Theory Of Fractional Differential Equations Research Articles

Analyzing crop production: Unraveling the impact of pests and pesticides through a fractional model

The continuous growth of the human population raises concerns about food, fiber, and agricultural insecurity. Meeting the escalating demand for agricultural products due to this population surge makes protecting crops from pests becomes imperative. While farmers use chemical pesticides as crop protectors, the extensive use of these chemicals adversely affects both human health and the environment. In this research work, we formulate a nonlinear mathematical model using the Caputo fractional (CF) operator to investigate the effects of pesticides on crop yield dynamics. We assume that pesticides are sprayed proportional to the density of pest density and pests not entirely reliant on crops. The feasibility of every possible nonnegative equilibrium and its stability characteristics are explored utilizing the stability theory of fractional differential equations. Our model analysis reveals that in a continuous spray approach, the roles of pesticide abatement rate and pesticide uptake rate can be interchanged. Furthermore, we have identified the optimal time profile for pesticide spraying rate. This profile proves effective in minimizing both the pest population and the associated costs. To provide a practical illustration of our analytical findings and to showcase the impact of key parameters on the system’s dynamics, we conducted numerical simulations. These simulations are conducted employing the generalized Adams–Bashforth–Moulton method, which allowed us to vividly demonstrate the real-world implications of our research.

Analyzing crop production: Unraveling the impact of pests and pesticides through a fractional model

The continuous growth of the human population raises concerns about food, fiber, and agricultural insecurity. Meeting the escalating demand for agricultural products due to this population surge makes protecting crops from pests becomes imperative. While farmers use chemical pesticides as crop protectors, the extensive use of these chemicals adversely affects both human health and the environment. In this research work, we formulate a nonlinear mathematical model using the Caputo fractional (CF) operator to investigate the effects of pesticides on crop yield dynamics. We assume that pesticides are sprayed proportional to the density of pest density and pests not entirely reliant on crops. The feasibility of every possible nonnegative equilibrium and its stability characteristics are explored utilizing the stability theory of fractional differential equations. Our model analysis reveals that in a continuous spray approach, the roles of pesticide abatement rate and pesticide uptake rate can be interchanged. Furthermore, we have identified the optimal time profile for pesticide spraying rate. This profile proves effective in minimizing both the pest population and the associated costs. To provide a practical illustration of our analytical findings and to showcase the impact of key parameters on the system’s dynamics, we conducted numerical simulations. These simulations are conducted employing the generalized Adams–Bashforth–Moulton method, which allowed us to vividly demonstrate the real-world implications of our research.

Mathematical modelling of COVID-19 outbreak using caputo fractional derivative: stability analysis

The novel coronavirus SARS-Cov-2 is a pandemic condition and poses a massive menace to health. The governments of different countries and their various prohibitory steps to restrict the virus's expanse have changed individuals' communication processes. Due to physical and financial factors, the population's density is more likely to interact and spread the virus. We establish a mathematical model to present the spread of the COVID-19 in worldwide. In this article, we propose a novel mathematical model (‘ S L I I q I h R P ’) to assess the impact of using hospitalization, quarantine measures, and pathogen quantity in controlling the COVID-19 pandemic. We analyse the boundedness of the model's solution by employing the Laplace transform approach to solve the fractional Gronwall's inequality. To ensure the uniqueness and existence of the solution, we rely on the Picard-Lindelof theorem. The model's basic reproduction number, a crucial indicator of epidemic potential, is determined based on the greatest eigenvalue of the next-generation matrix. We then employ stability theory of fractional differential equations to qualitatively examine the model. Our findings reveal that both locally and globally, the endemic equilibrium and disease-free solutions demonstrate symptomatic stability. These results shed light on the effectiveness of the proposed interventions in managing and containing the COVID-19 outbreak.

Analyzing vegetation pattern formation through a time-ordered fractional vegetation-sand model: A spatiotemporal dynamic approach

<p>This paper contributes to the field by developing a fractional-order vegetation-sand model that incorporates memory effects into the traditional integer-order framework. By studying the spatiotemporal dynamics of a time-order fractional vegetation-sand model, the research aimed to deepen our understanding of the complex interactions between vegetation and sand environments, providing insights for effective management and conservation strategies in arid and semi-arid regions. First, using the linear stability theory of fractional differential equations, we conducted a stability analysis of the spatially homogeneous fractional-order vegetation-sand model and provided the parametric conditions for stability and instability. Next, we performed a stability analysis of the spatiotemporal model, utilizing Turing instability to reveal the effects of diffusion and fractional order on vegetation distribution. Through numerical simulations, we demonstrated the spatiotemporal evolution patterns of the model under different environmental conditions and discussed the implications of these dynamic changes for ecological restoration and land management.</p>

BIFURCATION CONTROL STRATEGY FOR A DELAYED FRACTIONAL-ORDER POPULATION DYNAMICS MODEL WITH INCOMMENSURATE ORDERS

This paper excogitates a bifurcation control strategy for a delayed fractional-order population dynamics model with incommensurate orders. First and foremost, by using stability theory of fractional differential equations, the sufficient conditions for the stability of the positive equilibrium are established. It is not difficult to find that the fractional-order system has a wider stability region than the traditional integer-order system. Second, taking time delay as bifurcation parameter, the sufficient criteria for Hopf bifurcation are obtained. In the next place, it is interesting to introduce a delayed feedback controller to guide Hopf bifurcation. The results reveal that the bifurcation dynamics of the model could be effectively controlled as long as the delay or fractional order is carefully adjusted. In conclusion, numerical simulations are carried out to confirm our theoretical results.

Complex Dynamics Analysis and Chaos Control of a Fractional-Order Three-Population Food Chain Model

The present study investigates the stability analysis and chaos control of a fractional-order three-population food chain model. Previous research has indicated that the predation relationship within a long-established predator–prey system can be influenced by factors such as the prey’s fear of the predator and its carry-over effects. This study examines the state evolution of fractional-order systems and compares their dynamic behavior with integer-order systems. By utilizing the Routh–Hurwitz condition and the stability theory of fractional differential equations, this paper establishes the local stability conditions of the model through the application of the Jacobi matrix and eigenvalue method. Furthermore, the conditions for the Hopf bifurcation generation are determined. Subsequently, chaos control techniques based on the Lyapunov stability theory are employed to stabilize the unstable trajectory at the equilibrium point. The theoretical findings are validated through numerical simulations. These results enhance our understanding of the stability properties and chaos control mechanisms in fractional-order three-population food chain models.

On Solutions of Fractional Integrodifferential Systems Involving Ψ-Caputo Derivative and Ψ-Riemann–Liouville Fractional Integral

In this paper, we investigate a new class of nonlinear fractional integrodifferential systems that includes the Ψ-Riemann–Liouville fractional integral term. Using the technique of upper and lower solutions, the solvability of the system is examined. We add two examples to demonstrate and validate the main result. The main results highlight crucial contributions to the general theory of fractional differential equations.

Ulam Stability of Fractional Hybrid Sequential Integro-Differential Equations with Existence and Uniqueness Theory

In this paper, a variety of boundary value problems (BVPs) known as hybrid fractional sequential integro-differential equations (HFSIDs) with two point orders (p,q) are investigated. The uniqueness and existence of the solution are discussed via Banach fixed-point theorems. Certain particular theorems associated with Hyers–Ulam and Hyers–Ulam–Rassias stability to the solution, as well as the uniqueness and existence of the solution of the BVPs are studied. The results are illustrated with some particular examples, and the numerical data are analyzed for confirmation of the results. The results obtained in this work are simple and can easily be applicable to physical systems. Furthermore, symmetry analysis of fractional differential equations and HFSIDs are also presented. This is due to the fact that the aforementioned analysis plays a significant role in both the optimization and qualitative theory of fractional differential equations.

Stability of Boundary Value Discrete Fractional Hybrid Equation of Second Type with Application to Heat Transfer with Fins

The development in the qualitative theory of fractional differential equations is accompanied by discrete analog which has been studied intensively in recent past. Suitable fixed point theorem is to be selected to study the boundary value discrete fractional equations due to the properties exhibited by fractional difference operators. This article aims at investigating the stability results in the sense of Hyers and Ulam with application of Mittag–Leffler function hybrid fractional order difference equation of second type. The symmetric structure of the operators defined in this article is vital in establishing the existence results by using Krasnoselkii’s fixed point theorem. Banach contraction mapping principle and Krasnoselkii’s fixed point theorem are employed to establish the uniqueness and existence results for solution of fractional order discrete equation. A problem on heat transfer with fins is provided as an application to considered hybrid type fractional order difference equation and the stability results are demonstrated with simulations.

Power Flow Calculation in Smart Distribution Network Based on Power Machine Learning Based on Fractional Differential Equations

Abstract Based on the theory of fractional differential equations, this paper proposes a simple recursive, iterative scheme for power flow calculation in pure radial networks. The paper determines the network hierarchy formed by the ADT stack through breadth theory. This helps us define the branch sequence of the forward and backward generation in the power flow calculation of the smart distribution network. We ensure that the Jacobian matrix remains unchanged in the smart distribution grid power flow calculation. The interval model is more practical and computationally simpler than the point model. The research results show that the power flow calculation method is efficient based on the fractional differential equation.

Stability for the Systems of Ordinary Differential Equations with Caputo Fractional Order Derivatives

Fractional calculus has paid much attention in recent years, because it plays an essential role in many fields of science and engineering, where the study of stability theory of fractional differential equations emerges to be very important. In this paper, the stability of fractional order ordinary differential equations will be studied and introduced the backstepping method. The Lyapunov function is easily found by this method. This method also gives a guarantee of stable solutions for the fractional order differential equations. Furthermore it gives asymptotically stable.

Nonlinear fractional differential equations with advanced arguments

In this paper, we develop the existence and uniqueness theory of fractional differential equation involving Riemann-Liouville differential operator of order $0 < alpha< 1$, with advanced argument and integral boundary conditions. We investigate the uniqueness of the solution by using Banach fixed point theorem, we apply the comparison result to obtain the existence and uniqueness of solution by monotone iterative technique also by using weakly coupled extremal solution for the nonlinear boundary value problem (BVP). As an application of this technique, existence and uniqueness results are obtained.

On solutions of nonlinear BVPs with general boundary conditions by using a generalized Riesz–Caputo operator

In this work, we study the existence, uniqueness, and continuous dependence of solutions for a class of fractional differential equations by using a generalized Riesz fractional operator. One can view the results of this work as a refinement for the existence theory of fractional differential equations with Riemann–Liouville, Caputo, and classical Riesz derivative. Some special cases can be derived to obtain corresponding existence results for fractional differential equations. We provide an illustrated example for the unique solution of our main result.

EXISTENCE RESULTS FOR ABC-FRACTIONAL DIFFERENTIAL EQUATIONS WITH NON-SEPARATED AND INTEGRAL TYPE OF BOUNDARY CONDITIONS

This paper presents a study on the existence theory of fractional differential equations involving Atangana–Baleanu (AB) derivative of order [Formula: see text], with non-separated and integral type boundary conditions. An existence result for the solutions of given AB-fractional differential equation is proved using Krasnoselskii’s fixed point theorem, while the uniqueness of the solution is obtained using Banach contraction principle. Some conditions are proposed under which the given boundary value problem is Hyers–Ulam stable. Examples are given to validate our results.

Asymptotic stability analysis of Riemann-Liouville fractional stochastic neutral differential equations

The novelty of our paper is to establish results on asymptotic stability of mild solutions in $p$th moment to Riemann-Liouville fractional stochastic neutral differential equations (for short Riemann-Liouville FSNDEs) of order $\alpha \in (\frac{1}{2},1)$ using a Banach's contraction mapping principle. The core point of this paper is to derive the mild solution of FSNDEs involving Riemann-Liouville fractional time-derivative by applying the stochastic version of variation of constants formula. The results are obtained with the help of the theory of fractional differential equations, some properties of Mittag-Leffler functions and asymptotic analysis under the assumption that the corresponding fractional stochastic neutral dynamical system is asymptotically stable.

S-Asymptotically ω-Periodic Solutions of Fractional-Order Complex-Valued Recurrent Neural Networks With Delays

In this paper, we consider the problem of the S-asymptotically ω-periodic synchronization of fractional-order complex-valued recurrent neural networks with time delays. Firstly, we can not explicitly decompose the fractional-order complex-valued systems into equivalent fractional-order real-valued systems, by means of the contraction mapping principle and some important features of Mittag-Leffler functions, we obtain some sufficient conditions for the existence and uniqueness of S-asymptotically ω-periodic solutions for this class of neural networks. Then, by constructing an appropriate Lyapunov functional, the theory of fractional differential equation, and some inequality techniques, sufficient conditions are obtained to guarantee the global Mittag-Leffler synchronization of the drive-response systems. Finally, two examples are given to illustrate the effectiveness and feasibility of our main results.

Population Coding of Generative Neuronal Cells for Collaborative Decision Making in UAV-Based SLAM Operations

This study presents a modeling method for control system of multi-coupled nonlinear cyber physical system with the help of second-order dynamics of communicating membrane system of neuronal population based on a population coding algorithm. Here, the communicating membrane system is modeled based on the theory of fractional differential equations. Here, the challenge relies on the development of a collaborative multi-coupled system with self-learning attributes, which is essential for finding a collaborative workspace between data streams thereby giving increase in high-dimensional decision space. Therefore, to curb this issue, a gradient-based approach is appropriate with co-simulation features to enable several distributed units to be controlled and collaborative for decision-making applications. Here, at every interval of the sampling time, all the subsystems are optimally synchronized in P population system by multiset-rewriting rules including the effect of symport/antiport systems. This remains a vital problem, to classify and mathematically patch the borderline interjection between universality and non-universality of continuity of distributed control system. The presented study discusses the experimental proof of the algorithmic framework and model for SLAM operation. The consensus architecture in this domain will enable the development of evolving architecture of a nonlinear cyber physical system. The closed-loop stability and the recursive feasibility of the evolved architecture are also studied.

IP Traffic Modelling in Scalable Drone Network for Carrying Scalable Logistic Operations

This study presents a self modelling method for control system of multi-coupled non-linear scalable drones based logistic services with the help of second-order dynamics of communicating membrane system by realizing consensus. This usually requires a layered multiple network control architecture designed mainly for coordinating access of various unmanned aerial vehicles to control airspace, package delivery, rescue, traffic surveillance and more. Here, the communicating membrane system is modelled based on the theory of fractional differential equations. At every interval of the sampling time, all the subsystems are optimally synchronized in P population system by multiset-rewriting rules including the effect of symport/antiport systems. Model of the minimal symport or antiport is also derived to ensure the universality for P systems with multiset-rewriting rules. Further research in this domain will enable the development of evolving architecture of a non-linear cyber physical system. The close loop stability and the recursive feasibility of the evolved architecture are also studied. Comparison are also drawn predictive control to prove the effectiveness of the proposed system.

Nonlinear Boundary Value Problem of Fractional Differential Equations with Arguments under Integral Boundary Condition

In this paper, we develop the existence and uniqueness theory of fractional differential equation involving Riemann-Liouville di¤erential operator of order 0 &lt; &lt; 1, with advanced argument under integral boundary condition. We show the uniqueness of solution by using Banach xed point theorem with a weighted norm. We apply the comparison result and obtain the existence and uniqueness of solution by monotone iterative technique also by use weakly coupled extremal solution of (1.1).

Fractional Ostrowski Type Inequalities via Generalized Mittag–Leffler Function

If we study the theory of fractional differential equations then we notice the Mittag–Leffler function is very helpful in this theory. On the contrary, Ostrowski inequality is also very useful in numerical computations and error analysis of numerical quadrature rules. In this paper, Ostrowski inequalities with the help of generalized Mittag–Leffler function are established. In addition, bounds of fractional Hadamard inequalities are given as straightforward consequences of these inequalities.