Differential Equations Of Fractional Order Research Articles

Fractional order forestry resource conservation model featuring chaos control and simulations for toxin activity and human-caused fire through modified ABC operator

In this work, we proposed a nonlinear mathematical model with fractional-order differential equations employed to illustrate the impacts of depleted forestry resources with the effect of toxin activity and human-caused fire. The numerical and theoretical outcomes are based on the consideration of using a modified ABC-fractional-order depleted forestry resources dynamical system. In the theoretical aspect, we examination of solution positivity, existence, and uniqueness it makes use of Banach’s fixed point and the Leray Schauder nonlinear alternative theorem. The consecutive recursive sequences are purposefully designed to verify the existence of a solution to the depletion of forestry resources as delineated. To showcase the specificity and stability of the solution within the Hyers–Ulam framework, we employ the concepts and findings of functional analysis. Chaos control will stabilize the system following its equilibrium points by applying the regulate for linear responses technique. Using Lagrange polynomials insight of modified ABC-fractional-order, we conduct simulations and present a comparative analysis in graphical form with classical and integer derivatives. Results also demonstrate the impact of different parameters used in a model that is designed on the system, they provide more understanding and a better approach for real-life problems. Our results demonstrate the significant effects of toxic and fire activities produced by humans on forest ecosystems. More accurate management techniques are made possible by the modified ABC operator’s effectiveness in capturing the long-term effects of these disturbances. The findings highlight how crucial it is to use fractional calculus in ecological modeling to comprehend and manage the intricacies of forest preservation in the face of human pressures. To ensure the sustainable management of forest resources in the face of escalating environmental difficulties, this research offers policymakers and environmental managers a fresh paradigm for creating more robust and adaptive conservation policies.

Dynamic analysis of the fractional-order logistic equation with two different delays

In this paper, we analyze the stability and Hopf bifurcation of the fractional-order logistic equation with two different delays τ1,τ2>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au _{1}, \ au _{2}>0$$\\end{document}: Dαy(t)=ρy(t-τ1)1-y(t-τ2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^{\\alpha }y(t)=\\rho y(t-\ au _{1})\\left( 1-y(t-\ au _{2})\\right) $$\\end{document}, t>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t>0$$\\end{document}, ρ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho >0$$\\end{document}. We describe stability regions by using critical curves. We explore how the fractional order α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}, ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document}, and time delays influence the stability and Hopf bifurcation of the model. Then, by choosing ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document}, fractional order α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}, and time delays as bifurcation parameters, the existence of Hopf bifurcation is studied. An Adams-type predictor–corrector method is extended to solve fractional-order differential equations involving two different delays. Finally, numerical simulations are given to illustrate the effectiveness and feasibility of theoretical results.

Modeling monkeypox virus transmission: Stability analysis and comparison of analytical techniques

Abstract Monkeypox is a highly infectious disease and spreads very easily, hence posing several health concerns or risks as it may lead to outbreak. This article proposes a new mathematical model to simulate the transmission rate of the monkeypox virus-infected fractional-order differential equations using the Caputo–Fabrizio derivative. The existence, uniqueness, and stability under contraction mapping of the fixed point of the model are discussed using Krasnoselskii’s and Banach’s fixed point theorems. To verify the model proposed, we employ data that record the actual dynamics, and based on these data, the model can capture the observed transmission patterns in Ghana. Also, the analytic algorithm is used to find the result applying the Laplace Adomian decomposition method (LADM). Performance analysis of LADM is made regarding Runge-Kutta fourth order, which is the most commonly employed method for solving second-order ordinary differential equations. This comparison therefore offers information on the truth and reliability of the two techniques toward modeling the transmission pattern of the monkey pox virus. The information obtained through this study provides a better understanding of the antibodies linked to monkeypox virus spreading and provides effective strategies to doctors and politicians. This article helps shape better strategies about combating the impact of monkeypox virus in public health since it makes it easy to predict and prevent the occurrence of the disease.

Laplace Adomian Decomposition Method for Fractional Order SIS Epidemic Model

This research manuscript addresses finding a solution to the SIS Epidemic model using the Caputo derivative. To achieve the required results, Laplace transform together with the Adomian Decomposition Method (LADM) were utilized. This method is a powerful tool for dealing with various linear and nonlinear problems of fractional-order differential equations (FODEs). Additionally, some results related to the qualitative theory of the model of interest were studied. The computational results showed the verification of the performed analysis. The investigation of the approximate solution computed in the form of an infinite series was carried out. The results were graphically displayed to analyze the adopted procedure for solving nonlinear FODEs using the Caputo fractional derivative.

On the Qualitative Analysis of Solutions of Two Fractional Order Fractional Differential Equations

In applied sciences, besides the importance of obtaining the analytical solutions of differential equations with constant coefficients, the qualitative analysis of the solutions of such equations is also very important. Due to this importance, in this study, a qualitative analysis of the solutions of a delayed and constant coefficient fractal differential equation with more than one fractional derivative was performed. In the equation under consideration, the derivatives are the Riemann–Liouville fractional derivatives. In the proof of the obtained results, Laplace transform formulas of the Riemann–Liouville fractional derivative and some inequalities are used. We also provide some examples to check the accuracy of our results.

Solvability of a boundary value problem with integrable conditions for partial differential equations of fractional order

Differential equations have a remarkable ability to predict the world around us. They are used in a wide variety of disciplines, from biology, economics, physics, chemistry and engineering. The study of existence and uniqueness, periodicity, asymptotic behavior, stability, and methods of analytic and numerical solutions of fractional differential equations have been studied extensively in a large cycle works; Especially, The study of existence and uniqueness of solution of fractional partial differential equations are then proved by mani methods as: the well-known Lax–Milgram theorem, Energy estimate and fixed point theorem... Among them, we only mention here the apriori estimate method (energy inequality method). The current piece of work concentrates on exploring the existence and uniqueness of a solution for a non-linear boundary value problem that has integrable conditions in the case of fractional partial differential equations. For this we split the proof into two sections: linear and non-linear problem; for the associated linear problem, we derive the a priori bound and demonstrate the density of the operator generated by the problem posed; we solve the non-linear problem by introducing a iterative process. The results show the efficiency of energy inequality method in the case of time fractional order differential equations with integrable conditions our results illustrate the existence and uniqueness of the continuous dependence of solution on fractional order. The work of the authors could be considered as a contribution to the development of the functional analysis method. And to conclude we argue that the research carried out in this article could serve as a contribution for possible studies in the field of applied mathematics.

A numerical study for non-linear multi-term fractional order differential equations

In this article, a numerical method is presented for solving non-linear multi-order fractional differential equations (FDEs) using orthonormal Boubaker polynomials as basis functions. The operational matrix is constructed for the product of basis functions. The fractional derivative is considered in the Caputo sense. Using this technique, non-linear multi-order FDEs converted into a system of non-linear algebraic equations, which can then be solved numerically. The proposed method is implemented using MATLAB for numerical analysis. Using numerical results, validity, and accuracy of the proposed method is given and compared the results with existing literature

On the Boundary Functional of a Semi-Markov Process

In this paper, we consider the semi-Markov random walk process with negative drift, positive jumps. An integral equation for the Laplace transform of the conditional distribution of the boundary functional is obtained. In this work, we define the residence time of the system by generalized exponential distributions with different parameters via fractional order integral equation. The purpose of this paper is to reduce an integral equation for the Laplace transform of the conditional distribution of a boundary functional of the semi-Markov random walk processes to fractional order differential equation with constant coefficients.

Solving Fractional Boundary Value Problems with Nonlocal Mixed Boundary Conditions Using Covariant JS-Contractions

This paper investigates the existence, uniqueness, and symmetry of solutions for Φ–Atangana–Baleanu fractional differential equations of order μ∈(1,2] under mixed nonlocal boundary conditions. This is achieved through the use of covariant and contravariant JS-contractions within a generalized framework of a sequential extended bipolar parametric metric space. As a consequence, we obtain the results on covariant and contravariant Ćirić, Chatterjea, Kannan, and Reich contractions as corollaries. Additionally, we substantiate our fixed-point findings with specific examples and derive similar results in the setting of sequential extended fuzzy bipolar metric space.

A Qualitative Analysis of Leukemia Fractional Order SICW Model

Using a series of fundamental differential equations, including the Caputo derivative, which makes it easier to specify the initial conditions of the differential equations, we present a fractional order concept of leukemia in this study. The universality, positivity, and boundedness of solutions are first established. The local stability properties of the equilibrium are studied using the fractional Routh-Hurwitz stability criteria. The differential equation system has been solved using unconventional finite difference techniques. The Leukemia Fractional Order SICW model introduces several innovative elements compared to traditional epidemiological and disease models. This stands out due to its integration of fractional-order differential equations, inclusion of leukemic cells and immune cells compartments, simulation of treatment strategies, consideration of waning immunity, and its application to leukemia-specific scenarios. These elements collectively make it a valuable tool for studying leukemia dynamics, exploring treatment options, and improving our understanding of how the immune system interacts with cancer cells in leukemia patients. Numerical simulations of the model are shown at the conclusion to interpret our theoretical outcomes in support of various fractional orders of derivative options. From there, we can observe how the evolution of the system components is impacted by the fractional derivative .

Admissible and dissipative synchronisation of fractional order singular systems with time delay using event-triggered feedback control

The purpose of this work is to study the admissible and dissipative synchronisation of fractional order singular systems with time delay. The key goal is to use as few network resources as possible while retaining the desired closed-loop performance. An event-triggered control method is used to accomplish this. The Lyapunov function method, linear matrix inequality approach, and mathematical techniques are used to develop new synchronisation criteria that assure the synchronisation error system is Mittag–Leffler stable. Because fractional order differential equations have memory characteristics, which are very different from ordinary differential equations, leading to difficulties in preventing Zeno behaviour related to the event-triggered mechanism of fractional order systems, especially of delayed fractional order singular systems. To solve this issue, we propose a new theoretical framework and a new condition to preclude Zeno behaviours. This derived conditions use inequality techniques and take advantage of a number of fundamental aspects of fractional order calculus. Furthermore, the dissipative synchronous problem of delayed fractional order singular systems is also solved for the first time using the suggested stable criterion in conjunction with some auxiliary characteristics of fractional calculus. Two numerical examples are provided to demonstrate the usefulness of the proposed strategy.

An improved fractional-order transmission model of COVID-19 with vaccinated population in United States

In this paper, we propose a new fractional-order differential equation model with latent and vaccinated population to describe the dynamics of COVID-19. Firstly, the theoretical mathematical model is established based on the transmission mechanism of COVID-19 in the population. Then, the data of the infected, the recovered and the death are collected from big data report of Baidu’s epidemic situation, and the parameters are estimated by piecewise fitting and nonlinear least square method based on collected data. The correlation coefficients between the infected and model simulation, between the recovered and model simulation, between the death and model simulation are 0.9868, 0.9948 and 0.9994, respectively and the accuracy of prediction are 96.05%, 99.33% and 99.88%, respectively. Additionally, the accuracy of prediction is compared between fractional-order differential equation model and integer-order differential equation model, and the results show fractional-order differential equation model can better predict the development trend of COVID-19. Finally, we analyze the sensitivity of the parameters through numerical simulations, and put forward the corresponding strategies to control the epidemic development according to the screened sensitive parameters.

FDE-Net: A memory-efficiency densely connected network inspired from fractional-order differential equations for single image super-resolution

Dense connection has proved an effective way to take full advantage of hierarchical features extracted from low-resolution images using deep neural networks (DNNs) for single image super-resolution (SISR). However, existing densely connected networks are often manually designed and overly depended on practical experience, thus leading to suboptimal performance. Moreover, due to their complicated connections, they are memory-consuming and lack of interpretability. To address all these problems with one stone, we propose to construct a new densely connected network for SISR from a dynamic system perspective. Following this idea, we cast the hierarchical transformation of DNNs into the state evolution of a fractional-order dynamic system, which empowers us to automatically construct two interdependent densely connected modules based on the system solution rather manual design. They are a prediction module that controls the system to iteratively predict the next state, and a correction module that iteratively refines the predicted state to improve the prediction accuracy. With these two modules as backbone, we establish a Fractional-order Differential Equations-based network (FDE-Net) for SISR. Since the iterative computational manner requires both densely connected modules to be of the recurrent structure, FDE-Net is memory-efficiency and good interpretability. In addition, we analyze the existence and uniqueness of the solution for FDE to theoretically guarantee the feasibility of FDE-Net. Experiments on four SISR benchmark datasets demonstrate the superiority of FDE-Net over existing densely connected networks and other baselines in terms of generalization capacity, especially with limited memory.

Fractional-Order Sequential Linear Differential Equations with Nabla Derivatives on Time Scales

In this paper, we present a general theory for fractional-order sequential differential equations with Riemann–Liouville nabla derivatives and Caputo nabla derivatives on time scales. The explicit solution, in the case of constant coefficients, for both the homogeneous and the non-homogeneous problems, are given using the ∇-Mittag-Leffler function, Laplace transform method, operational method and operational decomposition method. In addition, we also provide some results about a solution to a new class of fractional-order sequential differential equations with convolutional-type variable coefficients using the Laplace transform method.

Bernstein Operational Matrix for Solving Variable Order Fractional Differential Equations with Generalized Caputo-Type

In this research, we present novel generalized derivative founded on the newly constructed (NGCFVODs) novel generalized Caputo Fractional variable order derivatives. Utilizing these operators, a numerical methodology has been formulated to resolve the Fractional Variable Order Differential Equations (FVODEs). We estimate the solution for (FVODEs) by employing bernstein polynomials as foundational vectors. We have further expanded the derivative operational matrix of bernstein Polynomials (bPs) to generalized derivative an operational matrix in sense of (NGCFVODs). The efficiency of developed numerical methodology is tested by a taking a various test examples. We also a compare results of our suggested approach with the methodologies existing in academic papers. In this study the Fractional variable order differential operator of new generalized Caputo was described by three categories: (i) various value in ρ and Fractional variable order a parameter, (ii) various value in a fractional parameters while Fractional variable order and ρ parameter are fixed, and (iii) various value in Fractional variable order parameter controlling fractional and ρ parameter.

Positive Solutions and Their Existence of a Nonlinear Hadamard Fractional-Order Differential Equation with a Singular Source Item Using Spectral Analysis

Based on the spectral analysis method, Gelfand’s formula, and the cones fixed point theorem, some positive solutions with their existence of a nonlinear infinite-point Hadamard fractional-order differential equation is achieved on the interval [a, b] under some conditions, and particularly the nonlinear term allows singularities for time and spatial parameters in the present study. Finally, an analysis case is carried out to reveal the principal results.

Comparative Study of Solutions of Fractional Order Mixed KdV Burger’s Equation

Objective: The aim of this study is to obtain numerical solutions of fractional order mixed KdV Burger’s equation using iterative methods. Methodology: The study is centered on a time fractional mixed KdV Burger’s equation in the Caputo sense. The approximate solutions of this equation are obtained using Adomian decomposition method and Homotopy Perturbation method. Also, a Python code is developed to obtained numerical solutions and simulate graphically. Finding: The obtained solutions are compared with the exact solution and presented graphically using Python program for better analysis. Novelty: These methods shall be used to obtain approximate solution of such type of non-linear fractional order differential equations. Keywords: Fractional Differential Equation, Caputo Derivative, ADM, HPM, KdV Burger equation, Python

Bifurcation and Chaos in a Fractional-Order Cournot Duopoly Game Model on Scale-Free Networks

In this study, a Cournot duopoly model describing Caputo fractional-order differential equations with piecewise constant arguments is discussed. We have obtained two-dimensional discrete dynamical system as a result of applying the discretization process to the model. By using the center manifold theory and the bifurcation theory, it is shown that the discrete dynamical system undergoes flip bifurcation about the Nash equilibrium point. Phase portraits, bifurcation diagrams, and Lyapunov exponents show the existence of many complex dynamical behaviors in the model such as the stable equilibrium point, period-2 orbit, period-4 orbit, period-8 orbit, period-16 orbit, and chaos according to changing the speed of the adjustment parameter [Formula: see text]. The discrete Cournot duopoly game model is also considered on two scale-free networks with different numbers of nodes. It is observed that the complex dynamical networks exhibit similar dynamical behaviors such as the stable equilibrium point, flip bifurcation, and chaos depending on changing the coupling strength parameter [Formula: see text]. Moreover, flip bifurcation and transition chaos take place earlier in more heterogeneous networks. Calculating the largest Lyapunov exponents guarantees the transition from nonchaotic to chaotic states in complex dynamical networks.

Study of a class of fractional-order evolution hybrid differential equations using a modified Mittag-Leffler-type derivative

This work is devoted to using topological degree theory to establish a mathematical analysis for a class of fractional-order evolution hybrid differential equations using a modified Mittag–Leffler-type derivative. In addition, two kinds of Ulam–Hyers (U–H) stability results are deduced for the mentioned problem. A pertinent example is given to verify the results.

An efficient collocation technique based on operational matrix of fractional-order Lagrange polynomials for solving the space-time fractional-order partial differential equations

In this research, we propose a novel and fast computational technique for solving a class of space-time fractional-order linear and non-linear partial differential equations. Caputo-type fractional derivatives are considered. The proposed method is based on the operational and pseudo-operational matrices for the fractional-order Lagrange polynomials. To carry out the method, first, we find the integer and fractional-order operational and pseudo-operational matrix of integration. The collocation technique and obtained operational and pseudo-operational matrices are then used to generate a system of algebraic equations by reducing the given space-time fractional differential problem. The resultant algebraic system is then easily solved by Newton's iterative methods. The upper bound of the fractional-order operational matrix of integration is also provided, which confirms the convergence of fractional-order Lagrange polynomial's approximation. Finally, some numerical experiments are conducted to demonstrate the applicability and usefulness of the suggested numerical scheme.