Boletim da Sociedade Paranaense de Matemática

A Novel Fractional-Order Approach for Modelling Glucose Regulation with Meal Spikes and Periodic Noise

2025-10-02T22:06:16+00:00

In this work, we develop a novel fractional order model for glucose-insulin-lactate dynamics in diabetic patients, incorporating both time-varying noise and meal-induced glucose spikes to enhance the realism of the system. This framework is about non-linear fractional differential equations that capture the chaotic behaviour of glucose regulation in the presence of noise and periodic fluctuations. To simulate real-world conditions, time-varying noise is introduced as physiological variability, including noise levels that fluctuate based on circadian rhythms and metabolic processes. In addition, we introduce meal spikes as a sudden increase in glucose levels, reflecting the physiological response to food intake. The glucose surge is modelled using a Gaussian function, with intensity and duration adjustable to simulate different meal patterns. The proposed model successfully captures the complex, real-world behaviour of glucose metabolism, providing insights into the effectiveness of control strategies under realistic conditions. From this approach, we offer a more comprehensive representation of the metabolic control system in diabetic patients and provide a practical method to examine intervention strategies.

E Explicit Class‑Field Generation via Chains of Modular Polynomials

2025-10-02T22:08:16+00:00

We introduce an augmented Ihara zeta function for supersingular
$\ell$‑isogeny graphs that records both the degree label and the
orientation determined by dual isogenies. A Bass–Hashimoto style
determinant formula is proved, and we show that the resulting zeta
function factors as the characteristic polynomial of the Hecke operator
$T_{\ell}$ acting on weight‑$2$ cusp forms of level~$p$. Deligne’s
bound on Hecke eigenvalues then yields a \emph{uniform Ramanujan
property} for supersingular isogeny graphs with any prime
$\ell<p/4$. We extend the zeta formalism to non‑regular ordinary
\emph{isogeny volcanoes}, derive a rationality result, and relate the
dominant pole to the volcano height. Finally, explicit cycle‑counting
formulas lead to an equidistribution theorem for cyclic isogeny chains,
confirmed by numerical experiments for primes $p\le 1000$ and
$\ell\in\{2,3,5\}$.

A Note on Sumsets and Difference Sets in Groups of Order 12

2025-10-03T00:40:41+00:00

A subset $A$ of a group $G$ is referred to as a balanced set when $|A+A|=|A-A|$, MSTD (more sums than differences) when $|A+A|>|A-A|$, and MDTS (more differences than sums) when $|A-A|>|A+A|$. In this paper, we present a comparative study of MSTD and MDTS sets in groups of order 12 up to isomorphism. Additionally, we have completely categorized such sets in these groups and have provided a set $A$ with the current highest value of $ln(|A+A|)/ln(|A-A|).$

Impact of Socio-Economic Drivers on Environmental Complexity Through Chaotic Synchronization

2025-10-03T00:45:47+00:00

In the field of environmental research, investigating the complexity of interactions between environmental systems and socio-economic drivers, such as financial market fluctuations, requires a variety of sophisticated scientific methodologies. This study aimed to examine the synchronization and anti-synchronization phenomena between two distinct dissipative systems using an active control method, which will offer a prospect in modelling complex environmental interactions. The first chaotic system, introduced by Huang and Li (1993), and the second system, proposed by P. Kumar and S. Jha (2022), are analyzed in depth. By employing phase portraits and Poincaré sections across a range of parameters and initial conditions, we confirm the chaotic nature of both systems. Subsequently, a set of active control laws is designed and implemented to control the intended dynamical phenomena. To validate the theoretical results, comprehensive numerical simulations are conducted, which highlight the effectiveness of the proposed control scheme. This research highlights the relevance of coordination strategies of dissipative systems and managing the complexity inherent in environmental systems influenced by socio-economic dynamics.

Reflection and Transmission of Elastic Waves at the Permeable Interface Between Fractured Porous Solid Saturated with Two Immiscible Pore Fluids and a Liquid Medium

2025-10-06T15:19:48+00:00

The reflection and transmission of elastic waves at a permeable interface between a fluid half-space and a fractured porous solid (FPS) half-space saturated with two immiscible pore fluids, are examined within the framework of Volume Average Theory for porous solids. The FPS comprises a solid matrix saturated with two immiscible fluids within pores and a connected network of fractures in which five types of wave modes exist. Five transmitted waves in the FPS medium are attenuating waves. Energy partition across the interface is examined through the computation of reflection and transmission coefficients. Parametric analysis is conducted to evaluate the effects of partial pore opening at the interface, fractures, frequency, incidence angle, and pores permeabilities. The results of wave characteristics and energy distribution are highly sensitive to pore opening at the interface and multiphase fluid interactions within the fractured porous solid, thus offering key insights for subsurface imaging, reservoir evaluation, and acoustic monitoring.

On Xbeta Distribution: Properties, Estimation and Application

2025-10-03T00:53:48+00:00

BackgroundUseful instruments for evaluating the randomness of events in life are probability models. Although the
literature contains many statistical distributions, it is always possible to obtain more powerful, more adaptable distributions
with more general applications. For estimation and prediction, these probability models are priceless. Current probability
models do not always offer the best match since a great deal of new data is available. By generalizing or presenting novel
probability models that can be simply matched to these latest data sets, researchers have substantially added to the body of
knowledge. The objective of this research is introducing new probability distributions; this is done by several methods and
usually by adding new parameters to an existing distribution.
MethodsA flexible distribution called the Xbeta distribution is proposed for analyzing bounded data. Key characteristics,
including the shape of the model, survival and hazard functions, and analytical expressions of mode, quantile function,
ordinary moments, and stress-strength reliability, are comprehensively analyzed. Additionally, several famous entropy
measures are derived.
ResultsThe Xbeta model parameters have been estimated using four distinct methods: maximum likelihood estimation,
Anderson-Darling, Cramer-von Mises, and ordinary least squares. A detailed simulation study is used to evaluate the
behavior of all derived estimators. Finally, a dataset is used to demonstrate the utility of the proposed distribution.
ConclusionThe value of the new distribution is demonstrated using a dataset linked to flood level and polyester fiber
tensile strength measurements. Goodness-of-fit tests led to the conclusion that the Xbeta distribution effectively analyzed
these data sets relatively to competing distributions.

Connectedness via (1, 2)Sβ - open sets in Bitopological spaces

2025-10-03T00:57:19+00:00

Topology is a fundamental branch of mathematics thar provides a framework for studying spaces
and their properties. Central to topology is the concept of open sets, which are subsets of a given space defined by specific characteristics inherent to the spaces structure. These open sets form the foundation for various topological properties, allowing for a deeper understanding of connectivity within the space. In this work, we introduce a class of open sets in bitopological spaces namely (1, 2)S_β - open set by involving (1, 2) semi- open set and (1, 2)_β -closed set. In addition, we present the essential properties of this class and study its relationship with the other classes of open sets with the help of counter examples. Further, we introduce (1, 2)S_β - separated sets, (1, 2)S_β - connected sets and study their properties in bitopological spaces. Also, we
prove some results related to (1, 2)S_β - continuous functions.

Numerical Analysis of Thermoelastic Wave Behavior in a Micropolar Medium with Dual-Phase-Lag, Nonlocality, and Pre-Stress under Gravitational Influence

2025-10-03T01:01:11+00:00

his study investigates the transient wave propagation in a micropolar thermoelastic half-space
under the influence of gravity, initial stress, and nonlocal effects, within the framework of two-temperature
generalized thermoelasticity incorporating the dual-phase lag (DPL) model. The governing equations are for-
mulated considering a quiescent medium subjected to an inclined mechanical load and a gravitational field.
An analytical solution is derived using normal-mode analysis to obtain exact expressions for the thermome-
chanical field variables. Numerical simulations are performed for a magnesium crystal-like material to evaluate
the distributions of displacement, stress, and temperature. The results are presented graphically to illustrate
the influences of initial stress, non-locality, the two-temperature parameter, and the angle of loading inclina-
tion. Comparative analyses are also conducted to highlight the role of these factors on wave behavior, with
particular cases discussed as subsets of the generalized model.

Profiling English Language Writing Creativity with AI: Quotients Affecting Task Complexity and Repetition

2025-10-03T01:06:02+00:00

The Study investigates the Artificial Intelligence generated written articles. Writing majorly depends on idea generation and the quality extends with linguistic features. Human originality, fluency, flexibility and elaboration generates the judgments and assessment of the text. Language is developed with the mind, mouth, hand and eye. While writing, domain specific knowledge adds high quality to the write up, whereas the AI generated the similar frame of specific words into the drafts generated for the particular topic titles. University student’s drafts shows more quotients and their text is more persuasive and impressive whereas the Quotients are missing in the drafts generated out of the help of Chat GPT i.e. AI based application. The study is to examine the range and influence of techno- generated articles with the quality impact on the text as complex or repetitive by approach. The study employs Descriptive, Correlation and Wilcoxon Signed-Rank Test in order to evaluate the association, direction and magnitude of the changes between paired observations i.e. Pre-AI and Post-AI variables. The study reveals that the AI-blended instructional method significantly enhanced student learning outcomes compared to traditional methods. The study although proves that the quality of writing is improved with the usage of writing tools of AI. The results simply clarify that literacy is more effective with the cognitive approach of human brain.

Antisymmetric Electromagnetic Lorentz Force Tensor Formulation of Electrodynamics

2025-10-03T01:10:22+00:00

The usual Lorentz force q[E+] predicts an independent structure of electrodynamics in terms of electric force and magnetic force. As electric field E and magnetic field B are the building blocks of antisymmetric second rank electromagnetic field tensor . Correspondingly, electric force qE and magnetic force q are the building blocks of antisymmetric second rank electromagnetic force tensor . In other words, Lorentz force is a tensorial force by birth. A new type of electrodynamics is emerged that possesses the same structure as standard Maxwell’s theory such that electrodynamical laws are now force laws. It obeys principle of relativity, conservation law and symmetry. We designate this model as Lorentz force electrodynamics (LFE). Matrix method is employed in tensor notation. The model consists of force field equations, Lorentz force Maxwell’s equations, conservation law, 4D waves of electric force and magnetic force. Similarly, dual of LFE is worked out. Conservation law is valid in both cases. This model is expected to facilitate electrical, electronic, mechanical, space engineers, theoretical physicists in the study of astrophysics, cosmology, magnetohydrodynamics etc.

ART Rings

2025-10-07T22:03:03+00:00

A ring R is said to be an ART ring if each member of R can be represented as the sum of a regular element (in the von Neumann sense), a tripotent element, and a nilpotent element. This article introduces and studies ART rings, highlighting the role of tripotent elements in such decompositions. Several fundamental properties and key results related to ART rings are investigated.

EXPLORING FUZZY GEOGRAPHIC PROFILING THROUGH MVPP AND BMVP APPROACHES

2025-10-07T22:06:11+00:00

A fuzzy extension of Minimal Variance Projection Profiling (MVPP), a geospatial analysis technique for locating possible areas of interest in geographical profiling, is presented in this study. In traditional MVPP, the spatial distribution of criminal events is analyzed using statistical measures, linear algebra, and Euclidean geometry. A minimal variance line and a bounding polygon that is likely to contain an offender's hideout are constructed. By adding fuzzy matrices, fuzzy covariance, and fuzzy distances, we expand MVPP in this fuzzy form to address spatial uncertainty. This method accounts for imprecision in spatial data by treating crime event locations and projections as fuzzy data points with different levels of membership. In situations where there is insufficient or unclear evidence, the fuzzy MVPP framework efficiently captures regions of interest, providing a more adaptable and realistic option for comprehending illegal spatial behavior. Through the provision of a rigorous, fuzzy-based approach for evaluating ambiguous spatial data in criminal investigations, this contribution enhances the field of geographical profiling.

ELQTR: More Secure Encryption System Based on MAL-Eleven Algebra with Quaternion Coefficients

2025-10-07T22:09:06+00:00

Our traditional world is transforming into a digital world at a very rapid pace, which requires the need to encrypt data at the same or greater speed. In this paper, we will present an improved NTRU encryption scheme called ELQTR, which is based on the eleven-dimensional M_ae algebra. As a special case, we will take the coefficients from the quaternion algebra, which gives a significant difference in security compared to NTRU and some of its improvements. We will also provide some comparisons between the new scheme and some other schemes in terms of message security, key security, and speed

A 4-dimensional multi goal transportation framework in the decomposed fuzzy configuration

2025-10-07T22:13:28+00:00

Decomposed Fuzzy Sets (DFS), a recent advancement of Intuitionistic Fuzzy Sets, extend the traditional framework by integrating functional and dysfunctional viewpoints into the formulation of membership and non-membership functions. This article first introduces a score index for the ranking and defuzzification of DFS. Thereafter, a multi goal transportation framework in 4-dimensions under DFS configuration is formulated. The proposed framework aims to optimize overall transportation cost and travel time. In order to deal with the proposed 4-dimensional transportation network, fuzzy programming and devised score index is employed. A numerical computation accomplished to elaborate the efficiency of devised model in the real-world transportation systems.

Comparison of some a posteriori error estimators

2025-10-07T22:51:38+00:00

The idea of {\it a posteriori } error estimates based on the reconstruction of the equilibrated potential and/or equilibrated flux goes back to the Prager-Synge equality %\cite{Pra}
for the Poisson equation $ -\Delta p = f $. This identity is valid for all $ v \in H^1_0(\Omega) $ and all
$ \mathbf{u} \in H (\Div, \Omega) $ such that $ \Div \mathbf{u} + f = 0 $, and given by
\[
\|\mathbf{u}-\nabla v\|^2_{0,\Omega}=\| \mathbf{u}-\nabla p\|^2_{0,\Omega}+\|\nabla p-\nabla v\|^2_{0,\Omega}.
\]
It follows that, to obtain such estimate, we need to reconstruct a so-called equilibrated flux; $ \mathbf{u} \in H (\Div; \Omega) $ satisfying the equilibrium condition $ \Div \mathbf{u} + f = 0 $ and such that $ \vu- \nabla p $ is as small as possible, and/or reconstruct a potential $ v $ in $ H^1_0 (\Omega) $.\\
In all cases, to have an estimate, which is said "by reconstruction", it is necessary to have at the end an equilibrated, flux and potential. Now, the question is: is it better to work with a numerical method that allows us to have an equilibrated quantities and in this case there is no need to reconstruct, or else, do we use a method where, we do not have an equilibrated solutions such as the Discontinuous Galerkin method, and in this case it is necessary to reconstruct the two variables?
We first compare two types of error estimators: the classical residual-based estimators, which do not require any reconstruction, and the reconstruction-based estimators, in the context of a diffusion problem. Then, using various numerical approximation methods, we proceed to compare the different reconstruction-based estimators.

Novel Coding Inequalities for Mean Codeword Length and Generalized Entropy using Noiseless Communication

2025-10-07T23:03:01+00:00

Shannon’s entropy forms the basis for almost every aspect of information theory. It formulates the foundational stone for various source coding theorems assuming statistical independence and extensive systems. This research aims to investigate the possibility of deriving novel entropy
measures using noiseless coding theorem. The obtained results find a widespread application in information theory and applied mathematics. To accomplish this, a novel expression for mean codeword length has been illustrated. Besides, established relation between entropy measure and
its corresponding codeword length. The results obtained pave the way for a new avenue for entropy-based coding in non-extensive and information-rich environments.

An Isoperimitric problem perturbed by the potential of the reproduction kernel of a reproducing kernel Hilbert space.

2025-10-07T23:08:40+00:00

The purpose of This paper is to study the minimizer of the isoperi
metric perturbation problem, over measurable sets of fixed volume.
The problem is perturbed by an addition of repulsive nonlocal potentials of kernel K, which is a reproduction kernel of a reproducing kernel Hilbert space. We establish the existence of a minimizer of this problem. Besides, we study the geometric shape of a minimizer.

Reliable ECG Classification Using RNN and GRU Architectures for Arrhythmia Detection

2025-10-07T23:13:23+00:00

Cardiac disease has become a severe threat to public health as
it has become a leading cause of mortality in India. Electrocardiogram (ECG)
signal classification plays a pivotal role in early detection of cardiac arrhyth
mias, potentially reducing morbidity and mortality associated with cardiovas
cular diseases. This study presents a comparative analysis of Recurrent Neural
Network(RNN) and Grated Recurrent Unit(GRU) architectures for ECG sig
nal classification using the MIT-BIH Arrhythmia dataset. After preprocessing
and class balancing, both models were trained to classify five heartbeat types.
Experimental results show that GRU model achieved a significantly higher test
accuracy (98.61%) compared to the RNN (83.39%), demonstrating its potential
for real-time cardiac monitoring applications in diagnostic systems and wearable
devices.

Mathematical Foundations of Classical Arabic Prosody: A Group-Theoretic Analysis of the Tawil Meter’s Cognitive and Cultural Dominance

2025-10-07T23:17:20+00:00

This study provides a thorough mathematical analysis of Arabic poetry meters, focusing on the Tawil meter, and clarifies significant connections between abstract algebra, information theory, and classical Arabic prosody. By establishing a prosodic transformation group $\mathcal{G} \cong D_8 \rtimes \mathbb{Z}_2$, we demonstrate that Al-Khalil's system of 16 taf'ilat constitutes a comprehensive group-theoretic framework wherein meter transformations are non-commutative operations that preserve poetic validity. The research establishes an entropy-based complexity metric $ C(B) = \alpha H(P_t) + \beta \log_2|\text{Aut}(B)| + \gamma R(B) $, illustrating that traditional metrics attain an optimal balance between predictability and information density ($ 0.4 \leq R \leq 0.6 $). Our examination of the Tawil meter $(10110, 1011010)^2$ reveals its distinct mathematical characteristics. Information-theoretic analysis reveal that classical meters operate within a constrained complexity spectrum ($0 \leq H(B) \leq 3$ bits), with Tawil positioned at the core of the ``poetic sweet spot." Historical data indicates that Tawil was culturally preeminent, with an annual growth rate of 0.023, with 38\% of classical Arabic poetry originating from this region. Neuroaesthetic study correlates its mathematical structure with a 30\% increase in alpha-wave brain responses compared to non-phi metrics. The research develops a metric space $(V_T,d_T)$ for meter variations, defines a linear operator $\mathcal{T}$ with eigenvalues $\phi$ and $-1/\phi$, and provides constraints for composition stability $\|\mathcal{T}^n v_0\| \leq \phi^n \|v_0\|$. These mathematical structures provide new tools for the computational analysis of poetic tradition, clarifying the enduring cognitive and cultural appeal of ancient Arabic meters through their inherent mathematical harmony.

Statistical Modeling of Groundwater Pollution Using Logistic Regression

2025-10-07T23:20:44+00:00

Binary logistic regression is used to examine groundwater pollution and determine
the main contributing elements. Groundwater level, Northing coordinates, and Total
Dissolved Solids (TDS) were the hydrogeological and geographical variables that
were examined using SPSS. The study showed the significant impact of variables
on the state of pollution. The results provide important information for methods
to mitigate pollution and manage groundwater resources.

Designing an efficient cryptosystem via Tripternion Algebra and Nitrosophic Integers

2025-10-07T23:25:29+00:00

The exchange of information between two parties always requires protecting its confidentiality from any unauthorized third party, which requires sending it encrypted using a secure cryptographic system. In this paper, we present a cryptosystem based on quaternion algebra with neutrosophic integer coefficients, an evolution of the QTRU system with features that make this system effective and desirable for many organizations concerned with information security.

Developing MRSA Encryption Based on Triptrion Algebra

2025-10-07T23:33:49+00:00

Encryption algorithms are a pressing need for protecting sensitive information and data, as well as for establishing infrastructure in technology security, electronic communication methods, and internet security. The RSA algorithm is one of the most important algorithms for protecting information through the process it performs. In this paper we present the development of the MRSA by adopting the triptrion algebra in building the proposed system.

Boundary Layer Analysis of MHD Heat and Mass Transfer with Soret and Dufour Effects on a Wedge Surface

2025-10-07T23:39:00+00:00

This study investigates the influence of magnetohydrodynamics (MHD), Soret (thermal diffusion), and Dufour (diffusion-thermo) effects on heat and mass transfer in an electrically conducting nanofluid flow past a wedge. The governing equations for momentum, energy, and species transport are derived from first principles and transformed into a system of nonlinear ordinary differential equations through similarity transformations. These equations are solved numerically using a Runge–Kutta-based shooting technique. The findings indicate that increasing the magnetic parameter suppresses velocity while enhancing thermal and concentration boundary layers. The Soret effect elevates concentration distributions, whereas the Dufour effect increases the thermal field, demonstrating strong cross-diffusion coupling. The Prandtl and Schmidt numbers were found to control the thinning of thermal and solutal boundary layers, respectively. A comparison of the present numerical results with previously reported benchmark solutions shows excellent agreement, validating the accuracy of the method and extending earlier studies by incorporating coupled MHD, Soret, and Dufour effects.

CERTAIN SUBCLASS OF ANALYTIC FUNCTIONS DEFINED BY q−ANALOGUE GENERALIZED DIFFERENTIAL OPERATOR

2025-10-07T23:42:25+00:00

In the present work, we define a subclass of uniformly starlike functions corresponding to the class of uniformly convex functions involving the $q$-analogue of a generalized differential operator. Furthermore, we discuss coefficient estimates, neighborhoods, partial sums, integral means inequality, and Radii of close-to-convexity and Starlikeness results related to the defined class.

Orthogonal Jordan Derivations on $\Gamma$-semihyperrings

2025-10-07T23:52:38+00:00

The present paper introduces the concept of orthogonal derivation on $\Gamma$-semihyperrings and explores some fundamental properties of orthogonal Jordan derivation. It is shown that a non-zero derivation is not necessarily orthogonal to itself and the specific condition under which such a derivation becomes orthogonal to itself is established. Furthermore, the sum of two orthogonal derivations remains orthogonal to each of its summands is proved. A necessary and sufficient condition for two derivations to be orthogonal is also derived. The study concludes with an in-depth examination of orthogonal Jordan derivations, revealing important results regarding their behaviour in the context of $\Gamma$-semihyperrings.

Building a More Secure Cryptosystem Using Tripternion Algebra and Polynomials

2025-10-07T23:57:55+00:00

There are many modern encryption methods, but the rapid development in the field of software and algorithm design has made the development of these methods or the design of new encryption methods a necessity. In this paper, we have developed a multidimensional encryption system based on the analysis of polynomials and triangular algebra to obtain an efficient system.

Reforming Decision Models and Assessing the Impact of State Schemes on Rural Women's Social Development

2025-10-08T00:01:21+00:00

The Government of Tamil Nadu has introduced numerous welfare schemes to empower women and uplift their socio-economic status. While these initiatives play a vital role in fostering inclusion and development, their effectiveness often varies across different dimensions. This study presents a comparative analysis of five major women-centric schemes using rhotrices-based decision-making techniques. The proposed framework enables a structured and quantitative evaluation by incorporating multiple criteria and offering a comprehensive visual representation of performance. By systematically comparing the schemes, this approach highlights their relative strengths and weaknesses, ensuring objectivity in assessment. The findings provide valuable insights that can guide policymakers in refining existing programs and formulating future interventions to maximize their impact on women’s empowerment and socio-economic progress.

Exploring gb-Compactness and gb-Connectedness through Generalized Topologies with Applications

2025-10-08T00:05:24+00:00

This paper presents and explores two extended topological structures g*b-compactness and g*b-connectedness which serve as generalizations of classical compactness and connectedness by incorporating the concepts of g-open and b-open sets. Beyond their theoretical significance, these generalized spaces offer valuable tools for modeling and analyzing complex systems where classical topological assumptions may not hold. Potential applications include the design of resilient network topologies, analysis of digital images and data clusters, control systems in engineering, and non-standard models in theoretical physics. The characteristics of c g*b-ompact and g*b-connected spaces examined in this study establish a strong foundation for advancing topological analysis in contexts where generalized notions of openness and continuity are fundamental.

ON ANALYTIC FUNCTIONS SUBCLASS DEFINED BY DIFFERENTIAL OPERATOR

2025-10-08T00:13:40+00:00

In this work, we present a novel differential operator-defined a specific set of negative-coefficient analytic univalent functions. For this class, we get subordination results, integral means inequalities, extreme points, and coefficient inequalities.

Artificial Intelligence-Based Optimization of PID Controllers for Two-Area AGC Systems Using Particle Swarm Optimization

2025-10-08T00:17:08+00:00

Automatic Generation Control (AGC) is vital for maintaining constant and reliable power in interconnected systems, and AGC is also responsible for distributing loads between generators optimally as well as maintaining frequency stability, and controlling exchanges over tie lines. This study proposes the use of an AI-based optimisation approach to enhance the performance of AGC in a two-area power system. More specifically, the tuning settings of the Proportional-Integral-Derivative (PID) controller are optimised by employing the AI algorithm called Particle Swarm Optimisation (PSO), which is motivated by the social behaviour of swarms occurring naturally. Two scenarios are evaluated: AGC without PID control in the two areas, and AGC with PSO-optimised PID controllers in both areas. The simulation results obtained from this study demonstrate that the PSO-optimised PID controllers result in a significant reduction in frequency deviations, improve the integral of time-weighted absolute error, and provide better dynamic performance of the system. The study results also identified that AI-based optimisation methods incorporated in AGC design achieve better stability, frequency regulation, and tie-line power control with rapid response, thus preparing an intelligent and resilient operation of the power system.