This research article is mainly focussed on a general problem of fractal-fractional differential equations (FFDEs) with double discrete delays. For This article investigates a class of fractal-fractional differential equations (FFDEs) involving two discrete delays. To establish the existence and uniqueness of solutions, we employ tools from functional analysis, including Krasnoselskii’s and Banach’s fixed point theorems. The stability of the proposed model is analyzed within the framework of Hyers-Ulam (H-U) stability theory. Furthermore, we construct a numerical approximation using an Adams-Bashforth-type method combined with Lagrange interpolation to handle the delayed arguments. Finally, we apply the theoretical results to biologically relevant models, including a two-delay logistic equation and a housefly population model, verifying the existence and stability of solutions and interpreting the numerical outcomes graphically. We have presented comparison between the Adams-Bashforth method and Rungge-Kutta method of order for (RK4).

This paper examines the existence and stability of an m-cyclic coupled system of higher-order fractional differential equations with non-singular kernels. Sufficient conditions for the existence and stability of solutions are obtained using fixed-point techniques. Two numerical examples involving coupled and triply coupled systems are presented to validate the theoretical results, and simulations of the triply coupled case illustrate the influence of different fractional orders on the system dynamics.

This paper develops a nonlinear fractal-fractional predator-prey model that incorporates logistic prey growth and immigration effects. The predator-prey interaction is characterized by a Holling type II functional response, capturing the saturation phenomenon in the predator's feeding rate. Using the Caputo-Fabrizio (CF) fractional operator, the model integrates memory effects into the population dynamics. Theoretical investigations establish the existence and uniqueness of solutions by applying Krasnoselskii's fixed point theorem and Banach's contraction principle, followed by the stability analysis of equilibrium points. For the numerical approximation, a modified Adams-Bashforth method adapted to the CF operator is employed. Simulation results reveal that small yet positive immigration rates promote asymptotically stable coexistence between prey and predator populations, emphasizing the stabilizing influence of immigration on ecosystem dynamics. The study demonstrates how fractal-fractional calculus can provide deeper insight into ecological stability and long-term behavior of interacting species.

Rotavirus remains a leading cause of gastroenteritis in children under five in low-and middle-income countries due to waning immunity and incomplete vaccine coverage. To address this, we propose a mathematical model to analyze the transmission dynamics with primary and booster vaccination strategies. The model is formulated using the fractal-fractional derivative in the Caputo-Fabrizio sense, which allows for the incorporation of memory effects and hereditary properties in disease evolution. The population is structured into five compartments, including booster-immunized individuals. We derive the disease-free and endemic equilibrium points and analyze their local stability. The basic reproduction number is computed to determine the threshold conditions for disease persistence. We establish the existence and Hyers-Ulam (H-U) stability of the model, and validate the results through numerical simulations using the Adams-Bashforth method (ABM), confirmed by comparison with Runge-Kutta 4th Order (RK-4) solution plots to assess the booster vaccination impact. The results reveal that booster immunization plays a significant role in reducing the infection burden, thereby highlighting its relevance in public health planning.

In this research article, we investigate a three-dimensional dynamical system governed by fractal-fractional-order evolution differential equations subject to terminal boundary conditions. We derive existence and uniqueness results using Schaefer’s and Banach’s fixed-point theorems, respectively. Additionally, the Hyers–Ulam stability approach is employed to analyze the system’s stability. We employ vector terminology for the proposed problem to make the analysis simple. To illustrate the practical relevance of our findings, we apply the derived results to a numerical example and graphically illustrate the solution for different fractal-fractional orders, emphasizing the effect of the derivative’s order on system behavior.

This research investigated a tri-trophic food chain model, incorporating carrying capacity and Holling-type predation, formulated using the fractal-fractional Caputo derivative. The four equilibrium states---trivial, prey-only, prey-predator, and coexistence---presented and their stability was discussed. Existence and uniqueness of solutions were established using Schaefer's and Banach's fixed point theorems. Also, stability requirements in the sense of Hyers-Ulam (H-U) were investigated. Numerical simulations were performed using the extended numerical method of Adams-Bashforth-Moulton (ABM), and comparative results were graphically presented to demonstrate the impact of varying fractal-fractional orders. A sensitivity analysis revealed how perturbations in individual parameters influence the model's outcome. The model accounts for memory and hereditary effects in ecological interactions. The proposed method enhances accuracy, stability, and convergence for long-time simulations compared to classical models.

We investigate a class of piecewise variable-order fractional differential equations with impulsive and nonlo cal conditions in Banach space. The nonhomogeneous term in the proposed system is given in terms of variable kernel which has flexibility property. We formulate appropriate equivalent integral equations to the considered evolution problem, then we show the solvability results by using mainly fractional calculus and fixed point techniques. Further, we study Hyers-Ulam stability analysis by adapting suitable conditions. The concerned area has numerous applications in those evolution processes and phenomenon, where abrupt changes occur. At the end, we support our obtained theory by illustrative and computational example.

In this work, we extended the classical Susceptible-Exposed-Vaccinated-Infected-Recovered (SEVIR) framework by incorporating an asymptomatic class, leading to the formulation of a new Susceptible-Exposed-Vaccinated-Asymptomatic-Infected-Recovered (SEVAIR) model that more accurately reflects the transmission characteristics of adenovirus. To account for memory effects and capture complex temporal behavior, the model was developed using the fractal-fractional Caputo-Fabrizio derivative with a power-law kernel. Existence and stability of solutions based on fixed point theory and Hyers-Ulam stability criteria were derived. Both the disease-free and endemic equilibrium states were derived, and their local stability properties were examined. Additionally, the basic reproduction number was computed to understand the disease's spread threshold. The theoretical results were supported by numerical simulations, which were performed using a modified Adams-Bashforth approach tailored for the fractional framework.

Fractal-fractional calculus has attracted much attention in the last few years due to its wide range applications in various disciplines of science and technology. The class of differential equations with order lying in (2, 3] has significant importance in physics because the concerned equations are usually used to represent jerk or jolt-type phenomenon in real-world applications. Therefore, we consider a class of fractal-fractional differential equations with fractional order p is an element of (2, 3] and fractals dimension q is an element of (0, 1]. The mentioned problem involves proportional-type delay term called Pantograph. We also develop the existence and stability theory for the problem under our consideration using fractal-fractional Riemann-Liouville differential operator. On using the fixed point approach due to Schaefer's and Banach's, we investigate appropriate conditions for the existence of at least one solution and its unique. In addition, the Hyers-Ulam (HU) approach of stability is used for stability analysis. Keeping in mind the importance of numerical analysis, Haar wavelets procedure is used to compute some numerical results for the considered problem. Finally, the established results are applied to a numerical problem to explain its validity. Hence, various graphical results related to different fractals and fractional orders have been presented.

This paper investigates a general class of variable-kernel discrete delay differential equations (DDDEs) with integral boundary conditions and impulsive effects, analyzed using Caputo piecewise derivatives. We establish results for the existence and uniqueness of solutions, as well as their stability. The existence of at least one solution is proven using Schaefer’s fixed-point theorem, while uniqueness is established via Banach’s fixed-point theorem. Stability is examined through the lens of Ulam–Hyers (U-H) stability. Finally, we illustrate the application of our theoretical findings with a numerical example.

This paper addresses a class of multi-point initial value problems with impulses and proportional delays. The framework is based on the Atangana–Baleanu–Caputo (ABC) fractional derivative, which allows the model to incorporate hereditary memory effects absent in standard integer-order systems. Using suitable fixed-point arguments, conditions ensuring the existence and uniqueness of solutions are derived. The reliability and robustness of the obtained solutions is analyzed through the Hyers–Ulam (H-U) method and generalized H-U stability. To demonstrate the theoretical findings, a general numerical example model and a fractional-order housefly population model are considered that incorporate impulsive effects and delay terms reflecting real ecological feedback. Numerical simulations illustrate how variations in the fractional order and impulse intensity influence the dynamic behavior of the adult population. The results reveal that impulsive interventions can effectively regulate population oscillations, while the fractional order governs the rate of decay and long-term stability of the system.

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.

In this study, we proposed a modified SEVIR-S (susceptible, exposed, vaccinated, infected, recovered) model for the transmission dynamics of adenovirus by incorporating the effects of immunity waning and reinfection. Unlike the classical SEVIR framework, the extended model accounted for the possibility that recovered individuals may lose immunity over time and become susceptible again — a critical feature for accurately modeling diseases like adenovirus. To better capture the disease's memory effects and temporal dynamics, the model used the fractal-fractional Caputo-Fabrizio derivative with a power-law kernel. The paper analyzed the model's existence and stability using fixed point theory and Hyers-Ulam (H-U) stability. Furthermore, both the disease-free and endemic equilibrium points and their stability were analyzed. Also, the basic reproduction number was provided. The findings were validated through numerical simulations using an extended Adams-Bashforth method.

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.

In the present research, we consider a biological model of serum hepatitis disease. We carry out a detailed analysis of the mentioned model along with a class with asymptomatic carriers to explore its theoretical and numerical aspects. We initiate the study by using the piecewise fractal–fractional derivative (FFD) by which the crossover effects within the model are examined. We split the time interval into subintervals. In one subinterval, FFD with a power law kernel is taken, while in the second one, FFD with an exponential decay kernel of the proposed model is considered. This model is then studied for its disease-free equilibrium point, existence, and Hyers–Ulam (H-U) stability results. For numerical results of the model and a visual presentation, we apply the Lagrange interpolation method and an extended Adams–Bashforth–Moulton (ABM) method, respectively.

This research article investigates a tripled system of nonlinear fractional differential equations with n terms. The study explores this novel class of differential equations to establish existence and stability results. Utilizing Schaefer’s and Banach’s fixed point theorems, we derive sufficient conditions for the existence of at least one solution, as well as a unique solution. Furthermore, we apply Hyers–Ulam stability analysis to establish criteria for the stability of the system. To demonstrate the applicability of the main results, a detailed example is provided.

In this paper, we study human liver disease with a different approach of interval-based investigation by introducing subintervals. This investigation may be referred to as a short memory investigation. Such concepts are useful in problems where a transition is observed when transitioning from one subinterval to the other one. We use the classical and fractal-fractional-order derivative in each subinterval. We study the existence of solutions by using Banach’s and Krasnoselskii’s fixed-point theorems. Their stability is analyzed by adopting the Hyers–Ulam (H-U) stability approach. Also, using the extended Adams–Bashforth–Moulton (ABM) method, we simulate the results that visually present the numerical solutions for different fractal-fractional-order values.

In recent years, the fractals (Hausdorff) derivatives with fractional order under various types kernel have gained attention from researchers. The aforesaid area has many applications in the description of intricate and irregular geometry of various processes. Numerous studies utilizing the fractional derivatives (HFDs) for initial value problems have been carried out. But the boundary value problems using the said concepts have been very rarely studied. Thus, a coupled system with non-homogenous boundary conditions (BCs) is examined in this study by using fractals fractional derivative in Caputo Fabrizio sense. To establish the required conditions for the existence and uniqueness of solution to the considered problem, we apply the Banach and Krasnoselskii’s fixed point theorems. Furthermore, some results related to Hyers-Ulam (H-U) stability have also deduced. We have included two pertinent examples to verify our results.

In this research paper, we study a coupled system of piecewise-order differential equations (DEs) with variable kernel and impulsive conditions. DEs with variable kernel have high flexibility due to the freedom of changing the kernel. We study existence and stability theory and derive sufficient conditions for main results of the proposed problem. We apply Scheafer’s fixed point theorem and Banach fixed point theorem for the result of at least one and unique solution, respectively. In addition, stability results based on the Ulam–Hyers concept are derived. Being a coupled system of piecewise fractional-order DEs with variable kernel and impulsive effects, the obtained results have multi-dimension applications. To demonstrate the applications, we apply the derived results to a numerical problem.

In this research work, we investigate a class of Atangana–Baleanu–Caputo (ABC) differential equations (DEs) having proportional and discrete delay terms accompanied by integral boundary conditions. We study the two important aspects, that is, the existence of the solution and stability analysis for the problem mentioned above. We confirm the existence of at least one solution via Schaefer's fixed‐point theorem and its uniqueness by Banach's fixed‐point theorem. Moreover, for stability analysis, we use the concept of Hyers–Ulam stability. For application purposes, we apply the main results to a numerical problem for specific values of the parameters involved.