https://mathco.journals.pnu.ac.ir/ Control and Optimization in Applied Mathematics en daily 1 Thu, 01 Jan 2026 00:00:00 +0330 Thu, 01 Jan 2026 00:00:00 +0330 https://mathco.journals.pnu.ac.ir/article_12370.html In this paper, we introduce a new continuous quantum evolutionary optimization algorithm designed for optimizing nonlinear convex functions, non-convex functions, and efficiency evaluation problems using quantum computing principles. ‎ Traditional quantum evolutionary algorithms have primarily been implemented for discrete and binary decision variables‎. ‎The proposed method has been designed as a novel continuous quantum evolutionary optimization algorithm tailored to problems with continuous decision variables‎. ‎ To assess the algorithm’s performance, several numerical experiments are conducted‎, ‎and the simulated results are compared with the Grey Wolf Optimizer and Magnet Fish Optimization search algorithm‎. ‎The simulation results indicate that the proposed algorithm can approximate the optimal solution more accurately than the two compared algorithms. https://mathco.journals.pnu.ac.ir/article_12381.html This paper addresses multi-attribute group decision-making (MAGDM) where linguistic assessments are represented by both positive and negative interval type-2 fuzzy numbers (IT2FNs)‎, ‎capturing the intrinsic uncertainty of group evaluations more accurately. ‎We introduce a novel ranking method for IT2FNs that simultaneously utilizes the mean and standard deviation of the upper and lower membership functions‎, ‎ as well as the IT2FN's height‎. ‎This enhances its discriminatory capability‎. ‎The theoretical foundations of this ranking— encompassing zero‎, ‎unity‎, ‎and symmetry properties— are rigorously established‎, ‎and its superiority over existing techniques is demonstrated through comparative analyses on seven benchmark datasets‎. ‎Building on this ranking‎, ‎we develop an integrated fuzzy MAGDM framework that can handle both positive and negative IT2FN assessments for criteria and weights‎. ‎The framework’s practicality and effectiveness are validated through two case studies‎: ‎one with exclusively positive linguistic terms and another with mixed positive and negative scales‎. ‎Results indicate that the proposed ranking and decision framework yield more rational and robust group decisions under substantial uncertainty‎. ‎They outperform conventional fuzzy methods and offer a nuanced solution for real-world MAGDM scenarios‎. https://mathco.journals.pnu.ac.ir/article_12384.html The conjugate gradient ({CG}) method is one of the simplest and most widely used approaches for unconstrained optimization, and our focus is on two-dimensional problems with numerous practical applications. We devise three hybrid {CG} methods in which the hybrid parameter is constructed from the Barzilai–Borwein process, and in these hybrids, the weaknesses of each constituent method are mitigated by the strengths of the others. The conjugate gradient parameter is formed as a linear combination of two well-known CG parameters, blended by a scalar, enabling our new methods to solve the targeted problems efficiently. Under mild assumptions, we establish the descent property of the generated directions and prove the global convergence of the hybrid schemes. Numerical experiments on ten practical examples indicate that the proposed hybrid {CG} methods outperform standard {CG} methods for two-dimensional unconstrained optimization. https://mathco.journals.pnu.ac.ir/article_12385.html In this study‎, ‎we fabricate and investigate a three-species intraguild predation model with a ratio-dependent functional response‎. ‎We also incorporate harvesting efforts into both intraguild prey and intraguild predators‎. ‎Then‎, ‎we analyze the dynamical behavior of the proposed model by taking the harvesting rate as the bifurcation parameter‎. ‎We precisely outline the prerequisites for the proposed model's existence‎, ‎stability‎, ‎and bifurcation near the equilibrium points‎. ‎It contributes to a better understanding of the impacts of harvesting on the survival or extinction of one or more species in the proposed model‎. ‎Furthermore‎, ‎we derive the suggested model's bionomic equilibrium and optimum harvesting policy by using the \textit{Pontryagin's maximum principle}‎. ‎Finally‎, ‎we provide some numerical simulations to validate the analytical results‎. ‎In addition‎, ‎we give some graphical representations to validate our results. https://mathco.journals.pnu.ac.ir/article_12373.html The advection-dispersion, variable-order differential equations have a vast application in fluid physics and energy systems. ‎In this study, ‎we propose a Ritz-approximation method using shifted Legendre polynomials to construct approximate numerical solutions for these equations‎. ‎The proposed method discretizes the original problem, converting it into a system of nonlinear algebraic equations that can be solved numerically at selected points‎. We discuss ‎the error characteristics of the proposed method‎. ‎For validation‎, ‎the presented examples are compared with exact solutions and with prior results. ‎The results indicate that the proposed method is highly effective‎.‎‎ https://mathco.journals.pnu.ac.ir/article_12441.html This paper introduces a robust hybrid adaptive control framework for stabilizing chaotic systems under persistent, potentially large time delays. The controller is based on an enhanced Lyapunov–Krasovskii functional that integrates an energy-capturing integral term with a bounded trigonometric term. The integral term accounts for historical effects by quantifying cumulative energy over the delay period, while the trigonometric term attenuates nonlinear oscillations. Embedding these components in a single control law yields stabilization of all state variables to the equilibrium despite substantial delays. We establish Uniform Ultimate Boundedness, showing that trajectories enter a compact neighborhood of the equilibrium after a finite transient and subsequently converge. Adjustable gains enable practitioners to determine the convergence radius and the size of the attraction region according to practical requirements. The method is validated on the delayed Lorenz system; simulations with a 20-second delay demonstrate rapid convergence to a small neighborhood of the equilibrium, with the Lyapunov functional derivative remaining non-positive. A comparative study with established controllers underscores the proposed approach’s favorable trade-offs among computational cost, oscillation suppression, and explicit stability guarantees. Overall, the proposed framework delivers a practical, robust, and high-performance solution for controlling chaotic systems in the presence of large time delays. https://mathco.journals.pnu.ac.ir/article_12477.html In irregular coloring, each vertex is labeled with a unique color code, a tuple consisting of its assigned color and the number of neighbors in each color class‎. ‎This work proposes a local search algorithm as a metaheuristic approach to the irregular face coloring problem in planar graphs‎, ‎with a particular focus on fullerene molecular structures‎. ‎Additionally‎, ‎a linear programming model is utilized to validate the performance of the proposed algorithm‎. ‎The methodology demonstrates efficient solutions for irregular coloring in fullerene graphs‎, bridging combinatorial optimization with practical applications in chemistry and materials science‎.‎ https://mathco.journals.pnu.ac.ir/article_12409.html ‎This study employs a two-stage analytical framework to assess efficiency, comprising a standard SBM evaluation and a novel weighted SBM model. Unlike conventional SBM-DEA applications, the proposed weighted model uses an enhanced slack-based mechanism that prioritizes strategic inputs (R&D investment, number of employees, and funding) and clearly distinguishes input redundancies (e.g., excessive R&D expenditure or staffing) from output deficiencies (e.g., weak revenue performance). This separation yields more precise and targeted diagnostic insights. Additionally, the model incorporates sector-specific efficiency differentiation, supported by ANOVA, enabling assessment of cross-firm inefficiencies and their statistical significance in terms of systemic versus sector-specific phenomena. The methodology is applied to a distinctive panel of 146 technology-based firms (TBFs) in Iranian science and technology parks from 2021–2023, a context rarely explored with DEA in emerging markets. The study combines quantitative DEA results from both models with qualitative follow-up analyses of factors such as marketing strategies, private investment initiatives, and certification achievements, producing a robust mixed-methods approach and actionable policy recommendations. A comparative analysis reveals that fully efficient firms comprise 2.7\% under the unweighted model and 3.4\% under the weighted model, indicating that weighting yields a small, non-significant change in overall efficiency. About 97.3\% of firms display efficiency gaps due to input redundancies or output shortfalls. Sectoral tests show no statistically significant inter-sector differences, pointing to systemic inefficiencies across industries. Qualitative insights identify firm-level success factors—effective marketing, certification, and investment strategies—that align with the detected inefficiency patterns. Collectively, these findings offer measurable strategies for improvement, such as reducing redundant investment and enhancing revenue-generation mechanisms, to inform evidence-based policy aimed at the commercialization and growth of TBFs in emerging markets. https://mathco.journals.pnu.ac.ir/article_12371.html ‎The prediction of chaotic time series is essential for understanding highly nonlinear and sensitive systems, with the Lorenz system serving as a standard benchmark due to its intricate and non-periodic dynamics‎. ‎Classical forecasting approaches often struggle to capture such irregularities‎, ‎ motivating a shift toward deep learning–based strategies‎. ‎In this study‎, ‎we develop two hybrid models—Feedback Long Short-Term Memory (FB-LSTM) and Feedback Variational Stacked LSTM (FBVS-LSTM)‎, ‎specifically designed for multivariate prediction of the Lorenz system‎. ‎‎‎‎‎‎By embedding feedback structures into LSTM networks‎, ‎the proposed methods deliver enhanced short-term prediction performance without substantial computational costs. ‎Comparative simulations indicate that our frameworks surpass traditional RNNs and baseline LSTM models‎, ‎ achieving prediction accuracies up to 94%‎. ‎These findings indicate that feedback-enhanced architectures offer effective and practical tools for forecasting chaotic systems‎, ‎with potential applications in both scientific research and engineering practice‎. https://mathco.journals.pnu.ac.ir/article_12527.html This study proposes the New Two-Parameter Weibull–Lindley Distribution (NTPWLD), a flexible lifetime model generated through a transformation of a one-parameter baseline survival function. Owing to its general structure, the NTPWLD accommodates diverse hazard rate shapes, including increasing, decreasing, and bathtub forms, and captures both light- and heavy-tailed behaviors relevant to survival analysis, engineering reliability, and biomedical applications. The work provides a full mathematical treatment of the distribution, deriving closed-form expressions for its density, distribution, survival, hazard, and quantile functions, along with ordinary and incomplete moments, the moment generating function, mean deviations, and Rényi entropy. Several reliability measures, such as mean residual life and stress–strength reliability, are also obtained. Parameter estimation is examined under various inferential approaches, with particular focus on maximum likelihood estimation. A Monte Carlo simulation study shows that the maximum likelihood estimator performs well across settings, displaying low bias, stability, and consistency. To incorporate uncertainty in lifetime data, fuzzy reliability measures are constructed using Zadeh’s extension principle and α-cut techniques. Applications to two real datasets demonstrate that the NTPWLD provides superior goodness-of-fit compared with several competing models based on AIC, BIC, AICC, and −2 log L, highlighting its practical value in both precise and fuzzy reliability environments.‎ https://mathco.journals.pnu.ac.ir/article_12492.html This study develops a nonlinear dynamic modeling framework to analyze and predict performance behavior in industrial environments using competitive-intelligence-related variables. Four organizational resource components are formulated as elements of a discrete-time state vector, and their influence on system output is modeled through a nonlinear state-transition function. Empirical observations collected from a steel manufacturing company were used to identify the unknown dynamics through a feed-forward artificial neural network trained via a gradient-based optimization procedure. Reliability of the measurement instrument was verified using Cronbach’s alpha coefficients of 0.92 and 0.86 for the independent and dependent constructs, respectively. The identified model demonstrates stable convergence, with the minimum prediction error achieved near iteration 1500, and outperforms a linear baseline in mean-squared error and correlation accuracy. The proposed formulation provides a mathematically oriented approach for reconstructing performance-driven system behavior and establishes a foundation for future extensions involving adaptive estimation, robust analysis, and optimal control strategies in industrial systems. https://mathco.journals.pnu.ac.ir/article_12530.html Enterprise architecture (EA) offers an integrated framework for strategic planning and organizational governance. Implementing EA effectively requires prioritizing a concise set of criteria within a complex system, leveraging mathematical modeling and optimization to inform decisions under uncertainty. This study introduces a hierarchical decision-making approach using Analytic Hierarchy Process (AHP) to extract and weight the most impactful criteria from an extensive literature base and expert opinions, with a focus on control-theoretic and optimization perspectives. Using insights from 18 experts from various fields and the proposed approach, key criteria of successful enterprise architecture deployment were identified and quantified: commitment (0.1143), governance (0.1082), infrastructure (0.0751), organizational management (0.0589), and senior management support (0.0484). The methodology integrates weights with objective-function considerations, sensitivity analyses, and optimization-oriented interpretations to ensure robust prioritization under uncertainty. The resulting framework supports decision-makers in (i) controlling and steering EA initiatives, (ii) optimizing resource allocation and process efficiencies, and (iii) designing data-driven, scenario-based decision models for dynamic organizational environments. These findings offer actionable guidance for managers aiming to enhance performance, reduce costs, and secure competitive advantage through disciplined governance, rigorous modeling, and evidence-based decision support. https://mathco.journals.pnu.ac.ir/article_12374.html This study analyzes the growth dynamics of melanoma tumor cells and develops a model predictive controller (MPC) using four well-known optimizers to suppress tumor growth, proposing an MPC framework that integrates multiple metaheuristic algorithms for regulating tumor size. All modelling, control design, and simulations are performed in MATLAB, and results indicate that a PSO-based MPC offers satisfactory response and rapid convergence, achieving effective tracking and disturbance rejection. The study assumes precise drug dosing is feasible and demonstrates substantial tumor-size reduction through the integration of MPC with metaheuristic optimization. Simulation findings reveal that the PSO-based MPC achieves notable improvement in tumor reduction and overall control performance, outperforming other metaheuristic approaches, as evidenced by comparative error metrics: ITAE ≈ 1.9377 × 10^3, IAE ≈ 244.45, MSE ≈ 4.6863 × 10^3. https://mathco.journals.pnu.ac.ir/article_12478.html This paper offers the idea of (anti) (m‎, ‎n)-fuzzy BL-subalgebras as a novel extension of classical BL-algebras within the fuzzy mathematical framework. ‎The proposed structures generalize various types of fuzzy subalgebras, including (anti) intuitionistic, (anti) Pythagorean, (anti) Fermatean, and (anti) q-rung orthopair fuzzy BL-subalgebras for q >= 1. Fundamental algebraic properties and equivalent characterizations of (m,n)-fuzzy BL-subalgebras are established through the notion of value-cuts. Furthermore, the concept of power-implication preserving (PIP) BL-algebras is introduced, and it is shown that a PIP BL-algebra exists for every prime number. Several closure properties of (m,n)-fuzzy BL-subalgebras under combination operations are also derived within this framework. From an applied perspective, the developed theoretical results can serve as a mathematical foundation for modeling and reasoning in fuzzy control systems and optimization processes, particularly in decision-making environments characterized by uncertainty and graded information. https://mathco.journals.pnu.ac.ir/article_12574.html In this study, we examine solutions to Optimal Tracking Control (OTC) problems for both Linear Quadratic (LQ) and nonlinear systems. Classical approaches to OTC rely on formulating and solving the Hamilton-Jacobi-Bellman (HJB) equation, which typically requires numerical solutions of the state, co-state, and stationary equations using the forward-backward method. Such methods often involve intricate mathematical analysis and substantial computational effort. To address these challenges, we explored the use of Physics Informed Neural Networks (PINN) as an alternative framework for solving OTC problems. The PINN approach is implemented by constructing a problem-specific loss function that directly incorporates the governing dynamics and control objectives. This method is comparatively simpler and more flexible to implement. The performance of PINNs is evaluated through quantitative error analysis and benchmarked against the classical Runge-Kutta (RK) method. A detailed comparison is presented using tabulated error metrics and time-domain plots of absolute errors. Numerical results demonstrate that PINNs achieve lower approximation errors than Runge-Kutta method for both LQ and nonlinear tracking problems, indicating their effectiveness as a viable alternative solution strategy for OTC problems.  https://mathco.journals.pnu.ac.ir/article_12620.html This study develops a mathematically informed optimization framework for decision-making in reverse supply chain management, with an application to Apple’s MacBook product line. The proposed framework integrates Failure Mode and Effects Analysis (FMEA) with deep learning, based sentiment analysis in a multi-stage structure designed to quantify risk factors and predict consumer-driven outcomes. The dataset consists of 91 days of Twitter user feedback on Apple notebooks, processed using supervised learning algorithms to extract sentiment scores and thematic indicators of product performance. The analysis identifies “power and battery” and “storage” as the most critical components contributing to user dissatisfaction and elevated risk severity. These data-driven insights are incorporated into an optimization model that supports decisions on product recycling, refurbishment, and reuse. The hybrid framework enhances decision stability and accuracy compared with conventional reverse logistics models, while improving operational efficiency and environmental performance. The results demonstrate the model’s suitability as a scalable, machine-learning-supported optimization tool for reverse supply chain systems. https://mathco.journals.pnu.ac.ir/article_12621.html  This paper introduces and analyzes, for the first time, the \emph{fractional Pauli operator}, a non-local generalization of the fundamental quantum mechanical operator describing spin-1/2 particles in magnetic fields. The operator is defined through the spectral theory of the magnetic fractional Laplacian $(H_{\vecA})^s$, with s ∈ (0,1), and acts on spinor-valued wavefunctions. We formulate the associated eigenvalue problem on a bounded domain Ω ⊂ ℝ^2 subject to exterior Dirichlet conditions. The intrinsic non-locality of the model is addressed via a variational formulation in suitable magnetic fractional Sobolev spaces. Under appropriate assumptions on the vector potential $\vecA$ and the magnetic field B, we establish the existence of a discrete spectrum. For a constant magnetic field on \R^2, we derive explicit eigenvalues exhibiting a nonlinear B_0^s scaling of the Landau levels. In addition, a finite element–based numerical scheme is developed to compute the spectrum on a disk, illustrating the combined effects of spatial confinement and non-locality. The physical implications of fractional kinetic effects on Landau quantization and spin-dependent phenomena are discussed, highlighting the relevance of the fractional Pauli operator for modeling anomalous transport in bounded quantum systems. https://mathco.journals.pnu.ac.ir/article_12650.html In this paper, we address the problem of covering a given bounded domain in the plane using simple geometric figures. The proposed approach is based on a discretization of the domain, which leads to a corresponding discrete optimization problem. To solve this problem, we introduce a novel iterative algorithm that minimizes a given objective function by generating successive neighboring nodal points. As the covering elements, circular sectors with centers located outside the domain are considered. The objective is to determine the locations of the sector centers and their radii in such a way that the entire domain is completely covered, while the ratio of the total area of the covering sectors to the area of the domain is minimized. Finally, the algorithm is demonstrated on a representative example, and the resulting coverings are illustrated. https://mathco.journals.pnu.ac.ir/article_12665.html This paper presents a novel hybrid orthogonal polynomial method for solving optimal control problems governed by fractional parabolic PDEs. By strategically weighting and combining these polynomial bases, the method adaptively leverages their respective strengths to achieve superior approximation properties. The proposed approach combines the spectral accuracy of Legendre polynomials, the minimax properties of Chebyshev polynomials, and the flexibility of Jacobi polynomials to create a robust numerical framework. The hybrid orthogonal polynomial method is applied to discretize the fractional parabolic PDEs, and an efficient numerical scheme is developed to solve the resulting optimal control problem. Numerical experiments demonstrate the accuracy, efficiency, and applicability of the proposed approach, showing significant improvements over traditional radial basis function methods. The results highlight the potential of the hybrid orthogonal polynomial method for solving complex optimal control problems in science and engineering. https://mathco.journals.pnu.ac.ir/article_12689.html Efficient layout design in healthcare facilities is critical for operational effectiveness and patient care. This study addresses the healthcare facility layout problem using a multi-objective optimization approach. We propose a novel methodology based on graph theory, specifically planar adjacency graphs, to generate and evaluate department layouts. Nodes in the graph represent departments, while weighted edges represent the desired closeness based on patient flow and functional relationships. We introduce five strategies based on different weightings of these objectives and evaluate them using a real-world hospital case study. Our results show that a hybrid strategy, prioritizing patient flow while incorporating departmental relationships, yields the optimal layout. This approach provides a systematic and data-driven framework for healthcare planners to create efficient layouts that enhance workflow, reduce travel distances, and improve overall service quality. https://mathco.journals.pnu.ac.ir/article_12693.html  Multi-criteria decision-making (MCDM) often involves situations characterized by uncertainty, ambiguity, and vagueness. To address such complexities, MCDM techniques play a crucial role. This paper presents a comparative analysis of two widely used methods—Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) and VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR)—within a hesitant fuzzy environment. Hesitant fuzzy sets allow decision-makers to express hesitation by assigning multiple possible membership values to an element rather than a single value. In this framework, the TOPSIS ranks alternatives based on their closeness to the positive and negative ideal solutions, while the VIKOR identifies a compromise solution by balancing individual and collective regret measures. The effectiveness of the comparison is demonstrated through illustrative numerical examples. Moreover, some real life applications of these methods are discussed. https://mathco.journals.pnu.ac.ir/article_12778.html Homogeneous second-order Aw-Rascle-type models have demonstrated greater effectiveness than their non-homogeneous counterparts in traffic flow modeling. This study addresses the numerical solution of hyperbolic conservation laws governing these models by coupling the second-order HLLE Riemann solver, a Godunov-type finite volume approach, with the wave propagation algorithm. A novel wave-speed selection strategy is proposed by comparing characteristic velocities with Roe speeds, yielding solutions with guaranteed positive density and speed. The proposed IWP-HLLE method is applied to simulate shock, rarefaction, and contact discontinuity waves under homogeneous long-road conditions, eliminating the influence of external source terms and ensuring the homogeneity of the governing hyperbolic equations. Its performance is benchmarked against the MacCormack scheme supplemented by two standard stabilization techniques, namely artificial viscosity (AV) and central differencing (CD). Spatiotemporal distributions and density profiles are examined across four representative traffic scenarios: free flow, congested traffic flow, queue dissolution, and congested flow with non-equilibrium velocity and uniform density. The results demonstrate that the IWP-HLLE approach substantially suppresses numerical oscillations compared to both AV and CD methods while maintaining stability across all test cases. https://mathco.journals.pnu.ac.ir/article_12779.html Classical inverse location models aim to modify problem parameters such that pre-specified facility locations become optimal with respect to a given objective. This paper addresses a fundamentally different variant: the inverse balanced    facility location problem in the Euclidean plane, in which parameters are adjusted so as to achieve an equitable distribution of client demand between two given facilities. Specifically, given a set of n weighted points in the plane and two predetermined facility locations, the objective is to minimally modify either the weights or the coordinates of the client points such that the absolute difference in total demand assigned to each  facility-referred to as the unbalancing number-is minimized. For the weight-modification case, we establish that the planar problem is structurally equivalent to its network counterpart and is therefore solvable in O(n Log n) time under any Lp norm, via an existing linear programming formulation. For the coordinate-modification case under the Euclidean norm, we exploit the isometric property of orthogonal rotations to prove that thetwo-dimensional problem reduces, without loss of generality, to a one-dimensional problem along the perpendicular bisector of the segment joining the two facilities. Leveraging this reduction, we design three novel greedy algorithms-IFLP1, IFLP2, and IFLP3-that prioritize minimization of the unbalancing number, minimization of the total transfer cost, and a hybrid criterion balancing both objectives, respectively. Under uniform weights and identical modification costs, all three algorithms are proven to yield optimal solutions and operate within O(n2) time  complexity. Extensive computational experiments on standard benchmark datasets and randomly generated instances demonstrate that IFLP1 achieves the lowest CPU time and smallest unbalancing number, while IFLP3 yields superior performance in termsof total transfer cost and is recommended for practical applications https://mathco.journals.pnu.ac.ir/article_12797.html This paper develops and analyzes a class of double pre-test shrinkage estimators for the reliability function of the Pareto distribution based on progressively Type-II censored samples. The proposed approach combines a preliminary test of the shape parameter against a prior target value with shrinkage toward the corresponding prior reliability, yielding four reliability estimators with fixed and data-dependent shrinkage weights. Closed-form analytical expressions are derived for the bias and bias ratio of the proposed reliability estimators, as well as for their risk functions under the Precautionary Loss Function (PLF) and the associated relative risk with respect to the classical pooled estimator. Numerical results are obtained by direct numerical evaluation of the derived analytical expressions, including one- and two-dimensional integrals and special functions, implemented in Python. Across a wide range of design settings and reliability levels, the proposed estimators reduce PLF-risk and improve relative efficiency, with the most pronounced gains typically occurring when the prior ratio λ = θ₀/θ  is close to unity. In addition, the proposed framework can be viewed as an optimization problem under uncertainty, where the PLF-risk acts as the objective function and the design parameters, including the shrinkage weight, significance level, and stage sample sizes, define the feasible decision space. https://mathco.journals.pnu.ac.ir/article_12800.html This paper presents a systematic comparative study of two widely used numerical solvers --- HOFiD_bvp (high-order finite difference scheme) and bvp4c (collocation-based) --- for solving singular second-order ordinary differential equations (ODEs) with first-kind (regular) boundary singularities. Four representative benchmark problems drawn from fluid dynamics, materials science, and radially symmetric diffusion models are used to evaluate solver performance across key metrics: maximum residual, maximum error, mesh point count, and ODE/BC function call counts. Results show that HOFiD_bvp consistently achieves lower residuals and errors with fewer function evaluations, making it computationally more efficient. Conversely, bvp4c demonstrates superior robustness for nonlinear singular problems and offers better adaptive mesh refinement capabilities. These findings provide practical guidance for selecting the appropriate numerical technique in applied science and engineering contexts, with implications for optimization of computational simulation workflows.