Asian Journal of Mathematics and Computer Research

Development of a Generalized Quadrature Formula Using Anti-gaussian Methods

Sanjit Kumar Mohanty — 2026-05-25

In this paper, we propose a novel generalized quadrature rule, denoted by SM₁₅ (f), constructed by combining the Anti-Lobatto 4-point rule and the Anti-Gauss 3-point rule through a generalized quadrature framework. A detailed theoretical investigation of the proposed rule is carried out, including convergence analysis and the derivation of appropriate truncation error estimates. The analytical results reveal that the proposed quadrature rule possesses a higher degree of precision and significantly improved accuracy compared with its constituent quadrature rules. To assess the practical performance of the method, several numerical experiments are performed on a variety of test integrals. The obtained results demonstrate that the proposed rule yields highly accurate approximations with considerably reduced truncation errors, thereby confirming its reliability, stability, and computational efficiency. Comparative error analysis and numerical illustrations further establish the superiority of the proposed quadrature rule over the existing component rules. Consequently, the generalized quadrature rule SM₁₅ (f) emerges as an efficient and powerful technique for high-precision numerical integration problems arising in applied mathematics and scientific computing.

Polynomially Stable of a Thermoelastic Timoshenko System with Cattaneo Heat Conduction Law

Hui Chang — 2026-05-27

This paper investigates the polynomial stability of a thermoelastic Timoshenko system with Cattaneo’s heat conduction law. The system consists of coupled hyperbolic-parabolic equations governing the transverse displacement, rotation angle, temperature, and heat flux. Previous work established the lack of exponential stability regardless of the equal wave speeds (EWS) condition. It is proved in this paper that when the EWS condition is satisfied, the associated C0-semigroup exhibits polynomial stability. Specifically, it is demonstrated that solutions decay at a rate of t^−1/4 as t → ∞, with the decay rate uniform for initial data in the domain of the generator. The analysis employs energy methods combined with semigroup theory, leveraging the structural properties induced by the EWS condition to establish polynomial decay estimates. This result extends previous stability analyses and highlights the critical role of wave speed matching in stabilizing Timoshenko systems with second-sound thermal effects.

Eco-Performance Analytics: A Multi-Objective Optimization Framework for Sustainable Sports Engineering

S. Ranjith Kumar — 2026-05-30

The sports engineering domain has traditionally emphasized performance maximization, often overlooking environmental sustainability and long-term ecological impact. With increasing global attention toward climate responsibility and sustainable product design, there is a pressing need for engineering frameworks that simultaneously address athletic performance and environmental efficiency. This paper proposes Eco-Performance Analytics (EPA), a multi-objective optimization framework that integrates performance metrics, biomechanical efficiency, material sustainability, and energy consumption into a unified decision-making model. The framework employs data-driven analytics, life-cycle assessment (LCA), and evolutionary multi-objective optimization algorithms to identify optimal trade-offs between performance enhancement and environmental impact. A conceptual implementation is demonstrated through a sustainable sports equipment design and athlete–equipment interaction model. The proposed framework aims to support designers, coaches, and sports technologists in developing high-performance yet environmentally responsible sports systems.

Properties of ΔLG Transformation via Optimal Bounds on Connectivity, Independence and Domination

Ghadeer Khudhair Obayes — 2026-06-01

In this study, we present a balanced graph (L_Δ (G)), which is an edge-based graph transformation that improves classical linear graphs by incorporating the vertex degree as a structural node in (L_Δ (G)). The two edges are adjacent only if they are not adjacent and have a length distance of 2 and are connected via an intermediate vertex w of deg(w)≥3.Through this formula, adjacencies mediated by vertices with reduced degrees are excluded, making \(\bar{L(G}\)) is a sub graph. It is clear to us from the definition that (L_Δ (G)), is continuous if and only if δ(G) ≥ 3, which reflects on the adjacency property, which is very weak, giving it a special uniqueness not found in traditional linear Apleyan graphs. We also define lower limits for the dominance number: γ(L_Δ (G)) ≥ \(\frac{|E(G)|}{2(Δ(G)−1)^2+1}\)

An upper bound on the matching number: μ(L_Δ (G)) ≤|E(G)|- α(L_Δ (G)) and for the independence number α(L_Δ (G))≥ μ(G)

The results obtained for these limits are accurate for third-order regular graphs, confirming that (L_Δ (G)), embodies the correlation between degree heterogeneity and edge independence. This transformation provides a solid foundation for the design of global networks, communication systems, and performance, and addresses the fundamental shortcomings of L_Δ (G)and (\(\bar{L(G}\))) linear graph models, which fail to accommodate the structural heterogeneity based on the degree of the vertex. It provides the first solid foundation for modeling structurally constrained systems where local degree centrality is a critical factor making it uniquely suitable for designing nested networks.

Tchebychev Polynomials of Second Kind on the Ellipse and Approximations

Abdelhamid Rehouma — 2026-06-05

We study the orthogonality of Tchebychev polynomials of second kind {U_n (z)}_{n=0,1,2,3,...}with respect to the Lebesgue planar measure concentrated on the ellipse D: b²x² + a²y² < a²b²where a > b,a system of orthogonal polynomials, given by : \(U_n(z)=\frac{T_{n+1}^{\prime}(z)}{n+1}=\frac{\sin \left((n+1) \cos ^{-1} z\right)}{\sqrt{1-z^2}},\) n = 0, 1, 2, 3, ...

where Tn (z) = cos ( n cos⁻¹ z ) , n = 0, 1, 2, 3, ...is a polynomial of degree n . T_n (z) is called the Tchebychev polynomial of degree n of first kind.They satisfies

\(\iint\limits_D U_n(z) \overline{U_m(z)} d x d y=\frac{4(n+1)}{\pi\left(\rho^{n+1}-\rho^{-n-1}\right)} \delta_{n, m}\) , n,m = 0, 1, 2, 3, ...

where δ_n,m , is the symbol of Kronecker and (a + b)² = ρ.

We study extremal properties and minimization and Fourier development involving of these orthogonal Tchebychev polynomials of second kind with respect to the Lebesgue planar measure concentrated on the ellipse.General expressions are found for the kernels polynomials associated to orthonormalized Tchebychev polynomials of second kind on the ellipse.These kernel polynomials can be used to describe the approximation of continuous functions and to solve some area extremal problems by Tchebychev polynomials of second kind on the ellipse .They can be used for the representation of the n-th partial sum of the Fourier series expansion of orthonormalized Tchebychev polynomials of second kind in the form of an integral.

Convergence Analysis of Generalized Non-Smooth Equations Using Gauss-Type Proximal Point Method

Md. Asraful Alom — 2026-06-05

This work discusses the Gauss-type proximal point algorithm for solving non-smooth generalized equations like 0 ∈ q(x) + Q(x), where a set-valued mapping Q : X ⇉ 2^Y acting between two real or complex Banach spaces X and Y with closed graph and q: U ⊆ X→Y is a single-valued mapping. In order to ensure the existence as well as convergence of any sequence produced by this algorithm under appropriate circumstances, we develop the convergence criteria of this approach by utilize metric regularity condition and point-based approximation. Lastly, we present a numerical example to validate the semi-local convergence of this algorithm.

Closed-Form Solutions of Leonardo-Type Sequences: Pell-Padovan, Jacobsthal-Padovan, and Narayana Families as Homogeneous Counterparts

Yüksel Soykan — 2026-06-06

Recurrence sequences are widely used mathematical models with applications across many disciplines. Beyond classical second-order sequences, higher-order families such as Tribonacci, Tetranacci, and Pentanacci reveal richer algebraic structures, with the study of their characteristic roots and recurrence relations advancing symbolic recurrence theory.

In this paper, we present unified closed-form solutions for third-order nonhomogeneous linear recurrence relations of Leonardo-type sequences, where the input term is taken as a polynomial. By decomposing each recurrence into homogeneous and particular components, we obtain explicit formulas that depend jointly on the multiplicity of the characteristic roots and the degree of the input polynomial. Although the general framework accounts for resonance phenomena arising from repeated roots, our illustrative examples focus on the non-resonant case r = 0, where all three roots of the characteristic equation are distinct from 1.

Within this setting, we investigate several notable families of generalized Tribonacci numbers, which appear as homogeneous analogues of the original nonhomogeneous relations. Classical sequences such as the adjusted Pell-Padovan, third-order Lucas-Pell, third-order Fibonacci-Pell, Pell-Perrin, Pell-Padovan, adjusted Jacobsthal-Padovan, Jacobsthal-Perrin (Jacobsthal-Perrin-Lucas), Jacobsthal-Padovan, Narayana, and Narayana-Lucas numbers arise naturally as special cases of the Leonardo-type framework. These examples illustrate how closed-form expressions clarify the interaction between characteristic roots, polynomial inputs, and resonance effects, while also providing templates for applications in discrete mathematics, combinatorics, computational number theory, algorithmic analysis, cryptography, and discrete models in physics and biology. A further illustration is given by considering the case where the input polynomial has degree s = 3, which serves as a natural extension of the classical situations with s = 0. Under identical initial conditions, the homogeneous dynamics reproduce the well-known Pell-Padovan, Lucas-Pell, Fibonacci-Pell, Pell-Perrin, and Pell-Padovan sequences, while the cubic input enriches the particular solution. This demonstrates the continuity of the framework across polynomial degrees and emphasizes the role of initial values in shaping the resulting closed forms.

Beyond their theoretical contribution, the explicit constructions offer pedagogical value by enabling students to engage directly with nonhomogeneous recurrences through accessible formulas rather than lengthy computations. Thus, the study demonstrates both the novelty and interdisciplinary reach of generalized Leonardo-type sequences, furnishing researchers with extended tools for higher-order recurrence analysis and educators with clear examples for teaching advanced recurrence methods.

Existence of Solutions for a Class of \(p\)-biharmonic Equations with Navier Boundary Conditions

Sha-Sha Liao — 2026-06-09

The p-biharmonic equation with Navier boundary conditions is an important class of higher-order nonlinear elliptic equations that arises in applications such as beam vibration theory and suspension bridge dynamics. Extensive research has employed variational methods and critical point theory to establish the existence and multiplicity of solutions, highlighting its significance in both mathematical analysis and engineering applications. This paper studies the existence of solutions for a class of p-biharmonic equations with Navier boundary conditions. Using Morse theory, critical point theory and the G-link theorem, we establish the existence and multiplicity of solutions under the nonquadratic type conditions and the asymptotic noncrossing condition.

A Novel 4D Nonlinear Interaction Football Model: Stability Analysis

I. C. Nwokike — 2026-06-13

A novel nonlinear mathematical model is formulated to study the effects of defense, midfield, and attack on an overall football team performance. Bilinear inhibitory effects are considered for match-based factors such as defensive breakdown under attacking pressure, midfield congestion, and failure to convert attacking opportunities into match outcomes. Qualitative analysis is carried out to confirm positivity, boundedness, and well-posedness of solutions. Equilibrium points corresponding to stable tactical configurations are obtained and analyzed. Local stability analysis is carried out through analysis of the Jacobian, while global stability is established with the use of a Lyapunov function and LaSalle’s Invariance Principle. Analytical results indicate that nonlinear inhibitory factors are a major force in maintaining control over football tactics, as they ensure that none of the considered aspects are able to grow unbounded. In fact, it is demonstrated that synergy between effective defense, midfield, and attack leads to more sustainable tactics and higher rates of team performance.

On Computing the Lower and Upper Bounds of Γ-Graphs via Some Topological Indices

V. H. Narendra — 2026-06-26

Topological indices provide numerical descriptors of graph structure and are widely used to relate molecular graph properties to structural features. This manuscript studies lower and upper bounds for selected degreebased topological indices of the Γ -graph, defined through the semi-total point graph of the vertex corona construction for two finite, simple and connected graphs G and H. Using an edge-partition approach, the work derives bounds in terms of the vertex and edge cardinalities of G and H, together with their minimum and maximum degrees. The indices considered are the first Zagreb index, the Nirmala index, the Sombor index, the hyper-Zagreb index, the Y -index and the V L-index. For each index, the corresponding formula is supported by illustrative computations on representative graph pairs, including paths, cycles and treetype graphs. These examples show that the computed index values lie within the stated lower and upper estimates, thereby demonstrating the consistency of the proposed bounding framework for the selected cases. The presentation also clarifies how the degree contributions from different edge classes determine the final expressions. The study contributes a structured treatment of Γ-graph bounds for several commonly used degree-based descriptors and indicates how the same edge-decomposition method may be adapted to related graph invariants, subject to careful verification of the underlying graph parameters.

A Quadrature on Average Lobatto Nodes

Pritikanta Patra — 2026-06-29

This paper presents a closed mixed quadrature rule constructed by averaging the Lobatto three-point rule and an anti-Lobatto four-point rule derived from the Lobatto formulation. The proposed rule is developed on the standard interval and is designed to improve the degree of precision obtained from its two constituent rules without changing the closed-type structure of the approximation. The construction combines two rules of precision three so that the resulting averaged Lobatto rule attains precision five. The truncation error is examined through series expansions and compared with the error terms of the base rules. The analysis indicates that the leading error term of the proposed rule is of higher order than those of the individual Lobatto and anti-Lobatto rules under the required smoothness assumptions. An adaptive quadrature routine using the proposed rule as the core approximation formula is also considered. To assess the numerical behaviour of the method, five definite test integrals with known exact values are evaluated using the Lobatto rule, the anti-Lobatto rule, and the proposed averaged rule. The numerical results show that the averaged rule gives smaller absolute errors than the two constituent rules for the selected test cases. The adaptive implementation further reduces the number of subintervals required to achieve the prescribed tolerance in the worked-out examples. These findings indicate that the proposed averaged Lobatto quadrature rule can serve as an effective approach for approximating definite integrals when increased precision is required within a closed-type quadrature framework. The study is limited to the theoretical derivation and numerical verification of the proposed rule using the selected test integrals.

Applications of Power Increasing Sequences

Hikmet Seyhan Ozarslan — 2026-07-04

This paper establishes a general theorem on absolute matrix summability for infinite series by employing the class of quasi-f-power increasing sequences in place of almost increasing sequences. The study is framed within the φ − |B, p_n|k summability method associated with a positive normal matrix B = (bnv) and a sequence of positive coefficients (pn). After recalling the relevant Riesz mean, normal matrix transformations, and bounded variation assumptions, the paper states a theorem that extends Bor’s result on absolute Riesz summability factors. The theorem assumes standard monotonicity and boundedness conditions on the matrix entries, the sequence (λ_n), the auxiliary sequence (β_n), and the n-th Ces`aro mean (t_n) of (na_n). Under these conditions, it is shown that the transformed series a_nλ_n is summable by the φ−|B, p_n|k method for k ≥ 1. The proof proceeds through Abel’s transformation, decomposes the matrix transform into four terms, and establishes convergence estimates for each term by using H¨older’s inequality, the properties of the matrix, and the consequences of quasi-f-power increasing sequences. The result confirms that replacing almost increasing sequences with the broader quasi-f-power increasing framework preserves the required summability conclusion. Several special cases follow from the theorem, including results for quasi-δ-power increasing sequences, | ¯N , p_n|k summability, and |B, p_n|k summability under appropriate choices of parameters. Thus, the theorem provides a unified formulation for related absolute summability factor results within the stated hypotheses. No additional assumptions are introduced beyond those specified in the theorem, and the derived consequences are presented only as formal reductions of the main result. This maintains a close connection between the generalised theorem and the earlier summability factor results considered in the manuscript.

Generalized Fractional Differential Operators Associated with an Extended Mittag-leffler Type Function

Krishna Gopal Bhadana — 2026-07-07

This paper establishes two image formulae for generalized fractional differentiation involving an extended Mittag-Leffler type function. The work uses Marichev-Saigo-Maeda fractional differential operators, whose kernels involve Appell’s function, and expresses the derived identities in terms of a generalized Wright hypergeometric function. The preliminary section recalls the extended Mittag-Leffler type function, the associated extended beta function, the Wright hypergeometric function and relevant special cases, thereby preparing the notation and parameter conditions used in the main results. Two principal theorems are then formulated: the first concerns the left-sided generalized fractional differential operator, whereas the second concerns the corresponding right-sided operator. In each case, the proof applies the definition of the extended Mittag-Leffler type function, the relevant Marichev-Saigo-Maeda operator and known power-function formulae, followed by the rearrangement of terms into the generalized Wright hypergeometric form. Several corollaries are obtained by specifying parameter values and by reducing the extended Mittag-Leffler type function to known forms. These reductions show that previously reported fractional-calculus identities can be recovered as special cases. The results therefore provide a unified presentation of related fractional differential formulae within the stated analytic setting. The study is theoretical and contributes a structured set of fractional differential identities for extended special functions only.

A Theorem on General Matrix Summability Method

Bağdagül Kartal Erdoğan — 2026-07-09

This paper establishes a general summability factor theorem for the ϕ − |B; δ|_k summability of an infinite series associated with a positive normal matrix B. The study considers the sequence of partial sums of an infinite series and the corresponding matrix transformation determined by B. A generalised framework is developed for sequences (ϕ_n), (p_n), (λ_n), and (\(\gamma\)_n) under prescribed boundedness, monotonicity, and growth conditions. The main theorem replaces the earlier weighted mean setting with a broader positive normal matrix setting and assumes that ϕ_np_n ≍ P_n, together with additional conditions on the associated lower semimatrices \(\bar{B}\) and \(\hat{B}\). The proof is based on Abel’s transformation and estimates obtained through Holder’s inequality, after decomposing the transformed difference into four components. Each component is shown to satisfy the required convergence condition under hypotheses (12)–(18) and the assumptions inherited from the corresponding weighted mean theorem.
The result demonstrates that the series \(\sum\) a_nλ_n(\(\gamma\)n)⁻¹ is ϕ − |B; δ|_k summable for k ≥ 1 and 0 ≤ δ < 1/k. For δ = 0 with the specified choices of ϕ_n and b_nv, the theorem yields a weighted mean matrix analogue of the known result. The study is confined to positive normal matrices satisfying the stated structural conditions, leaving wider matrix classes for further investigation.

Counting Methods of Area Integrals on the Unit Disk

Abdelhamid Rehouma — 2026-07-10

Suppose that a simple closed curve C is defined parametrically in the complex w-plane

u = u (θ) , v = v (θ) , 0 ≤ θ ≤ 2π

Then the area A enclosed by C is given by,

\(A=\frac{1}{2} \int_0^{2 \pi}\left(u \frac{d v}{d \theta}-v \frac{d u}{d \theta}\right) d \theta\)

If there is a conformal mapping Ψ_r,from the exterior of D_r ,r > 1 to the exterior of C_r of the form

\(\Psi_r(z)=z+\sum_{n=0}^{\infty} \frac{b_n}{z^n}\)

then by Gronwall’s area formula, the area A_r of the region B_r enclosed by C_r is given by,

\(A_r=\pi\left(r^2-\sum_{n=1}^{\infty} n\left|b_n\right|^2 r^{-2 n}\right)\)

We use Gronwall’s area formula to find the area of some differents regions as circles with radius r centred ot the origin lying in the complex w-plane , ellipses and lemniscates A lemniscate shapped with any number of leaves .The boundary of m-leafed symmetric lemniscate is the set

\(\left\{w \in \mathbb{C},\left|w^m-1\right|=0\right\}, m=2,3,4 \ldots . .\)

We use Laurent and Taylor series expansions of conformal mapping from the exterior of the unit disk to either of these regions to compute the area of them.

Quantum Search using Grover’s Algorithm: Principles, Performance, and Applications

Anggit Gusti Nugraheni — 2026-06-23

Quantum search is one of the most important algorithmic developments in quantum computing because it demonstrates how quantum principles can improve the efficiency of solving unstructured search problems. Grover’s algorithm provides a quadratic speed-up over classical linear search by reducing query complexity from O(N) to O(√N), where N represents the size of the search space. This review examines the principles, performance, and applications of Grover’s algorithm, with emphasis on its theoretical foundation and practical relevance. The discussion covers the roles of superposition, quantum interference, oracle operations, diffusion operators, and amplitude amplification in increasing the probability of measuring a desired solution. The manuscript also reviews the computational performance of quantum search in comparison with classical search methods and considers its relevance to database searching, combinatorial optimisation, cryptography, machine learning, bioinformatics, quantum chemistry, and materials science. The review highlights that Grover’s algorithm is not limited to database retrieval, as many computationally intensive problems can be reformulated as search or optimisation tasks. However, practical implementation remains constrained by current quantum hardware limitations, including decoherence, quantum noise, limited qubit availability, circuit depth, and the difficulty of designing efficient oracles. Despite these challenges, continuing progress in quantum hardware, error correction, cloud-based quantum platforms, and hybrid quantum-classical architectures suggests that quantum search will remain an important area of research. Overall, Grover’s algorithm represents a foundational quantum algorithm with significant theoretical value and potential long-term application in computational domains involving large and unstructured search spaces.

Fixed Point Theory in Partial Ordered Metric Spaces and Their Generalizations: A Systematic Review

Chandni Byapari — 2026-06-10

Fixed point theory has emerged as a fundamental area of modern functional analysis due to its significant role in establishing the existence, uniqueness, and approximation of solutions to a wide range of linear and nonlinear mathematical problems. Over the past two decades, substantial progress has been made in extending the classical Banach contraction principle through the introduction of generalized contractive conditions, novel metric structures, and auxiliary control functions. This review systematically examines recent developments in fixed point theory with particular emphasis on partial order metric spaces and their important generalizations, including partial metric, partial cone metric, partial b-metric, partial cone b-metric, and partial Aᵦ-metric spaces. Relevant literature published between 2000 and 2025 was identified through a structured literature review approach using major academic databases and predefined inclusion and exclusion criteria. The review is organized into four thematic areas: the role of fixed point theory in functional analysis; advances in generalized metric spaces; fixed point results in partial order metric spaces and their extensions; and recent developments in hybrid, integral-type, and generalized contraction mappings. The synthesis highlights significant theoretical advancements and their applications to differential equations, integral equations, optimization, and dynamic programming. Furthermore, several research gaps are identified, including the lack of a unified framework for partial-type spaces, limited investigations in partial Aᵦ-metric spaces, insufficient development of constructive iterative methods with convergence guarantees, and relatively few applications to fractional and Volterra-type integral equations. The review concludes by outlining promising directions for future research aimed at unifying existing theories and expanding the applicability of fixedpoint techniques to emerging mathematical and applied problems.

The Evolution of Software Containerization: Docker's Impact on DevOps and Microservices Architecture

Sanika Pravin Surve — 2026-06-29

Software containerization has changed how applications are built, packaged, deployed and operated, and Docker has been the principal force behind that change. This article presents a critical narrative review of the academic literature on the technical foundations of containerization, its orchestration through platforms such as Kubernetes, and its relationship with two closely associated movements in contemporary software engineering: DevOps and microservices architecture. The review traces the conceptual line from hypervisor-based virtualization to lightweight, operating-system-level isolation, then examines the empirical evidence on performance, resource efficiency, security and organisational adoption. Particular attention falls on how container technology has served as an enabling substrate for continuous integration and delivery pipelines, for the decomposition of monolithic systems into independently deployable services, and for the extension of cloud-native practice into edge, Internet of Things and serverless settings. The review also engages critically with contested claims in this literature, including the durability of reported performance advantages, the persistence of security and supply-chain risk, and the uneven evidence base behind many widely repeated benefits of DevOps and microservices adoption. The synthesis suggests that Docker's importance lies less in any single technical innovation than in its function as a standardising artefact that lowered the practical threshold for adopting distributed, service-oriented and automation-intensive delivery models. The article closes by identifying gaps in empirical rigour, a shortage of longitudinal and industrial-scale studies, and promising directions including container security taxonomies, AI-assisted operations and the convergence of serverless and container paradigms.