In this paper, we present a mathematical model that describes the dynamics of a hepatitis C virus infection. The model takes two classes of target cells (hepatocytes and extrahepatic cells) into account as well as a saturated Cytotoxic T lymphocytes (CTLs) immune response in the liver. We found that the system solutions are positive and bounded, and the equilibrium points are computed. Through a careful examination of the model, we demonstrated that it exhibits global threshold dynamics that are fully captured by the threshold parameters. Numerical simulations were carried out to demonstrate the stability results.

We consider a discrete-time consensus dynamics of an opinion model with stubbornness under fixed and switching Topologies. Regarding the underlying network structures, both static and dynamically switching interaction topologies are considered. For the fixed topology, we prove that as long as the system parameters and initial states meet certain conditions, the presence of a rooted directed graph suffices for consensus. For the dynamically switching topology, under similar parametric conditions, consensus is guaranteed provided that the union of switched directed graphs over a bounded time interval contains a directed spanning tree. Numerical simulations are established to show the consensus and non-consensus phenomenon.

In this work, we present and analyze a high-precision numerical scheme to solve a class of variable-order fractional diffusion equations involving the Caputo-Fabrizio (CF) fractional derivative. The proposed scheme employs a finite difference method for temporal discretization and a local discontinuous Galerkin (LDG) method equipped with generalized alternating numerical fluxes for spatial discretization. For any variable-order function $\alpha(t) \in(0,1)$, we demonstrate the stability and L² convergence of the proposed scheme. Moreover, we show that the method attains an accuracy order of O($\tau^{2}+h^{k+1}$), where τ, h, and k denote the temporal step size, spatial step size, and degree of the piecewise P^k polynomials, respectively. A series of numerical experiments are carried out to verify the theoretical findings and the effectiveness of the proposed algorithm.

This paper investigates the existence and uniqueness of mild solutions for a class of fractional Langevin equations subject to multi-point closed boundary conditions. By applying Krasnosel’skii fixed point theorem and the Banach contraction mapping principle, we establish sufficient conditions which ensure existence and uniqueness. Furthermore, we analyze the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of the solutions within this framework. Illustrative examples are provided to demonstrate the applicability of the main results and to verify the stated hypotheses.

In this study, we investigate the dynamics of a within-host model of hepatitis B virus (HBV) and hepatitis C virus (HCV) coinfection with two distinct transmission pathways: virus-to-cell and cell-to-cell transmission. The model incorporates both HBV-specific and HCV-specific humoral immunity. Using the next-generation matrix method, we derive the basic reproduction numbers for HBV mono-infection ($\mathcal{R}_{0}^{B}$), HCV mono-infection ($\mathcal{R}_{0}^{C}$), and HBV/HCV coinfection ($\mathcal{R}_{0}$ = max{$\mathcal{R}_{0}^{B}$, $\mathcal{R}_{0}^{C}$}). We establish the fundamental qualitative properties of the model, identify eight possible equilibrium points, and investigate their global stability. The stability analysis is carried out using Lyapunov functions together with the Lyapunov–LaSalle invariance principle. The model is further formulated as a nonlinear control system, where anti-HBV and anti-HCV therapies serve as the control inputs. For each virus, two antiviral treatment strategies are considered: one that reduces virus-to-cell transmission and another that reduces cell-to-cell transmission. The results show that neglecting cell-to-cell transmission may underestimate the treatment effort required for viral clearance, highlighting the importance of incorporating this transmission pathway to obtain more realistic descriptions of within-host HBV/HCV dynamics. The optimal control problem is formulated to maximize the population of uninfected hepatocytes while minimizing treatment costs. Numerical simulations illustrate the theoretical results and demonstrate that the proposed control strategies can reduce viral infections and increase the population of uninfected hepatocytes under the considered parameter scenarios. Finally, sensitivity analysis of $\mathcal{R}_{0}^{B}$ and $\mathcal{R}_{0}^{C}$ identifies the parameters that have the greatest influence on the basic reproduction numbers, providing insights into the model mechanisms associated with viral persistence and clearance.

In the present article, a higher-order NIPG method on a Bakhvalov-type mesh is developed to solve a second-order singularly perturbed Volterra integro-differential equation. Drawing on the distinctive properties of Bakhvalov-type meshes, we construct a composite interpolation operator, which enables the derivation of parameter-uniform error estimates for the proposed NIPG method. It is shown that the accuracy of this method attains an order of k in the energy norm, where k denotes the degree of the piecewise polynomials employed in the finite element space. Numerical experiments are presented to confirm our theoretical results.

This paper investigated strongly nonlinear anisotropic elliptic problems with singular terms in variable exponent Sobolev spaces. The problem involves the Finsler p(·)-Laplacian operator $\Delta_{p(\cdot)}^{H} u=$ $\operatorname{div}\left(H^{p(\cdot)-1}(\nabla u) \nabla H(\nabla u)\right),$, where H is a uniformly convex Finsler norm. We considered the differential inclusion

with zero Dirichlet boundary conditions. Here ξ is a maximal monotone operator, G satisfies natural growth conditions, $f \in L^{\infty}(\Omega)$ is nonnegative, and 0 < Υ ≤ 1. Under log-Hölder continuity of p(·) and uniform convexity of H, we established existence of weak solutions via an approximation scheme combining truncation, Yosida regularization, and energy estimates. Key innovations include anisotropic monotonicity estimates, $L^{\infty}$ bounds, strict positivity, and strong convergence of gradients. Our results unify and extend previous work to encompass variable exponents, anisotropy, multivalued operators, and singular nonlinearities.

In this paper, we introduce a new class of polynomials called the mixed-type hypergeometric Bernoulli and Euler r-Dowling polynomials. These polynomials extend the classical Bernoulli and Euler polynomials, together with their hypergeometric (level-m) versions, while incorporating the combinatorial structure of the r-Dowling framework within a single generating function. Using exponential generating functions, we derive their main properties such as explicit formulas, recurrence relations, differential identities, and integral representations. In addition, we examine the zeros of these polynomials numerically and graphically. The results show that the zeros follow symmetric and structured patterns in the complex plane and display regular behavior as the degree increases. Overall, this study provides a natural extension of classical polynomial families and offers new insights for further research in approximation theory and related areas.

We present a dynamic analysis of the Gnedenko repairable system with multiple working vacations, which is comprised of N operating units and one warm standby unit. The system dynamics are governed by partial differential equations with integral boundary conditions. By applying the C₀-semigroup theory, we establish the well-posedness of the system and the strong convergence of the time-dependent solution to its steady-state solution. Numerical examples illustrate the effect of key parameters on time-dependnet reliability indices, providing practical insights into the system performance.

A robust multiscale finite element method is proposed for singularly perturbed convection–reaction–diffusion problems, featuring high accuracy, computational efficiency, and numerically observed parameter-uniform superconvergence. Within the multiscale finite element framework, multiscale basis functions are enriched with adequate local fine-scale information to enable global model reduction. The nontrivial relation between standard finite element basis functions and multiscale basis functions is analyzed and encoded in a global mapping matrix for practical implementation. By utilizing parameter-independent layer-adapted graded meshes, large errors induced by boundary layers are effectively suppressed. Numerical comparisons against existing approaches confirm the efficiency, reliability, and applicability of the presented scheme. Further numerical tests demonstrate this parameter-uniform superconvergence on graded meshes, as well as its promising capability for simulations in complex media and high-dimensional singularly perturbed systems.

This paper investigates influence of fractional order on the dynamic behavior of a cross- reaction diffusion model. Stability analysis determines the existence conditions of equilibrium points, and bifurcation analyses are performed. A numerical method combining high-order time-precision generating functions and a nine-point difference spatial discretization scheme is proposed, incorporating the short memory principle to improve computational efficiency. Numerical simulations demonstrate the spatial pattern formation of the system at different parameters, validating the effectiveness of the theoretical results. Some novel patterns are shown.

In this paper we developed an algebraic and geometric study of Fibonacci – Cartan numbers and Fibonacci Pauli–Cartan numbers by integrating Fibonacci sequences with the algebraic structure of Cartan numbers in pseudo–Euclidean spaces. Building on the concept of Fibonacci–Cartan numbers, we investigated their fundamental algebraic and geometric properties, including conjugation, character, norm, and causal classification, proving that they are spacelike elements and establishing their geometric interpretation within a Minkowski–type framework. Matrix, polar, and exponential representations were obtained, enabling the derivation of De Moivre-type identities, formulas for powers and roots, and characterizations in terms of the determinant and trace of real 2 × 2 matrices. In addition, the powers of Fibonacci–Cartan numbers were represented in terms of Chebyshev polynomials of the first and second kinds, yielding explicit polynomial expressions for their scalar and vector parts. Scalar and vector products were also analyzed, providing a consistent geometric interpretation in a three–dimensional Cartan space. Furthermore, Fibonacci Pauli–Cartan numbers were defined and studied through recurrence relations, Binet formulas, ordinary and exponential generating functions, summation formulas, and real matrix representations. Finally, rotational properties associated with lightlike directions were examined, showing that parabolic rotations degenerate into linear trajectories within the Cartan geometric framework.

This work develops and examines a fully discrete local discontinuous Galerkin (LDG) approach on graded temporal partitions for the tempered fractional advection-diffusion problem. The tempered fractional derivative is approximated via an L1-type formula over a nonuniform time mesh, which efficiently resolves the weak singularity near the initial instant. In space, an LDG discretization with purely alternating fluxes is employed, and its stability is established through an energy estimate. It is demonstrated that the resulting scheme is unconditionally stable and converges with order O($h^{k+1}+N^{-\min \{2-\alpha, r \alpha\}}$), where h denotes the spatial step, N is the number of time intervals, k stands for the polynomial degree, and r ≥ 1 is the grading parameter. Several numerical experiments are performed to validate the theoretical findings and to illustrate the benefit of graded meshes over uniform ones.