This paper investigates the controllability of semilinear neutral stochastic Atangana-Baleanu (AB) Caputo fractional differential equations (FDEs) under integral boundary conditions with impulses. The primary results are derived from principles and techniques of measures of noncompactness, semigroup theory, [Formula: see text]-set contraction, fractional calculus, and stochastic theory. Finally, the key theoretical findings are verified with an illustration.
This paper presents a fractional-order financial model integrating interest rates, investment demand, and price dynamics. The model is formulated as a three-dimensional system of Caputo fractional differential equations involving key economic parameters for saving amount, investment cost, elasticity of demand, A government control parameter to model economic stimulus and time varying critical rate to account for periodic economic shocks or policy cycles. Stability analysis via linearization and eigenvalue techniques confirms the local stability of the set of some equilibrium points. Notably, the fractional order [Formula: see text] introduces memory effects, expanding the system’s dynamic behavior beyond classical models. Existence of solution of governing equations model is provided using Banach fixed point theorem. Numerical simulations indicate conditional stability under the criterion [Formula: see text], and reveal cyclical patterns driven by complex eigenvalue pairs. These findings highlight the nuanced influence of fractional derivatives in financial modeling.
Malware propagation in the Industrial Internet poses a serious threat to critical infrastructure. Classical homogeneous models often overlook node-level differences in spreading capability. To address this gap, a DomiRank-[Formula: see text] heterogeneous SIS model is proposed. The model maps each node’s neighborhood dominance score to a node-specific infection rate. This construction provides an interpretable link between network structure and propagation capability. A series of theoretical analyses is carried out to investigate the dynamical behavior of the system. The model is evaluated through numerical and stochastic simulations, comparisons with centrality-based heterogeneous baselines, empirical validation on Witty and Code Red traces, and benchmarking against data-driven methods. The results show that topology-informed infection-rate heterogeneity changes outbreak thresholds and propagation trajectories. They also show that the proposed construction reveals sustained spreading risks that may be underestimated by homogeneous and conventional centrality-based models. This framework provides a new perspective for studying malware propagation.
In this work, we introduce a novel co-infection mathematical model of HIV and MTB infections by combining ART and MDR-TB. The population is divided into eight classes: susceptible individuals ([Formula: see text]), HIV-infected ([Formula: see text]), latent TB ([Formula: see text]), active TB ([Formula: see text]), co-infected ([Formula: see text]), MDR-TB cases ([Formula: see text]), individuals on ART ([Formula: see text]) and recovered individuals ([Formula: see text]). The well-posedness and boundedness of the model are established. Stability of the equilibrium points is analyzed via the basic reproduction number and LaSalle's invariance principle. Sensitivity analysis identifies the most influential parameters through numerical and graphical methods. Simulations using the RK4 method illustrate the model's dynamics. Furthermore, the impact of ART scale-up and MDR-TB on HIV, TB and co-infection classes is examined. The results highlight effective strategies, emphasizing ART's role in reducing HIV progression and TB susceptibility, and the importance of MDR-TB. This framework provides insights into the dynamics of co-infection and supports public health interventions.
In this study, a new numerical approach is presented by combining the differential quadrature method with particle swarm optimization for an efficient solution of nonlinear partial differential equations. The proposed methodology is illustrated using the Burgers’ equation, which is used to model complicated physical phenomena including convection, diffusion, and reaction processes. Optimization methods are used in the presented technique to find the best parameter selection in the spline-based basis functions that considerably increase the accuracy and stability of the solution. The effectiveness of the framework is guaranteed through a wide range of numerical experiments, which are in excellent agreement with analytical/exact solutions. Stability analysis according to the Courant-Friedrichs-Lewy criterion ensures the stability of the proposed method. The ability of method to capture detailed nonlinear behaviors renders it a potential agent for treating a broad array of convection-diffusion-reaction problems in applied science and engineering.
This work investigates wave symmetry in bacterial pattern formation for the generalized Chavy-Waddy-Kolokolnikov (CWK) model, a nonlinear framework based on partial differential equations (PDEs). In microbiology, investigating how bacterial colonies develop spatial structures is crucial to analyzing their adaptive behaviors. By employing the Hirota bilinear transformation, we constructed numerous symmetric wave structures such as solitons, homoclinic breathers, periodic cross-kink, lumps, and M-shaped waves with kink and rogue properties. These structures help in explaining how bacterial populations organize and react to stimulation. This work employs 3D, contour, and density plots to reveal the complex dynamics of such structures. In a nutshell, this work gives new a perspective into the physical behavior of bacterial colonies and contributes to a broader understanding of pattern formation in biological systems with potential relevance to ecological and evolutionary studies.
This paper presents an innovative parallel computational methodology employing adaptive fractional differential operators to address Monkeypox epidemiological modeling challenges. The complex dynamics of such models often present computational challenges. To address these, we develop a productive hybrid graphics processing unit-central processing unit (GPU-CPU) parallel approach. Using fractional differentiation operators, the presented approach focuses on managing the memory component of disease transmission dynamics. Our parallel method significantly reduces the calculation time and improves the overall performance of solving the Monkeypox disease model by utilizing the computing capabilities of both GPU and CPU cores. In this paper, we provide a hybrid variable-order fractional derivative that combines the variable-order fractional Caputo derivative with the integral of Riemann-Liouville. A predictor-corrector method with discretization of the Caputo proportional constant variable-order fractional hybrid operator is applied for numerical solutions. Julia, a high-level programming language, was chosen to implement the hybrid parallel technique. According to the results of the simulation, parallel approaches significantly increase productivity and efficiency.
Vertex and edge operations are very popular tools in studying several properties of graphs, as they help us to calculate complex statements by means of easier or well-known ones. Deletion is probably the most important graph operation. If the vertex to be deleted is a cut vertex, the situation is much simpler as the number of components is increased and the remaining components are easier to deal with. The classical vertex removal operation is studied in relation to several topological graph indices, independence number, chromatic number and many other graph parameters. Here, similar methods are applied to study the magnetic separation and vertex separation of graphs.In this paper, motivated by the usefulness of this deletion operation, we introduce two more novel vertex removal operations, named as magnetic separation and vertex separation and study the effect of these three operations for several indices and graphs.
Inverse rising factorial moments M-n are fundamental quantities for positive integer-valued random variables, with broad applications in probability and combinatorics. Despite their importance, there is a lack of explicit, closed-form algebraic identities connecting M-n to established combinatorial quantities and a limited framework for their multivariate extension. Addressing these gaps, this study investigates M-n to derive more computationally efficient alternatives. We establish both a novel finite sum identity and an infinite series expression for M-n. Furthermore, we provide a closed-form expression for the geometric random variable and extend these results to multivariate random variables. Finally, we extend the concept of M-n to positive real-valued continuous random variables, deriving a finite sum identity for the beta random variable.
In this paper, within the framework of unsigned degenerate r-Stirling numbers, we establish connections among Cauchy polynomials, degenerate Bernoulli polynomials, and degenerate hyperharmonic numbers. The key results include: (i) expressing the values of Cauchy polynomials at nonnegative integers using the unsigned degenerate r-Stirling numbers of the first kind, and inverting these expressions via the inversion relation with the degenerate r-Stirling numbers of the second kind; (ii) representing the values of Cauchy polynomials at nonnegative integers in terms of the values of fully degenerate Bernoulli polynomials, and vice versa, using either kind of degenerate r-Stirling numbers; (iii) introducing degenerate Cauchy polynomials and deriving an identity that connects them to degenerate hyperharmonic numbers.
Chemical mechanical polishing (CMP) is a critical process in semiconductor fabrication, where controlling slurry behavior at the pad-wafer interface is essential for achieving uniform material removal. This study investigates how variations in the relative rotational speed between the CMP pad and polishing head affect slurry transport characteristics, including replacement rate, coverage, and spatial uniformity. CFD simulations were performed, modeling the slurry as an incompressible Newtonian fluid, and the pad-wafer interface was divided into five concentric zones for zone-specific analysis. Increasing pad speed improved slurry replacement by up to 42.6% but reduced coverage by about 20%. This zone-based approach revealed localized nonuniformities not captured by global averages, emphasizing the need for spatially resolved evaluation. Complementing these quantitative results, qualitative flow analyses clarified how relative motion impacts slurry behavior, particularly highlighting nonuniformities in the outer interface zones. Our robust evaluation framework offers a practical basis for improving slurry dynamics in the CMP.
Recently, there has been significant research on the generalization of various numbers using probabilistic methods. In this paper, we introduce the probabilistic higher order Frobenius-Euler polynomials which are a generalization of Frobenius-Euler polynomials using probabilistic methods and demonstrate that these polynomials can be expressed as a linear combination of probabilistic Stirling numbers of the second kind and falling factorial sequences which provide both practical applications and profound insights into these polynomials. Furthermore, we show that when Y is a Bernoulli, gamma, or Poisson random variable, it can be expressed as a linear combination of the Stirling numbers of the second kind, Bell polynomials, or rising factorial sequences, respectively, and derive several new and interesting identities of those polynomials. We also investigate properties of the polynomials Hn(k,Y)(x|u) using graphs for different values of the random variable Y, two different integers k, and three different integers n.
This paper presents a mathematical model to study diabetes mellitus disease by incorporating fractional derivatives in the Caputo sense, which generalizes classical integer-order models through improved representation of memory effects. The model incorporates fractional time delay and treatment function to enable a more comprehensive analysis of the disease. First, we determine the model's essential equilibria and establish its positivity and boundedness. A qualitative analysis of approximate solutions has been carried out using fixed point theory which is well-suited for handling non-local and nonlinear aspects of fractional order systems. We have performed computational simulations by using an Adams-Bashforth-Moulton-type predictor-corrector approach for different fractional order values. These findings facilitate to advance our understanding of the dynamics of diabetes and can inform the development of effective public health initiatives.
In this paper, we generalize the classical Hermite-Hadamard, Ostrowski, and Simpson type integral inequalities by using Lidstone interpolation polynomials and their associated Green functions. By incorporating the Riemann-Liouville fractional integral operator, we construct novel Ostrowski and Simpson-type inequalities whose remainder estimates are explicitly expressed in terms of fractional integrals and Lidstone-Green kernels. Our results contribute to the interplay between fractional calculus, interpolation theory, and classical analysis.
This study offers novel (k,Psi)-Hilfer version of Wirtinger-type estimates in the context of Lp spaces for p>1 by using H & ouml;lder's inequality. Numerous special cases are also presented to emphasize the generality of the results. The consistency of the presented results is verified through graphical demonstrations. Moreover, the work examined Wirtinger-type estimates for arithmetic and geometric mean inequalities, with mathematical implications that extend to economic analyses where such means characterize growth and distributional properties. The results show that these inequalities have underlying symmetric structures that connect fractional calculus to balanced mean relations. In addition to improving theoretical comprehension, these symmetric viewpoints offer a cohesive foundation for applications in both economics and mathematics.
Hepatitis B virus (HBV) remains a major public health concern because of its chronic progression and severe liver complications. We propose a compartmental model describing HBV transmission that incorporates vaccination, antiviral treatment, liver transplantation and environmental contamination, accounting for both direct and indirect transmission pathways. The disease-free and endemic equilibria are derived and their local and global stability properties are analyzed. A sensitivity analysis is conducted to assess the impact of key parameters, including the infection rate, vaccination coverage and environmental persistence, on the basic reproduction number R0 and system dynamics. Numerical simulations using the fourth-order Runge-Kutta method and a nonstandard finite difference scheme validate the theoretical results and illustrate the effectiveness of combined control strategies. These findings provide useful insights for improving HBV prevention and control policies.
This study invokes the interval valued analysis of factional order linear Volterra integro-differential equations in Riemann Liouville senses. To validate the presented results, particular cases are discussed with graphical illustrations.
This study develops a mathematical model for the transmission dynamics of the Chikungunya (CHIK) virus using the Caputo-Fabrizio fractal-fractional derivative (CFFFD), which incorporates memory effects and complex transmission patterns not captured by traditional models. The fractal-fractional approach accounts for memory effects and long-range dependence in disease transmission, allowing more accurate representation of real-world dynamics in which current states depend on past history. This framework improves the prediction of outbreak patterns and supports a more thorough assessment of control measures such as vaccination and vector control. The existence and uniqueness of solutions are established using the Krasnoselskii and Banach fixed-point theorems. Numerical simulations are performed via the Adams-Bashforth method, together with stability and sensitivity analyses. Graphical results illustrate the effects of the fractional order (theta) and fractal dimension (kappa) on CHIK transmission dynamics. The findings provide useful insights for public health policymakers.
Building on Carlitz’s foundational work with degenerate Euler and Bernoulli polynomials, recent research has introduced and studied various degenerate special numbers and polynomials. This paper focuses on exploring degenerate Stirling numbers of the second kind. Specifically, we investigate properties, identities, recurrence relations, and explicit expressions related to these numbers. We derive several expressions for a degenerate version of sums of powers of consecutive integers and the sums related to them.
The energy transition calls for innovative thermal power generation systems with low- or zero-emission and highly flexible operation. Dynamic modelling and simulation are key enabling factors in this field, as controlling such plants is a difficult task for which there is no previous experience and very short design times are expected. The steady-state initialization of such dynamic models is, unfortunately, a critical task involving the solution of large systems of nonlinear equations with iterative methods, which are often prone to failure. In this work, several strategies and methodologies are discussed to robustly achieve steady-state initialization of first-principles equation-based, object-oriented models of advanced thermal power generation systems. These are presented with reference to the Modelica language, but are applicable to any equation-based, object-oriented modelling and simulation environment. The successful application of such strategies and methodologies to the SOS-CO2 advanced power generation system is presented.
