Solution of Nonlinear Reaction-Diffusion Model in Porous Catalysts Arising in Micro-Vessel and Soft Tissue Using a Metaheuristic

The steady-state reactive transport model (RTM) is a generalization of the nonlinear reaction-diffusion model in porous catalysts. The RTM is expressed as a non-linear ordinary differential equation of second-order with boundary conditions. Artificial neural network (ANN), Particle swarm optimization (PSO), and hybrid of PSO-SQP (Sequential Quadratic Programming) are used to obtain accurate, approximate solutions to the non-linear RTM. The proposed technique is applied to three different cases of non-linear RTM. The properties of the nonlinear reactive transport model in porous catalysts are investigated by considering various cases based on variation in the half-saturation concentration “<inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>” and the characteristic reaction rate “<inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula>.” The stability, reliability, and exactness of the proposed technique are established through comparison with the outcomes of the standard numerical procedure with the RK4 method and along with the different performance indices, which are Root-Mean-Square Error (RMSE), (TIC), Absolute Error (AE), and Mean Absolute Deviation (MAD).


I. INTRODUCTION
Problems arising in microvessels and fluid are multiscale model of nonlinear steady-state one-dimensional reactive transport (RTM) which is also known as reaction-diffusion model in porous catalysts are used to solve those types of problems [1]- [3]. Reactive transport model dynamics are essential for the study of physical and biological processes because they are used to develop behavior in various Earth-related studies [4], [5]. The development of reactive transport models provides a forum for testing and integrating new theoretical information on transport, geochemical, and biological processes. The effect of changes in air transport and temperature, as well as water pollutants, due to diffusion and convection characteristics play an important role in human lives in the reactive transport model phenomenon of heat and mass transfer [6]- [8].
The associate editor coordinating the review of this manuscript and approving it for publication was Vivek Kumar Sehgal . The reactive transport model, which governs in a onedimensional steady-state can be stated as follows [9]: with boundary conditions as follows: the advective velocity is denoted as V, the reaction process is denoted as r(θ) and the diffusivity parameter is denoted asD. Moreover the parameters are discussed in [3] and [1] in detail. By using the non-dimensional quantities θ(x) = θ (X ) θ s , x = X L , and the non-dimensional reaction term r(x) into Eq. (1), we get where P = LV D is so-called Péclet number. Without advection of transport, we have P = 0, and the model was used to study VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ porous catalyst pellets as a diffusion and reaction model in this case. The r(x) is assumed by Michaelis-Menten [3] to be a nondimensional reaction, then Eq. (4) is modified in the following way: the boundary conditions as: where α, is the concentration of half saturation, which cannot be negative and β is characteristic reaction rate, when β < 0, then instead of product reactions we looks at the reactives. The RTM problem in Eq. (3) has been extensively studied without advective transport (P = 0), whereas the RTM problem in Eqs. (4,5) is a fluid and solute transport model arising from soft tissue and microvessel research. The model (3)- (4), which was recently introduced by Ellery and Simpson [3], is a modification of the primer model, referred to as the nonlinear reaction-diffusion model in porous catalysts, which has been used to study porous catalyst pellets and has been analysed using a variety of techniques [10]- [12]. The models (3)-(4) incorporate advective and diffusive transport, as well as the Michaelis-Menten reaction model, which is frequently used to describe biological processes [13]- [15]. This model encapsulates a variety of critical engineering processes, including various applications in chemical [16], [17] and environmental engineering [13], [15]. The boundary value problems (1)-(2) have a nonlinear fractional term, which makes them rather challenging to solve numerically. Ellery and Simpson [3] proposed a Taylor series solution for this model that is actually convergent if the Michaelis-Menten reaction term has finite derivatives.
The RTM problems (4,5) can be solved for different half-saturation concentration β and characteristic reaction rate α. The RTM's significant role motivates researchers to develop solutions for the model. Alves, Van Genuchten [18], and Torideet [19] provide analytical solutions for steadystate non-linear RTMs, such as the Homotopy analysis method [20], the Adomian decomposition method [21]- [24], and the Taylor-Galerkin methods for non-linear dynamical problems [25]. These RTM investigations use deterministic numerical and analytical methodologies, each with its own set of advantages, applicability, reliability, and limitations. On the other hand, artificial intelligence-based techniques have not yet been investigated for solving mathematical relations of nonlinear RTMs.
Additionally, the stochastic algorithms also handled problems arising in electromagnetics [39], astrophysics [40], electrical circuits [41], communication, signal processing, controls [42], plasma physics [43], bio-informatics [44], atomic physics [45], and nanotechnology [42]. These are inspirations for the author to investigate, analyze, and exploit research stochastic mathematical methodologies to establish a new, precise, robust, and dependable computing approach to investigate nonlinear reactive transport models that arise in soft tissue and microvessel studies.

A. SUMMARY OF THE STUDIES
The following is a summary of the study's findings in terms of critical features: 1) For solving nonlinear second-order reactive transport model systems, governing Earth and heat systems, the accurate modeling of ANNs that have been optimized with particle swarm optimization PSO) and PSO-SQP is efficiently exploited. 2) To determine reactive transport model dynamics for various cases based on of variations of the characteristic half-dynamic concentration and reaction rates, the proposed stochastic technique is applied with reasonable accuracy when compared to the RK4 solution. The proposed approach outperforms the other methods in terms of performance.
3) The techniques comparison indicates that the performance of PSO and PSO-SQP is better than the rest of the other numerical techniques for such a model. 4) PSO's high performance in investigating the governing mathematical equations of reactive transport models (RTMs) was further supported by a detailed evaluation of the findings using statistical performance from MAD, TIC, and RMSE.

II. ORGANIZATION OF THE PAPER
The rest of this paper is structured as follows: Sect. 3: Design is provided with computational intelligence paradigms for the non-linear transport models results.
Sect. 4: Presented the designed scheme of the proposed technique.
Sect. 5: Statistics results are presented on the basis of various tests.
Sect. 6: For the reactive transport model, numerical experiments are illustrated with graphical and numerical diagrams. Sect. 7: Conclusions and future research directions are discussed.

III. DESIGN METHODOLOGY
The nonlinear reactive transport model (RTMs) design scheme is divided into two parts. The first section describes the system's ANN-based modeling. The second section of the study provides an overview of the optimization strategies that are employed as dynamic methods for training ANN model weights. The proposed scheme's workflow diagram is visually shown in Fig. 2.

A. MATHEMATICAL MODELING
The non-linear reactive transport model (RTM) mathematical modeling is described in two sections. The first section constructs ANN simulations for the system's solution and derivative terms. The second section discusses the use of ANN simulations to formulate fitness functions.
We create an artificial neural network approximate solution as follows by continuously mapping the solutionθ(x) and its derivative up-to n th order as: Here adaptive parameters of networks, areγ i ,Ã i andω i are in i th form, f is an activation function and k represent total number of neurons. In artificial neural network, log-sigmoid f (τ ) = 1/ 1 + e −τ , is usually used for activation function, where the τ =ω i x +γ i . Using activation function the log-sigmoid in the set of equations from Eq.(6) to Eq.(8) and its derivatives can be represents as follow.
The ANN architecture of RTM is built using the appropriate network combinations described in Eq (4) in the form of single inputs, outputs and hidden layers as indicated in Fig 2. The RTM fitness function of Eqs (4, 5) is developed using the approximation principle in the mean square sense as: Although ε 1 represents the mean square error of the differential equation representing RTM.
and ε 2 is the boundary condition the mean square error as: forθ k =θ(x k ), x k = kh, h = 1/k. The network forθ k and its first and second-order networks are described in a set of equations from Eqs. (10) to (12). Now, the approximate solutionθ(x) will be overlapping with the exact solution θ(x) of the reactive transport model as defined in Eqs. (4,5), if the tuned weights of the network with ε RTM are appropriately close to zero.

IV. LEARNING METHODOLOGY
Once the fitness function for the non-linear reaction-diffusion model has been constructed using an artificial neural network, we applied PSO and PSO-SQP to obtain optimal weights for the model.

A. OPTIMIZATION PROCEDURE 1) PARTICLE SWARM OPTIMIZATION
At the end of the nineteenth century, Eberhart and Kennedy developed PSO, a global heuristic search optimization tool. Particle swarm optimization utilizes genetic algorithms [46] and has become one of the most popular optimization techniques due to its ease of implementation and lower memory requirements [47]. PSO provides improved performance on various standards and the spectrum of engineering issues to provide more optimal results. PSO is associated with the combined swarm success of bird flocking and fish schooling [48]. Multicast communication network routing problems [49], solar photovoltaic systems [50], vehicle-to-grid energy resource scheduling [51], high-dimensional data clustering [52], pathway optimization for humanoid robots [53], multilevel thresholding [54], collective robotic selection, and cancer classification gene selection [55], are some of the most recent PSO applications. Each candidate outcome is a particle that represents an optimization model in the search space. To form a swarm, randomly generated particles explore the problem in the PSO algorithm. Initial swarms are distributed to determine the technique's optimum efficiency on a larger scale. Each particle in the swarm has fitness values that define the problem's parameters, known as the objective function. An iteratively optimal solution provides parameter initialization in the particle swarm optimization algorithm. The velocity and position  of the swarm are restructured by using the global best and local positions of its previous point, P r−1 GB and P r−1 LB . The standard continuous velocity and position particle swarm optimization update form is provided as: the velocity vector represented by V i and i ih swarm particle denoted by vector X i . The random vectors is denoted by r l and r 2 , and the acceleration constant denoted by a l and a 2 , where as inertia weight is ω ∈ [0, 1]. The elements of the velocity vector is between [−v max , v max ], and maximum velocity is indicated by v max . Based on a predefined number of flights, the output of the algorithm is stopped. The global search performance of PSO is increased further with the use of the Sequential Quadratic Programming method (SQP), which is an efficient, speedy, and fast local search optimization technique.

2) SEQUENTIAL QUADRATIC PROGRAMMING
Sequential quadratic programming (SQP) has become a very reliable, effective, efficient and accurate method for optimizing the linear and nonlinear constrained optimization problems since early 1970s. SQP is the class of optimization algorithms with conceptual procedure of various specific methods that have evolved. The dominance of SQP algorithm is established through computational and theoretical experimentations. SQP algorithm is considered to be one of the fundamental techniques to be exploited in the domain of both public and commercial sector problems of practical significance. Few recent applications addressed effectively by SQP procedures are bilinear optimal control problem [7], finding the worst resonance response [56]. Furthermore, Fletcher [57] and Schittkowski [58] are good sources of reference material on SQP methods. Montoya [59], Kim et al. [60], Witkowska and Smierzchalski [61], Kouzoupis [62], and Welhazi [63] are just a few recent examples of how the technique has been applied to engineering or applied science problems. Using the MATLAB optimization toolbox, the SQP algorithm is used to optimize the artificial neural network model using parameters and initial starting point. Fig. (1) depicts the PSO and PSO-SQP algorithms' workflow, while the pseudocodes for reproducing the results are as follows:

V. PERFORMANCE INDICATORS
The performance of the designed model for solving nonlinear RTM models is examined in this research study by integrating various performance indices, with a focus on mean absolute deviation (MAD), the inequality coefficient of Theil (TIC) and Root Mean Squared Error (RMSE). The benefits of using these three metrics provide an in-depth analysis of precision, stability, and convergence for perfect modeling of different optimal values.
The mathematical output operators for the numerical solution θ m and the approximate solutionθ m is displayed as:  where n is a grid point for input. TIC, and RMSE error functions are defined mathematically as:   6) and (9).

VI. NUMERICAL RESULTS
The findings of the simulation results are shown here for three scenarios of reactive transport model (RTM) dynamics with variance in the half-saturation concentration denoted by α and the characteristic reaction rate represented by β using intelligent ANN-based computing methods optimized with PSO and PSO-SQP. The graphical illustration of weights is shown in Fig. (8). The approximate solution for all cases by using the optimal wights of the proposed techniques can be written in the form of: The reactive transport model (RTM) Eqs. (4,5) may be stated as follows in this case [9]: subject to the boundary conditions dθ dx = 0, θ(1) = 1.
The fitness function ε, for case 1 can be expressed as: Particle Swarm Optimization and Sequential Quadratic Programming is applied for optimizing the fitness function (25). In Eq. (22), the optimal weights of PSO and PSO-SQP are used to find the approximate solutions for case 2 of the reactive transport model as shown in Eq. (32) and Eq. (33) respectively. The approximated solutions for fitness function (25) are calculated, and results for inputs between 0 and 1  with h = 0.1 are shown graphically in Fig. (3) and numerically in Tab. (1). Furthermore, AEs are determined for PSO and PSO-SQP results, which are shown graphically in Fig. (3) and numerically in Tab. (2). The data shown in Tab. (2) and Fig. (3) indicate that the PSO and PSO-SQP absolute errors are about 10 −6 to 10 −7 and 10 −7 to 10 −8 respectively, which means that PSO achieved (6 − 7) and PSO-SQP achieved (7 − 8) decimal places of precision from the reference results. The remaining techniques PS-AST, and GA-AST [9] absolute errors lie at around 10 −5 and 10 −6 respectively, which means that the PSO and PSO-SQP is relatively better and more effective than the rest of the techniques for this case of the reactive transport model.   The reactive transport model given in Eqs. (4,5) in this case can be written as [9]: subject to the boundary condition dθ dx = 0, θ(1) = 1.
The fitness function ε, for case 2 can be expressed as: PSO and PSO-SQP is applied for optimizing the fitness function (28). In Eq. (22), the optimal weights of PSO are     The data are shown in Tab. (4) and Fig. (5), which indicate that the PSO and PSO-SQP absolute errors are about 10 −6 to 10 −7 , and 10 −9 to 10 −11 respectively, which means that the PSO and PSO-SQP results are overlapping with the RK4 results. From Tab. (4) and Fig. (5) it is evident that the PSO-SQP model is better and more effective than PSO in this case of the reactive transport model.  In this case the reactive transport model given in Eqs. (4,5) is written as [9]: subject to the boundary condition The fitness function ε for case 3 can be expressed as: PSO and PSO-SQP are used to optimize the fitness function (31). The optimal weights are used in Eq. (22), to find VOLUME 10, 2022    Fig. (7) and numerically in Tab. (4). In addition, AEs are determined for PSO, and PSO-SQP results, which are graphically illustrated in Fig. (7) and numerically in Tab. (5). The absolute errors of PSO and PSO-SQP are about 10 −6 to 10 −7 and 10 −7 to 10 −8 respectively, as shown in Tab. (6), implying that PSO and PSO-SQP achieve (6-7) and (7)(8) decimal places of precision from the reference results. The absolute errors of the remaining techniques, GA, PS-AST, and GA-AST [9] are around 10 −4 to 10 −5 , indicating that the PSO and PSO-SQP are better and more effective than the other techniques for this case of the reactive transport model.

VII. CONCLUSION
The following conclusions are summarized: The steady-state reaction-diffusion model that occurs in the studies of soft-tissues and microvessels in the fluid and solvent transport models is discussed using the approximation ability of ANN modeling via the heuristic technique. The PSO and PSO-SQP computational methodologies can investigate various cases of reaction-diffusion models by varying the half-saturation concentration rate and the characteristic reaction rate with reasonable precision. In Case I, the proposed technique show good agreement with RK4 by achieving an accuracy of 10 −8 . In Case II, we compared the results to an RK4 solution, demonstrating that the proposed technique achieves an accuracy of 10 −11 . In Case III, the accuracy for the proposed approach is also up to 10 −15 , which is a good approximation for the RTM model. Comparing the proposed method to RK4 and other numerical methods shows that the PSO and PSO-SQP performance for all cases of the reaction-diffusion model is better than the other methods.
For all three cases, histograms for the performance indicator and fitness function are shown in Figs. (4), (6), and (9) respectively. Similarly number of independent runs of MAD, TIC, FIT and RMSE are also done for reactive transport model as shown in Fig.(10). MAD values show that the majority are close to zero in Fig. (10a), TIC values show that more than 90% of the values are less than or equal to 10 −03 in Fig. (10b), and RMSE values show that more than 90% of the values are less than or equal to 10 −03 in Fig. (10d) in all three cases. Similarly, for all three cases the fitness (FIT) values in Fig. (10c) show that more than 90% of the values are less than or equal to 10 −03 . The preceding discussion validates the PSO-SQP algorithm's effectiveness in investigating the proposed model.  IRFAN UR RAHMAN received the M.Phil. degree in applied mathematics from Abdul Wali Khan University Mardan, Pakistan, in 2020, where he is currently pursuing the Ph.D. degree. His research interests include mathematical optimization techniques, real world problems, heuristics, metaheuristics, multi-objective optimization, and design engineering optimization problems. He has the potential to accept any challenging environment in research productivity. Mardan, where he has been working as an Associate Professor of mathematics, since August 2021. He is the author of two book chapters and more than 59 research articles. His research interests include mathematical optimization techniques, global optimization, and evolutionary algorithms, heuristics, metaheuristics, multi-objective optimization, design engineering optimization problems, structural engineering optimization problems, linear programming, linear and non-linear least squares optimization problems, evolutionary algorithms, nature-inspired metaheuristics, artificial neural networks, and differential equations. He is an Associate Editor of the journal COJ Reviews and Research and SCIREA Journal of Mathematics.

MUHAMMAD SULAIMAN
KAMSING NONLAOPON received the B.S. degree in mathematics from Khon Kaen University, Khon Kaen, Thailand, in 2000, and the Ph.D. degree in mathematics from Chiang Mai University, Chiang Mai, Thailand, in 2005. He is currently an Associate Professor with the Department of Mathematics, Faculty of Science, Khon Kaen University. He has authored or coauthored more than 100 international peer-reviewed journals. His main research interests include differential equations, fluid dynamics, fractional calculus, generalized functions, inequalities and applications, mathematical model, quantum calculus, and special functions. VOLUME 10, 2022