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Research Article
Existence and Uniqueness of Mild Solutions for Neutral Stochastic Fractional Sum-Difference Equations with Impulses Driven by a Rosenblatt Process
Yogesh Hanmant Shirole*,
Suryakant Muralidhar Jogdand
Issue:
Volume 14, Issue 5, October 2026
Pages:
277-286
Received:
7 June 2026
Accepted:
14 July 2026
Published:
9 September 2026
Abstract: This paper investigates the existence and uniqueness of mild solutions for a class of neutral stochastic fractional difference equations with impulsive effects in a Hilbert space framework, driven by a Rosenblatt process. The considered system is formulated using the Caputo fractional difference operator and incorporates delay terms, impulsive effects, and stochastic perturbations exhibiting long-range dependence. The primary objective is to establish sufficient conditions ensuring the existence and uniqueness of mild solutions for the proposed system. To achieve this objective, the theory of resolvent operators is combined with stochastic analysis techniques and the Banach fixed point theorem. In particular, an appropriate operator is constructed from the mild solution formulation, and suitable conditions are imposed to guarantee its contractive property. Consequently, the existence and uniqueness of a mild solution are established. The obtained results extend and generalize several existing results for fractional differential and stochastic systems to the discrete fractional setting involving Rosenblatt stochastic processes. The proposed framework effectively accounts for the combined influence of fractional memory, delays, impulsive effects, and long-range-dependent stochastic disturbances. Furthermore, an application to a class of impulsive stochastic partial fractional difference equations is presented to demonstrate the applicability and effectiveness of the theoretical results. The application verifies that the established assumptions can be satisfied in a relevant stochastic fractional model. Thus, the results contribute to the qualitative theory of neutral stochastic fractional difference equations and provide a useful framework for the analysis of discrete-time stochastic systems with memory, delay, impulsive phenomena, and long-range dependence. These findings may also serve as a basis for further investigations of more general stochastic fractional difference systems.
Abstract: This paper investigates the existence and uniqueness of mild solutions for a class of neutral stochastic fractional difference equations with impulsive effects in a Hilbert space framework, driven by a Rosenblatt process. The considered system is formulated using the Caputo fractional difference operator and incorporates delay terms, impulsive effe...
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Research Article
Controlling the Transmission Dynamics of Monkey Pox Infection: A Mathematical Model Approach
Issue:
Volume 14, Issue 5, October 2026
Pages:
287-294
Received:
3 August 2026
Accepted:
17 August 2026
Published:
18 September 2026
Abstract: The spread of Monkey Pox (Mpox) poses significant public health challenges, requiring effective intervention strategies to mitigate its impact. This study propose a mathematical model to explore the transmission dynamics of Mpox and assess the impact of quarantine as a control measure. The model divides the population into compartments representing Susceptible, Exposed, Infectious, Quarantined, and Recovered individuals. The system of ordinary differential equations describing the dynamics of the infection were derived. The equilibrium states of the model equations: Disease free equilibrium and Disease endemic equilibrium states were obtained. The stability analysis of the disease free equilibrium was analyzed and found it to be stable. The reproduction number (R0) was obtained and its numerical value was computed. Numerical simulations were performed to investigate how varying quarantine rates affect the progression of the disease across these compartments. The simulations explore the implications of different quarantine intensities on the number of infectious and exposed individuals over time. The results obtained indicate that quarantine can effectively reduce the transmission rate of the infection. Also, it revealed the potential of quarantine to reduce the disease spread and alleviate its overall impact on the population. The findings offered a framework for understanding the dynamics of Mpox transmission and assist public health authorities in designing effective intervention strategies.
Abstract: The spread of Monkey Pox (Mpox) poses significant public health challenges, requiring effective intervention strategies to mitigate its impact. This study propose a mathematical model to explore the transmission dynamics of Mpox and assess the impact of quarantine as a control measure. The model divides the population into compartments representing...
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Research Article
On the Structural Properties of Translations of Neutrosophic Fuzzy Ideals in B-Algebras
Issue:
Volume 14, Issue 5, October 2026
Pages:
295-302
Received:
18 July 2026
Accepted:
30 July 2026
Published:
20 September 2026
Abstract: In this article, we introduce and investigate the concepts of neutrosophic fuzzy ideals and their translations in B-algebras. We establish the fundamental properties of translations of neutrosophic fuzzy ideals and prove that the translation of a neutrosophic fuzzy ideal of a B-algebra is also a neutrosophic fuzzy ideal. Furthermore, we characterize neutrosophic fuzzy ideals through their level sets and show that a neutrosophic fuzzy set is a neutrosophic fuzzy ideal if and only if its corresponding level sets are also ideals. We also provide sufficient conditions under which the neutrosophic translation of a neutrosophic fuzzy ideal forms a neutrosophic fuzzy subalgebra. In addition, we introduce the concept of an extension of a neutrosophic fuzzy ideal and prove that the neutrosophic translation of a neutrosophic fuzzy ideal is a neutrosophic fuzzy ideal extension. An illustrative example is presented to demonstrate that not every neutrosophic fuzzy ideal extension is obtained as a translation of a neutrosophic fuzzy ideal. Moreover, we define the multiplication of neutrosophic fuzzy ideals and investigate its properties in B-algebras. We prove that the multiplication of neutrosophic fuzzy ideals is also a neutrosophic fuzzy ideal under appropriate conditions. The results establish meaningful relationships among neutrosophic fuzzy ideals, translations, level sets, extensions, and multiplication in B-algebras. These findings contribute to the development of neutrosophic fuzzy algebraic structures and provide a systematic framework for further investigations of ideals and their associated operations in B-algebras. The proposed characterizations also clarify structural behavior and broaden the existing theoretical framework for neutrosophic fuzzy algebraic systems.
Abstract: In this article, we introduce and investigate the concepts of neutrosophic fuzzy ideals and their translations in B-algebras. We establish the fundamental properties of translations of neutrosophic fuzzy ideals and prove that the translation of a neutrosophic fuzzy ideal of a B-algebra is also a neutrosophic fuzzy ideal. Furthermore, we characteriz...
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Research Article
Absolutely k-Harmonious Labeling of Product Graphs
Kailasam Annathurai*
,
Murugan Gomathi
Issue:
Volume 14, Issue 5, October 2026
Pages:
303-306
Received:
8 June 2026
Accepted:
8 June 2026
Published:
22 September 2026
Abstract: Graph labeling assigns integers to vertices, edges, or both of a graph subject to prescribed balance conditions, and has broad applications in coding theory, network design, and communication protocols. In this paper we study the absolutely k-harmonious labeling (ABH labeling) of product graphs. A vertex labeling assigns to every vertex an integer label drawn from the residues modulo k, and every edge is then given an induced label formed from the absolute difference between the sum of its two endpoint labels and k, reduced modulo k. A labeling is called absolutely k-harmonious if, across all label classes, the vertex counts carrying each label differ from one another by at most one, and the edge counts carrying each label likewise differ by at most one. We prove that the absolutely k-harmonious property is preserved under three principal graph product operations ? the Cartesian product, the tensor product, and the strong product ? of two absolutely k-harmonious graphs, provided suitable divisibility conditions hold. We further prove that the property is preserved under graph complementation, when the order and the size of the graph are each divisible by k, and under disjoint union. Concretely, we establish that grid graphs formed as Cartesian products of two paths, and complete bipartite graphs whose part sizes sum to a multiple of three, are absolutely 3-harmonious for all suitable orders. All theorems are derived rigorously from the defining balance conditions and each is accompanied by a fully worked numerical example. The restriction to the case k equal to 3 for concrete graph families is natural because the modulo-3 cyclic structure aligns with the ternary residue classes of path indices, and extensions to larger values of k are identified as directions for future research.
Abstract: Graph labeling assigns integers to vertices, edges, or both of a graph subject to prescribed balance conditions, and has broad applications in coding theory, network design, and communication protocols. In this paper we study the absolutely k-harmonious labeling (ABH labeling) of product graphs. A vertex labeling assigns to every vertex an integer ...
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Research Article
A Modified Algorithm for Broyden Family Using Natural Cubic Spline Interpolation Polynomial
Issue:
Volume 14, Issue 5, October 2026
Pages:
307-330
Received:
23 August 2026
Accepted:
5 September 2026
Published:
24 September 2026
Abstract: The approximation of the objective function's second-derivatives matrix underlies the Broyden family (BF) of unconstrained optimization methods, and richer gradient information generally yields a more accurate approximation. This paper proposes a new optimization technique that replaces the traditional two-point, secant-based linear model of the gradient with a Natural Cubic Spline Interpolation Polynomial (NCSIP), constructed using either three points (M=2) or four points (M=3). The proposed method was implemented in MATLAB and tested against the traditional Broyden family method (M=1) on a set of standard unconstrained test problems across the range The traditional method (M=1) recorded a total of 11564 iterations and 14867 function/gradient evaluations, while the four-point NCSIP model (M=3) achieved a clear efficiency improvement, with totals of 11063 iterations and 14082 function/gradient evaluations; the improvement achieved by the three-point model (M=2) was comparatively modest (11627 iterations and 14750 function/gradient evaluations). The best performance of the M=3 model was observed at higher values of the parameter Φ (near Φ=1), where it clearly outperformed the traditional method. The proposed method was also compared against the related Newton Divided Difference Interpolation (NDDI) method, using its corresponding three-point (M=4) and four-point (M=5) variants; the results showed a marginal numerical advantage of NCSIP over NDDI in total function/gradient evaluations when using four points (14082 vs. 14187). Taken together, these findings suggest that the number of gradient evaluations exploited, rather than the specific interpolation scheme, is the primary driver of efficiency gains. It should be noted that the algorithm's convergence properties are inferred from its algebraic reduction to the classical secant-based Broyden equation near the minimum, rather than established through a formal convergence proof.
Abstract: The approximation of the objective function's second-derivatives matrix underlies the Broyden family (BF) of unconstrained optimization methods, and richer gradient information generally yields a more accurate approximation. This paper proposes a new optimization technique that replaces the traditional two-point, secant-based linear model of the gr...
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Research Article
Numerical Investigation of Solutions of Chaotic Systems Via Domain Decomposition Technique
Issue:
Volume 14, Issue 5, October 2026
Pages:
331-338
Received:
14 August 2026
Accepted:
31 August 2026
Published:
27 September 2026
Abstract: Chaotic systems, such as the Lorenz and Chen systems, models exhibit high sensitivity to initial conditions. They require fine discretization for accurate long-time integration. Classical time-stepping methods suffer from accumulated truncation errors, whereas, global spectral methods, though highly accurate, yield dense coefficient matrices that are computationally expensive, limiting their efficiency. In this paper we propose an efficient numerical scheme that combines the non-overlapping Domain Decomposition Method (DDM) with the Spectral Relaxation Method (SRM) to solve the Lorenz and Chen systems. The time domain is decomposed into smaller sub intervals, and SRM is applied in each subdomain to linearize the nonlinear terms iteratively, thereby avoiding Taylor series truncation errors while maintaining spectral accuracy. The performance of the proposed Multistage Spectral Relaxation Method (MSRM) is investigated for different step sizes h = 0.01 and h = 0.001 over a long integration time T= 10. Numerical results are compared with the single-domain SRM in terms of accuracy, CPU time and memory usage. The results demonstrate that MSRM achieves comparable accuracy to the single-domain approach while significantly reducing computational time and memory requirements, especially for smaller step sizes. The proposed MSRM provides an effective and efficient framework for the simulation of chaotic systems over a long-time interval.
Abstract: Chaotic systems, such as the Lorenz and Chen systems, models exhibit high sensitivity to initial conditions. They require fine discretization for accurate long-time integration. Classical time-stepping methods suffer from accumulated truncation errors, whereas, global spectral methods, though highly accurate, yield dense coefficient matrices that a...
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Research Article
Mathematical Model Analysis of Marriage Divorce Incorporating Counselling and Medical Attention
Felicien Habiyaremye*
Issue:
Volume 14, Issue 5, October 2026
Pages:
339-347
Received:
28 August 2026
Accepted:
10 September 2026
Published:
27 September 2026
Abstract: This paper has investigated the mathematical model analysis of marriage divorce taking into consideration the effect of counselling and medical attention in order to minimize the divorce cases. The research aimed at developing the deterministic mathematical model of marriage divorce incorporating the counselling and medical attention(treatment), determining the divorce free and endemic equilibrium points, determining the sensitivity analysis and finding numerical simulation. The system of the differential equations of the model was proved to be well posed which showed that the mathematical model of marriage divorce was re-searchable. The differential equations were linearized using Jacobean matrix and It was found that the divorce free equilibrium point to be locally asymptotically stable when the reproduction number is less than one, and divorce endemic equilibrium point to be globally asymptotically unstable when the reproduction number is greater than one, . The sensitivity analysis was done to get the numerical values of the parameters involved in the model. The numerical simulations were done using Runge Kutta fourth order and MATLAB in order to generate the graphs. It was found that the number of divorced individuals reduced as results of individuals got counselling sessions and medical attention (treatment). It was equally observed, the active interaction between married and divorced individuals resulted in the increase of the divorced individuals due to the fact that the reproduction number is greater than one. The number of married individuals increased as a result of individuals got counselling sessions and separated individuals renewed their relationship. This research is of the great help of the community that with good communication, assume responsibility in the family, fidelity as couples, considering diversity in family and avoiding violence of any kind in the family as key factors of avoiding divorce in marriage.
Abstract: This paper has investigated the mathematical model analysis of marriage divorce taking into consideration the effect of counselling and medical attention in order to minimize the divorce cases. The research aimed at developing the deterministic mathematical model of marriage divorce incorporating the counselling and medical attention(treatment), de...
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Research Article
Backward One-Step Block Hybrid Numerical Method for Solving Second-Order Initial Value Problems
Issue:
Volume 14, Issue 5, October 2026
Pages:
348-358
Received:
18 August 2026
Accepted:
18 August 2026
Published:
28 September 2026
Abstract: This study develops a backward hybrid block method for solving oscillatory second-order initial value problems (IVPs), which arise frequently in engineering and the physical sciences. Existing hybrid methods often struggle with stiff-oscillatory systems, lacking the robustness needed for accurate and stable solutions. To overcome this limitation, the proposed method incorporates backward-step information to significantly improve both accuracy and stability. The derivation employs multistep collocation and interpolation techniques, with the Chebyshev polynomial of the first kind serving as the basis function. This choice of basis function is particularly effective for approximating oscillatory behavior. Unknown parameters are efficiently determined using Gaussian elimination. The resulting continuous scheme is then evaluated at selected points to obtain the discrete block method. A detailed theoretical analysis establishes the method's order, error constant, consistency, and zero-stability, collectively confirming its convergence. To validate its performance, the method is applied to standard oscillatory test problems. Numerical results demonstrate that the proposed method achieves superior accuracy compared to existing methods in the literature. The findings confirm that this backward-step hybrid block method offers a robust, reliable, and computationally efficient solution for oscillatory IVPs. Its accuracy and stability make it a valuable tool for researchers and practitioners dealing with oscillatory problems in engineering and the physical sciences.
Abstract: This study develops a backward hybrid block method for solving oscillatory second-order initial value problems (IVPs), which arise frequently in engineering and the physical sciences. Existing hybrid methods often struggle with stiff-oscillatory systems, lacking the robustness needed for accurate and stable solutions. To overcome this limitation, t...
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