features of the B-spline basis function but with the addition of a parameter that depicts the form of the piecewise polynomial. Many attempts have been made to solve diferent types of diferential equations using exponential basis functions, but they have always been limited by the requirement to incorporate a parameter, the value of which has only been approximated intuitively. An exponential B-spline in the piecewise domain is defned as follows: where h is the uniform space partition and other parameters are reported as follows: , , Te numerical values of the function and the derivatives at nodal point can be obtained as follows: Discrete Dynamics in Nature and Society Te basic functions at the border grid points are redefned to meet the criterion of a diagonally dominating matrix before the basis function is implemented, as illustrated in [18].

Artifcial Bee Colony Optimization.
To optimize numerical issues, Karaboga [37] proposed the artifcial bee colony (ABC) algorithm in 2005. Te ABC algorithm is a swarm-based metaheuristic algorithm. It was infuenced by how honey bees used intelligence in their foraging. Various algorithms, tailored to the unique intelligence of bee swarms, have been created throughout the years. A swarm of bees is a large group of bees that have gathered to form a colony [10]. Te primary feature of a swarm is the foraging behaviour of the bees. A bee will collect nectar from a fower and put it in its honeycomb when it discovers a good source of nourishment. Enzymes are produced, and nectar is poured into the hive's vacated cells to begin the honey-making process. Worker bees dance to communicate with each other and tell each other about the location of potential food sources. Tey dance in one of the defned ways, depending on how well the food supply is doing economically. Bees perform a dance called the waggle dance to indicate the direction of the sun and, by extension, the location of food sources. How quickly the dance moves indicate the distance between the hive and the food supply. A trembling bee is a sign that the bee is uncertain about the present food supply's potential for proft [38]. Tere are a number of parameters involved in defning the algorithm based on the performance of bees. Figure 1 explains the types of bees and their roles. Figure 2 depicts the role of unemployed bees in hunting for food.

Procedure.
To fnd optimal solutions, swarm-based algorithms rely on a cooperative process of trial and error. Peerto-peer learning's social colony behaviour is the engine that propels ABC optimization algorithms [39]. ABC generates a set of candidate solutions and then continuously selects the best one. Evolution in an ABC population is driven by variation and selection. Each iteration of the variational process investigates a new region of the search space. Te selection process ensures that prior knowledge will be put to good use.
Te ABC algorithm comprises four stages: the beginning, when no data are available, the working bees, and the observing phase. Each solution in ABC's initial population is a dimension vector, and the population is generated randomly. Te number of dimensions is proportional to the number of variables in the population's food-source optimization issue. Te seeded bees adjust the current solution based on their own experiences and the ftness benefts of the change. Te bee will replace the old food source with the new one if the ftness value of the new food source is greater than the old one. To update the position, the dimension vectors specifed in the introductory phase are being used together with the necessary step sizes to calculate the new coordinates. Te increment might range from −1 to 1.
Te employed bee phase and the position update procedure are shown in Figure 3. Here, X i indicates where the bee is right now, and the highlighted box indicates the randomly chosen direction. Te X k bee was chosen at random. o achieve this, the random bee's direction is subtracted from that of the chosen bees. Te step size is an arbitrary positive integer that is multiplied by this diference. Finally, this number is added to the vector of dimensions X i to determine the size of the new food location "V." Tis vector is created in the neighbourhood of X i and has the same metric values.
Bees that are hard at work in the hive share their position and information about the nectar quality of newly developed solutions (food sources). Uninvolved bees analyse the available information and choose the best options based on the ftness probability they ofer. Te worker bee's observer counterpart, meanwhile, updates the position in its memory, and evaluates the upcoming resource's suitability. If the new area is more suitable than the previous one, the bee will remember it and abandon the old one.
If the food source's location hasn't been updated in a certain amount of time, it has been deemed abandoned. If a food source is abandoned, the bee assigned to it becomes a scout bee and is sent to investigate a new food source elsewhere in the search area. Te ABC exit limit, or the predetermined number of cycles, is a key regulating factor. Figure 4 explains the workings of the ABC algorithm in the form of a fow chart that is also presented below as a given pseudocode (Algorithm 1).
Pseudocode of the algorithm:

Numerical Applications and Discussion
Consider the following boundary values for numerically investigating Burgers' equation using the provided methodology: and initial conditions With considerable certainty, working bees share their knowledge with their counterparts in the jobless bee population. Unemployed bees are responsible for summing together data gathered by employed bees and choosing a food source. They've broken up the jobless bees into two factions.
Some species of bee make advantage of readily accessible food sources. Workers bees are responsible for ensuring that the connected food source remains profitable, taking into account factors such as abundance, proximity, and direction from the hive.

Onlooker Bees
Scout Bees they are the bees that survey the colony's paid labourers for data, and then use that information to hunt for food.
They are in charge of searching for new food sources around the hive. (1) Input parameters � Population, ft, t, lb, ub, limit, Np.
(3) Set trial � 0. (4) for i � 1 to t Evaluate employed bee stage Determined probability Apply Onlooker bee phase for generating food source Memorise the best food source if trial > limit enter into Scout bee stage end end ALGORITHM 1: Pseudocode of the ABC algorithm.
from the exact solution on the domain [a, b]. When the exponential B-spline diferential quadrature technique is used to substitute the space derivatives in Burgers' equation, the equation is changed into a series of nonlinear ordinary diferential equations with time dependence, as shown in the following equation: with i � 1, 2, . . . , n.
In this paper, the MATLAB 2014 programming approach is used to determine the numerical solution of the equation while applying EDQ alongside ABC, and the results are displayed as errors. Te two test problems presented are considered for determining the numerical answer with the defned approach: where c � πx, β � −π 2 ] 2 t. Te numerical solution of the problem is achieved at various time levels for k � 0.01 at various v values. Te value of the parameter for the exponential B-spline is derived using ABC for application in the diferential quadrature technique. Te obtained numerical fndings are compared It can be observed from the results presented in the Tables 1 and 2 that the algorithm has played an important role in minimizing the errors even at the same value of the parameter. (1) to be solved in the domain [0, 1.2] with boundary and initial condition taken from the exact solution [19] for α � 1, given as follows:

Example 2. Consider the equation
Te numerical solution has been obtained with ] � 0.005 at diferent time levels for t � 1.7 and t � 3.1.
In Table 3, the errors are estimated at diferent time levels for time step, k � 0.01 and the number of domain partition as 121 and 151, the fndings are compared with available data in literature. Using ABC, results are calculated for the parameters in Table 3 considering swarm size: 5; maximum iterations: 20; inertia weight is linearly decreased; and social and cognitive coefcients are taken as c 1 � c 2 � 2.05 and k �  1. Te collected fndings show that the numerical results are afected by the parameter as well as the number of domain divisions. Te acquired fndings are equivalent to the exact solutions that are accessible. Figure 7 depicts the solution behaviour along with the physical behaviour of the equation over time.

Concluding Remarks
Te ABC technique is used to acquire the parameter for the exponential B-spline-based diferential quadrature approach, which is used to calculate the numerical solution to Burgers' problem. To demonstrate the accuracy of the procedure the problems are solved with the diferent values of the coefcient of kinematic viscosity ranging from small to high. To illustrate the errors, diferent time steps and the number of domain divisions are employed. Te obtained results are compared to the applied numerical method with particle swarm optimization algorithm in comparison to the ABC algorithm. It can be observed that the errors obtained by ABC algorithm are less as compared to the PSO algorithm when applied for minimizing the error while the value of parameter is same for diferent applied domain partitions. Te proposed strategy presented in the work may be experimented using some other optimization algorithm such as spider monkey and grey wolf optimization algorithm.

Data Availability
All data generated or analysed during this study are included in the article.

Disclosure
I hereby declare that the results provided are true and complete to the best of my knowledge and the paper has not been submitted, in whole or in part in any other journal.

Conflicts of Interest
Te authors declare that there are no conficts of interest.