ABSTRACT This study proposes a numerical framework for solving fractional differential equations with nonuniform step sizes using the Caputo derivative formulation. The suggested scheme, known as the Fractional Explicit Method (FEM), improves on the traditional third‐order Adams–Bashforth approach by integrating Lagrange interpolation to approximate the history term in fractional operators. The formulation allows for varied time increments, increasing flexibility for problems where uniform discretization is impractical. A theoretical examination of the method is performed, including derivation of its order of accuracy and convergence properties, and the method is compared numerically against an established fractional predictor–corrector method. As an illustrative example, the approach is applied to a fractional dynamical model of smoking habit, and the resulting solutions are studied using graphical representations. Conditional stability of the scheme is also discussed via the linear test equation, though a full characterization of the stability region for variable step sizes is left for future work. The results show that the suggested approach is a reliable and efficient tool for numerically treating fractional differential systems with nonuniform temporal discretization.
ABSTRACT In this article, we investigate an inverse problem of identifying a time‐dependent coefficient in a time‐fractional diffusion equation with two‐point boundary conditions. The unknown coefficient is recovered from an overdetermination condition prescribed at the midpoint of the spatial interval. Since the corresponding boundary conditions are not strongly regular, the direct Fourier method cannot be applied to the original spectral problem. We overcome this difficulty by decomposing the solution into symmetric and antisymmetric parts and reducing the original problem to two auxiliary problems.
ABSTRACT This paper presents a geometric and algebraic analysis of hyperbolic hybrid numbers, which combine structural features of the complex, dual, and hyperbolic number systems. Based on the algebraic properties of the hybrid number set, hyperbolic rotations in both the plane and space are investigated. The study constructs mutually orthogonal hyperbolic planes in the hybrid setting and examines the left‐hand, right‐hand, and sandwich multiplication maps that produce hyperbolic rotational transformations. In addition, Rodrigues and Cayley type formulas are formulated within the hybrid number system to obtain explicit hyperbolic rotation matrices in three dimensions. These results show that hyperbolic hybrid numbers provide a consistent algebraic setting for modeling Lorentzian‐type geometries and clarify the relationship between algebraic operations and geometric motions.
ABSTRACT This paper develops an efficient numerical algorithm for solving the generalized time‐fractional Burgers equation subject to three classes of boundary conditions, whose solutions possess weak singularities in the vicinity of the initial instant . Traditional numerical approaches employ uniform temporal grids, which inevitably results in deteriorated numerical accuracy. To overcome this defect, the nonuniform ‐ formula is utilized for the discretization of the Caputo fractional derivative. For the spatial component, the radial basis function finite difference scheme is adopted. The established algorithm attains a global convergence accuracy of , with representing the mesh grading parameter, and standing for the order of the Caputo fractional derivative. The stability of the presented scheme is strictly proven through energy analysis, and its convergence property is also investigated systematically. A series of numerical tests and comparative analyses with existing benchmark methods verify the remarkable performance of the proposed algorithm. Moreover, the computational efficiency is demonstrated by the CPU times.
ABSTRACT In this paper, we analyze the boundedness of solutions to nonhomogeneous first‐order linear delay dynamic equations on time scales containing a term with the forward shifted delay function, and we examine the stability of the corresponding homogeneous equation. Three kinds of stabilities are investigated with the application of an appropriate Lyapunov‐Krasovskii functional. In addition, a local jump condition is introduced to facilitate the results. Suitable examples are provided to check the applicability of the results.
ABSTRACT This article addresses the critical challenge of estimating time‐varying faults in nonlinear one‐sided Lipschitz systems under external disturbances. A novel disturbance‐decoupled adaptive sliding‐mode observer is proposed, integrating sliding‐mode robustness with gradient‐based adaptation laws to simultaneously track system states and dynamic fault parameters. By leveraging null‐space projection techniques, the observer decouples disturbances from fault signals, ensuring robust estimation performance. The design is underpinned by a Lyapunov‐based stability analysis, yielding linear matrix inequality (LMI) conditions that guarantee asymptotic state error convergence and ultimate boundedness of parameter estimation errors. Theoretical advancements are validated through a comprehensive quadrotor UAV case study, demonstrating efficiency in disturbance rejection and steady‐state fault estimation accuracy. Comparative analysis reveals 62% faster convergence than existing constant‐fault observers, while Monte Carlo simulations confirm robustness against 20% initialization errors and input noise. The framework bridges a significant gap in fault‐tolerant control by providing rigorous solutions for systems with evolving fault dynamics.
ABSTRACT We study a bioconvective flow model in a porous medium with a Brinkman‐Forchheimer nonlinear damping term. Such a model arises naturally in biomedical applications, including bacterial transport in tissues, infection propagation, and biofluid dynamics in porous biological structures. We prove the global‐in‐time existence of weak solutions. Under suitable assumptions on the Forchheimer exponent and the initial microorganism concentration, the solution depends continuously on the initial data and is therefore unique. Moreover, for a range of parameters, we obtain uniform‐in‐time bounds for the velocity and the concentration and show that the concentration vanishes as time tends to infinity.
ABSTRACT In this paper, we present a novel, highly accurate, and efficient numerical method for solving fractional partial differential equations with proportional delays. These equations pose significant challenges in both numerical and analytical analysis due to their inherent nonlinearity and delay parameters. The proposed approach addresses these difficulties by transforming this equation into an equivalent Volterra integral equation using a collocation method and the matrix representation of the fractional integral operator. This process creates a nonlinear system of algebraic equations, and solving it results in an approximate solution to the original problem. The error analysis is investigated, and the results confirm that the method is convergent. Numerical experiments validate the theoretical results and confirm the accuracy and computational efficiency of the proposed approach. Furthermore, comparative studies show that the method outperforms existing numerical techniques in terms of accuracy.
ABSTRACT We develop a framework for abstract integro‐differential equations with state‐dependent integration intervals and demonstrate its applicability to partial differential equations. Using Banach's contraction principle and the Schauder fixed‐point theorem, we prove local existence and uniqueness and establish local well‐posedness for both mild and strict solutions. The paper concludes with examples drawn from population dynamics that illustrate the theory.
ABSTRACT In this study, we explore the quasilinear two‐species chemotaxis system with two chemicals where () is a smooth bounded domain. The functions and exhibit asymptotic behavior of the form We prove that when is a ball, if and , there exist radially symmetric initial data and , such that the corresponding solutions blow up in finite time; for any general smooth bounded domain , if , all solutions are global. The result in [X. Huang and H. Zhong, Blow‐up curve for a quasilinear two‐species chemotaxis system with two chemicals in higher dimension, Mathematical Methods in the Applied Sciences, 2026] implies that all solutions are globally bounded when Combining these results, we point out that the system () possesses two critical lines and to classify three dynamics among global boundedness, finite‐time blow‐up, and global existence of solutions to system ().
ABSTRACT To address the limitations of classical fishery population models (e.g., Ricker and Beverton–Holt) in capturing complex interspecific interactions and lag effects, this article constructs an improved two‐species, discrete dynamic model, where the parameters are positive real numbers and the initial values are arbitrary nonnegative real numbers. We rigorously analyze the dynamic behaviors of the system, establishing that positive solutions are uniformly bounded and strictly persistent under specific intrinsic growth conditions ( and ). Furthermore, we provide a comprehensive classification of the boundary dynamics, exploring non‐hyperbolic and 1:1 resonant scenarios. We derive sufficient conditions for global asymptotic stability (GAS): (1) the trivial fixed point (extinction) is GAS via direct inequality bounding when intrinsic growth rates satisfy and , and (2) utilizing the mean value theorem (MVT) and Lyapunov function methods, the unique positive fixed point (coexistence) is GAS when growth rates lie within the non‐chaotic range () under weak coupling. Numerical simulations validate these theoretical derivations and demonstrate that the improved model provides a more generalized framework for describing mutual inhibition compared to previously proposed models. These findings offer quantitative support for balanced harvesting strategies in sustainable fishery management.
ABSTRACT In this article, we study an exact boundary controllability problem for a wave equation with a potential that varies in the space on non‐smooth domains of , . The potential considered is a positive space variable function which has the short‐range property, that is when . The obtained control function is of Neumann type, square integrable and acts on all boundary of the considered domain. The result is obtained using the controllability method established by David L. Russell, taking advantage of the local energy decay estimates for the problem established by Vodev, combined with trace theorems due to Daniel Tataru.
ABSTRACT This article investigates the existence of coexistence states for a strongly coupled cooperative elliptic system with cross‐diffusion, modeling two mutualistic species. By employing the method of upper and lower solutions coupled with monotone iterative techniques, we establish sharp sufficient conditions for the existence of positive coexistence states, revealing that strong interspecific cooperation and sufficiently large intrinsic growth rates can overcome the competitive exclusion induced by cross‐diffusion in hostile environments. A novel transformation is introduced to handle the strongly coupled structure, and numerical simulations are provided to validate the main theoretical findings.
ABSTRACT This paper investigates the dynamic behavior of a discrete‐time predator–prey model incorporating a square root functional response, which captures the herding behavior of prey populations. By applying Euler discretization to the continuous‐time model, we obtain a system of difference equations that serves as the foundation for a detailed bifurcation analysis. The study focuses on codimension‐two bifurcations, specifically the 1:2, 1:3, and 1:4 strong resonances, which are analyzed both analytically and numerically. Using center manifold theory and a series of affine transformations, we derive the normal forms associated with each resonance. These normal forms reveal the underlying bifurcation structures and provide conditions for the occurrence of complex dynamical phenomena. Numerical continuation methods, implemented via MATCONT, are employed to validate the analytical results and to visualize bifurcation curves and phase portraits. The findings contribute to the broader understanding of discrete dynamical systems in population ecology, highlighting the rich bifurcation scenarios that arise from parameter variations.
ABSTRACT In this paper, an efficient conformal multi‐symplectic structure‐preserving method is proposed for two‐dimensional damped nonlinear Schrödinger equation. The method decomposes the original equation into several tractable one‐dimensional subproblems by the splitting method, and then solves them separately and alternatively, and combines the resulting schemes by an appropriate method lastly. Moreover, the spatial derivatives are discretized with a high‐order compact scheme which uses fewer grid nodes to achieve higher accuracy and improve the computational efficiency. Based on this framework, an efficient conformal multi‐symplectic algorithm is developed for the equation. Some numerical experiments are reported to verify the effectiveness of the new scheme.
ABSTRACT We consider immuno‐epidemiological SIS models. The models are based on two factors: the dynamics of the immune status and interactions between susceptible and infected individuals. The starting point is an individual‐based model describing interactions between individuals. Through an appropriate limit passage with the number of individuals to infinity, we obtain a model acting on measures describing the distribution of immune status in both epidemic groups. We show that the densities of these measures satisfy a system of partial differential equations with a jump operator. This system generates a continuous semigroup of nonlinear operators on the space of densities. The main results are the existence and uniqueness of an endemic state and a method for determining stationary distributions. We also present simulations of the distribution of the number of infected people over time and the stationary distributions.
This paper investigates the well-posedness and vanishing viscosity limit for three-dimensional incompressible viscoelastic fluids in a 3D periodic slab. The fluid velocity is subject to Navier-slip boundary conditions, while the deformation tensor satisfies a specific boundary condition. We first establish the existence of global weak solutions. By deriving uniform estimates in Sobolev spaces that are independent of the viscosity coefficients, we analyze the convergence of the viscous solutions to the inviscid ones as the viscosities tend to zero. Furthermore, for sufficiently small initial data, we prove the global existence and uniqueness of classical solutions for both the viscoelastic Navier-Stokes equations and the elastodynamics system. A key result for the inviscid system is the exponential decay of its -norm. As an application of this decay property, we establish time-independent convergence rates for the vanishing viscosity limit.
This paper offers a thorough analytical and numerical analysis of nonlinear dynamics under the Ablowitz-Kaup-Newell-Segur (AKNS) water wave equation, particularly in its possible applicability to advanced fiber materials. Utilizing an appropriate wave transformation, the governing nonlinear partial differential equation is simplified into an ordinary differential equation, which is solved exactly using the generalized Riccati equation mapping method (GREMM). The closed-form solutions form the basis for a systematic study of the system's rich dynamical features such as equilibrium states, bifurcations, multi-stability, and chaos. Bifurcation analysis identifies the key transitions, demonstrating the routes from stable periodic oscillations to irregular and chaotic states. Numerical simulations also confirm these transitions, demonstrating the system's evolution between regular, quasi-periodic, and fully developed chaotic states. Illustrations like phase portraits, temporal evolution of series, Lyapunov exponents, Poincar & eacute; sections, and recurrence plots attest to the existence of chaos, fractal-like self-similarity, and huge sensitivity to initial conditions. Multi-stability, which is the coexistence of more than one attractor under the same parameter conditions, highlights the tenuous balance of nonlinear responses in complex systems. These results are most directly applicable to high-performance fiber materials, in which wave propagation, stress-strain instabilities, and nonlinear dynamic responses have a profound impact on performance and reliability. The revealed bifurcation and chaos mechanisms shed light on how fiber composites, smart fibers, and adaptive materials could switch between different mechanical states under different loads, impacts, or environmental inputs. Mapping the entire bifurcation topology of the system provides a predictive platform for designing fibers with improved vibration damping, impact resistance, and energy absorption properties. In general, this work demonstrates the strength of linking analytical exact solutions with numerical computations in revealing the subtle interaction among nonlinearity, parameter variation, and dynamic behavior, providing insightful viewpoints toward designing and controlling next-generation fiber materials and fiber-reinforced structures. The novelty of this work lies in the integration of exact analytical solutions with advanced chaos diagnostics, providing deeper insight into the interplay between nonlinearity and system dynamics, and extending beyond existing studies by offering both a broader solution spectrum and a comprehensive dynamical characterization.
Elastic waves provide improved dynamic properties and the ability to manipulate material behavior in micro and nano-structured materials. This novel work deals with the vibrations and wave propagation in such a structure using the continuum theory. The characteristics of wave propagation and longitudinal vibrations of nanorods and carbon nanotubes (CNTs) in spatiotemporal nanostructures are studied using the Eringen stress and gradient elasticity theory. Systems of differential equations are derived to describe the proposed physical problem. These equations are solved using computational methods to investigate the physical properties of the nonlocal nanomaterials, considering the effect of utilizing nonlocal microscopic parameters and dispersion properties that characterize material behavior at the nanoscale. The model equations are solved using analytical methods to investigate the physical properties of the nonlocal nanomaterials and their effects on wave propagation. Additionally, it involves the examination of elastic curvature and displacement, which signify the nonlocal response influenced by various effects from adjacent points. The nonlocal parameters are determined based on values adopted from established literature, and the proposed model is validated through rigorous comparisons with previously published theoretical and numerical investigations. This approach ensures the accuracy and reliability of the analytical solutions in predicting the dynamic response of nanostructured materials. Furthermore, the phase-plane and eigenvalue analyses demonstrate the coherence and mutual validation of Eringen's nonlocal stress theory and the strain-gradient elasticity models within a unified dynamical framework.
In this article, we provide complete asymptotic expansions of the linear positive operators associated with the generalized Laguerre and Touchard polynomials, respectively. We also provide specific methods to obtain better order of approximation by these operators. Moreover, we establish quantitative difference estimates for these operators with respect to their components through moduli of smoothness. Finally, we investigate their rates of convergence through graphical illustrations.
