Search Results (324)

Most existing fractional models of COVID-19 describe only the infection process without explicitly accounting for the role of vaccination. In this study, a refined Caputo fractional model is proposed that incorporates a vaccinated class to better understand how immunization influences disease progression. The mathematical formulation guarantees the existence, uniqueness, and positivity of solutions, ensuring that all system trajectories remain biologically valid. The equilibrium points are obtained, and the reproduction number is derived to identify the conditions for disease control. The stability investigation covers local behavior alongside Ulam–Hyers and its extended variants, ensuring the system remains stable under small perturbations. Numerical experiments performed with the Adams–Bashforth–Moulton algorithm illustrate that vaccination reduces infection peaks and shortens the epidemic duration. Overall, the proposed framework enriches fractional epidemiological modeling by providing deeper insight into the combined effects of memory and vaccination in controlling infectious diseases. Full article

The spread of computer viruses poses a critical threat to networked systems and requires accurate modeling tools. Classical integer-order approaches had failed to capture memory effects inherent in real digital environments. To address this, we developed a four-compartment fractional-order model using the Atangana–Baleanu–Caputo (ABC) derivative with Mittag-Leffler kernels. We established fundamental properties such as positivity, boundedness, existence, uniqueness, and Hyers–Ulam stability. Analytical solutions were derived via Laplace transform and homotopy series, while the Variation-of-Parameters Method and a dedicated numerical scheme provided approximations. Simulation results showed that the fractional order strongly influenced infection dynamics: smaller orders delayed peaks, prolonged latency, and slowed recovery. Compared to classical models, the ABC framework captured realistic memory-dependent behavior, offering valuable insights for designing timely and effective cybersecurity interventions. The model exhibits structural symmetries: the infection flux depends on the symmetric combination $L+I$ and the feasible region (probability simplex) is invariant under the flow. Under the parameter constraint $\delta =\theta$ (and equal linear loss terms), the system is equivariant under the involution $(L,I)\mapsto (I,L)$, which is reflected in identical Hyers–Ulam stability bounds for the latent and infectious components. Full article

The objective of this manuscript is to investigate the existence, uniqueness criteria and Ulam–Hyers stability of solutions to tripled systems of the Hilfer–Hadamard type supplemented with symmetric nonlocal multi-point Riemann–Liouville integral boundary conditions. By converting the considered problem into an equivalent fixed-point problem, the existence and uniqueness are proven by application of the Leray–Schauder nonlinear alternative and Banach’s contraction principle, respectively. In addition, we discuss the Ulam–Hyers stability and generalized Ulam–Hyers stability of the results, and illustrative examples are also presented to demonstrate their correctness and effectiveness. Full article

Fractional differential equations offer a natural framework for describing systems in which present states are influenced by the past. This work presents a nonlinear Caputo-type fractional differential equation (FDE) with a nonlocal initial condition and attempts to describe a model of memory-dependent behavioral adaptation. The proposed framework uses a fractional-order derivative $\eta \in (0,1)$ to discuss the long-term memory effects. The existence and uniqueness of solutions are demonstrated by Banach’s and Krasnoselskii’s fixed-point theorems. Stability is analyzed through Ulam–Hyers and Ulam–Hyers–Rassias benchmarks, supported by sensitivity results on the kernel structure and fractional order. The model is further employed for behavioral despair and learned helplessness, capturing the role of delayed stimulus feedback in shaping cognitive adaptation. Numerical simulations based on the convolution-based fractional linear multistep (FVI–CQ) and Adams–Bashforth–Moulton (ABM) schemes confirm convergence and accuracy. The proposed setup provides a compact computational and mathematical paradigm for analyzing systems characterized by nonlocal feedback and persistent memory. Full article

This article presents several key findings for fractional-order delay differential equations. First, we establish the existence and uniqueness of solutions using two distinct approaches, the Chebyshev norm and the Bielecki norm, thereby providing a comprehensive understanding of the solution space. Notably, the uniqueness of the solution is rigorously demonstrated using the Lipschitz condition, ensuring a single solution under specific constraints. Additionally, we examine a specific form of constant delay and apply Burton’s method to further confirm the uniqueness of the solution. Furthermore, we conduct an in-depth investigation into the Hyers–Ulam stability of the problem, providing valuable insights into the behavior of solutions under perturbations. Notably, our results eliminate the need for contraction constant conditions that are commonly imposed in the existing literature. Finally, numerical simulations are performed to illustrate and validate the theoretical results obtained in this study. Fractional-order delay differential equations play a crucial role in real-life applications in systems where memory and delayed effects are essential. In biology and epidemiology, they model disease spread with incubation delays and immune memory. In control systems and robotics, they help design stable controllers by accounting for time-lagged responses and past behavior. Full article

We propose a hybrid Caputo–Lagrange Discretization Method (CLDM) for the fractional-order modeling of glucose–insulin dynamics. The model incorporates key physiological mechanisms such as glucose suppression, insulin activation, and delayed feedback with memory effects captured through Caputo derivatives. Analytical results establish positivity, boundedness, existence, uniqueness, and Hyers–Ulam stability. Numerical simulations confirm that the proposed method improves accuracy and efficiency compared with the Residual Power Series Method and the fractional Runge–Kutta method. Sensitivity analysis highlights fractional order $\theta$ as a biomarker for metabolic memory. The findings demonstrate that CLDM offers a robust and computationally efficient framework for biomedical modeling with potential applications in diabetes research and related physiological systems. Full article

The purpose of this paper is to investigate a class of novel symmetric coupled fuzzy fractional partial differential equation system involving the Caputo–Katugampola (C-K) generalized Hukuhara (gH) derivative. Within the framework of C-K gH differentiability, two types of gH weak solutions are defined, and their existence is rigorously established through explicit constructions via employing Schauder fixed point theorem, overcoming the limitations of traditional Lipschitz conditions and thereby extending applicability to non-smooth and nonlinear systems commonly encountered in practice. A typical numerical example with potential applications is proposed to verify the existence results of the solutions for the symmetric coupled system. Furthermore, we introduce Ulam–Hyers stability (U-HS) theory into the analysis of such symmetric coupled systems and establish explicit stability criteria. U-HS ensures the existence of approximate solutions close to the exact solution under small perturbations, and thereby guarantees the reliability and robustness of the systems, while it prevents significant deviations in system dynamics caused by minor disturbances. We not only enrich the theoretical framework of fuzzy fractional calculus by extending the class of solvable systems and supplementing stability analysis, but also provide a practical mathematical tool for investigating complex interconnected systems characterized by uncertainty, memory effects, and spatial dynamics. Full article

This study introduces a new class of coupled differential systems described by fractal–fractional Caputo derivatives with both constant and state-dependent delays. In contrast to traditional delay differential equations, the proposed framework integrates memory effects and geometric complexity while capturing adaptive feedback delays that vary with the system’s state. Such a formulation provides a closer representation of biological and physical processes in which delays are not fixed but evolve dynamically. Sufficient conditions for the existence and uniqueness of solutions are established using fixed-point theory, while the stability of the solution is investigated via the Hyers–Ulam (HU) stability approach. To demonstrate applicability, the approach is applied to two illustrative examples, including a predator–prey interaction model. The findings advance the theory of fractional-order systems with mixed delays and offer a rigorous foundation for developing realistic, application-driven dynamical models. Full article

We introduce a class of triply coupled systems of differential equations with fractal–fractional Caputo derivatives and time-dependent delays. This framework captures long-memory effects and complex structural patterns while allowing delays to evolve over time, offering greater realism than constant-delay models. The existence and uniqueness of solutions are established using fixed point theory, and Hyers–Ulam stability is analyzed. A numerical scheme based on the Adams–Bashforth method is implemented to approximate solutions. The approach is illustrated through a numerical example and applied to a three-species food-chain model, comparing scenarios with and without time-dependent delays to demonstrate their impact on system dynamics. Full article

We investigate the Hyers–Ulam–Rassias stability of a generalized quadratic functional equation of the asymmetric four-function form $F(x+y)+G(x-y)=L\left(x\right)+M\left(y\right)$, where F, G, L, and M are unknown mappings. This study is conducted within the framework of non-Archimedean normed spaces over the p-adic numbers. Our approach employs a separation technique, analyzing the even and odd parts of the functions to establish stability results. We show that all four functions are approximated by a combination of a quadratic function and two additive functions. Full article

In this paper, we investigate the Hyers–Ulam–Rassias stability of reciprocal functional equations in non-Archimedean fuzzy normed spaces by using both the direct method and the fixed point alternative. In addition, we study a modified reciprocal type functional equation within the same framework using Brzdȩk’s fixed point method. A brief remark is provided on the incidental role of symmetry in the structure of such functional equations. Finally, a comparative analysis highlights the distinctive features, strengths, and limitations of each approach. Full article

This paper investigates linear two-sided fractional matrix delay differential equations (TSFMDDE). Firstly, the two-sided fractional delayed Mittag-Leffler matrix functions (TSFDMLMF) are constructed. Further, the representation of solutions of two-sided homogeneous and nonhomogeneous problems are studied, and Ulam–Hyers (UH) stability of a two-sided nonhomogeneous problem is discussed. Lastly, we provide a numerical example to demonstrate our results. In the numerical example, the fractional order $\beta =0.6$, delay $\varrho =2$, UH constant $u_h\approx 5.92479$, $n=2$, and $s\in [-2,4]$. Full article

We present a unified framework for fuzzy statistical convergence of Grünwald–Letnikov (GL) fractional differences in Bag–Samanta fuzzy normed linear spaces, addressing memory effects and nonlocality inherent to fractional-order models. Theoretically, we establish the uniqueness, linearity, and invariance of fuzzy statistical limits and prove a Cauchy characterization: fuzzy statistical convergence implies fuzzy statistical Cauchyness, while the converse holds in fuzzy-complete spaces (and in the completion, otherwise). We further develop an inclusion theory linking fuzzy strong Cesàro summability—including weighted means—to fuzzy statistical convergence. Via the discrete Q-operator, all statements transfer verbatim between nabla-left and delta-right GL forms, clarifying the binomial GL↔discrete Riemann–Liouville correspondence. Beyond structure, we propose density-based residual diagnostics for GL discretizations of fractional initial-value problems: when GL residuals are fuzzy statistically negligible, trajectories exhibit Ulam–Hyers-type robustness in the fuzzy topology. We also formulate a fuzzy Korovkin-type approximation principle under GL smoothing: Cesàro control on the test set $\{1,x,x^2\}$ propagates to arbitrary targets, yielding fuzzy statistical convergence for positive-operator sequences. Worked examples and an engineering-style case study (thermal balance with memory and bursty disturbances) illustrate how the diagnostics certify robustness of GL numerical schemes under sparse spikes and imprecise data. Full article

In this study, we investigate implicit fractional differential equations subject to anti-periodic boundary conditions. The fractional operator incorporates two distinct generalizations: the Caputo tempered fractional derivative and the Caputo fractional derivative with respect to a smooth function. We investigate the existence and uniqueness of solutions using fixed-point theorems. Stability in the sense of Ulam–Hyers and Ulam–Hyers–Rassias is also considered. Three detailed examples are presented to illustrate the applicability and scope of the theoretical results. Several existing results in the literature can be recovered as particular cases of the framework developed in this work. Full article