Caputo-Hadamard Fractional Derivative Research Articles

Analysis and Synchronization of the Chen System with Fractional Derivative

In this paper, we study the dynamic behaviors and chaos synchronization of the Chen system described by Caputo–Hadamard fractional derivative. First, the existence and uniqueness of a solution to the Chen system with Caputo–Hadamard derivative are proved by qualitative analysis. Further, the stability of equilibria of the considered system is analyzed with the aid of Routh–Hurwitz criteria. Meanwhile, the bifurcation condition of the Caputo–Hadamard Chen system is compared with the integer-order Chen system, where the differences between the two systems are demonstrated numerically. In the study of chaos synchronization of the drive–response Chen systems with Caputo–Hadamard derivative, two control schemes are developed: three nonlinear controllers and single linear controller. The feasibility of two control schemes is verified, and the synchronization performances of these two schemes are compared by numerical simulations. Based on this, the influence of the fractional-order on chaos synchronization performance is illustrated as well.

Superconvergence analysis of the nonconforming FEM for the Allen–Cahn equation with time Caputo–Hadamard derivative

This paper first introduces a high-order approximate formula for the Caputo–Hadamard fractional derivative using interpolation theory, and analyzes its convergence and accuracy. Although the formula bears resemblance to the logarithmic L2-1σ formula in form, it differs in the selection of mesh points, making it suitable for more complex models in numerical computations. Subsequently, a numerical method for the Allen–Cahn equation with time Caputo–Hadamard derivative is proposed. The temporal direction is discretized using the newly derived difference formula, while the spatial direction is approximated by the anisotropic nonconforming quasi-Wilson finite element method (FEM). It is proven that the derived scheme has optimal convergence accuracy in the L2-norm and global superconvergence property in the H1-norm. Numerical examples are provided to corroborate the theoretical claims.

Mittag‐Leffler stability of neural networks with Caputo–Hadamard fractional derivative

In this paper, a Hopfield‐type neural network system with Caputo–Hadamard fractional derivative is discussed. The importance of the existence of the equilibrium point in the analysis of artificial neural networks is well known. Another important investigation is the stability properties. So, the stability of a neural network system is dealt with in the present paper. First, a theorem that asserts the existence and uniqueness of the equilibrium point of the system is proven. Later, the conditions that ensure the Mittag‐Leffler stability of the equilibrium point is obtained by using the Lyapunov's direct method. In addition, an example is given with numerical simulations to show the effectiveness of our theoretical results.

On Some Impulsive Fractional Integro-Differential Equation with Anti-Periodic Conditions

We investigate a class of boundary value problems (BVPs) involving an impulsive fractional integro-differential equation (IF-IDE) with the Caputo–Hadamard fractional derivative (C-HFD). We employ some fixed-point theorems (FPTs) to study the existence of this fractional BVP and its unique solution. The boundary conditions (BCs) established in this study are of a more general type and can be reduced to numerous specific examples by defining the parameters involved in the conditions. In this way, we extend some recent nice results. At the end, we use an example to verify our results.

Efficient spectral collocation method for nonlinear systems of fractional pantograph delay differential equations

<abstract><p>Caputo-Hadamard-type fractional calculus involves the logarithmic function of an arbitrary exponent as its convolutional kernel, which causes challenges in numerical approximations. In this paper, we construct and analyze a spectral collocation approach using mapped Jacobi functions as basis functions and construct an efficient algorithm to solve systems of fractional pantograph delay differential equations involving Caputo-Hadamard fractional derivatives. What we study is the error estimates of the derived method. In addition, we tabulate numerical results to support our theoretical analysis.</p></abstract>

Finite Difference Approximation for the Space-Time Fractional Linear Diffusion Equation Involving the Caputo-Hadamard Fractional Derivative

Finite Difference Approximation for the Space-Time Fractional Linear Diffusion Equation Involving the Caputo-Hadamard Fractional Derivative

Existence criteria for fractional differential equations using the topological degree method

<abstract><p>In this work, we analyze the fractional order by using the Caputo-Hadamard fractional derivative under the Robin boundary condition. The topological degree method combined with the fixed point methodology produces the desired results. Finally to show how the key findings may be utilized, applications are presented.</p></abstract>

Stability analysis of Hadamard and Caputo-Hadamard fractional nonlinear systems without and with delay.

This paper handles with the Hadamard and the Caputo-Hadamard fractional derivative and stability of related systems without and with delay. Firstly, the derivative inequalities are obtained, which is indispensable in applying the theorems derived in this paper. Then, for systems without delay, we get the stability results by using the Lyapunov direct method and for systems with delay, we explore two useful inequalities to verify the stability. Examples are presented with numerical simulations to illustrate the effectiveness of our results.

Multiterm Impulsive Caputo–Hadamard Type Differential Equations of Fractional Variable Order

In this study, we deal with an impulsive boundary value problem (BVP) for differential equations of variable fractional order involving the Caputo–Hadamard fractional derivative. The fundamental problems of existence and uniqueness of solutions are studied, and new existence and uniqueness results are established in the form of two fixed point theorems. In addition, Ulam–Hyers stability sufficient conditions are proved illustrating the suitability of the derived fundamental results. The obtained results are supported also by an example. Finally, the conclusion notes are highlighted.

Fractional SEIR model and data-driven predictions of COVID-19 dynamics of Omicron variant.

We study the dynamic evolution of COVID-19 caused by the Omicron variant via a fractional susceptible-exposed-infected-removed (SEIR) model. Preliminary data suggest that the symptoms of Omicron infection are not prominent and the transmission is, therefore, more concealed, which causes a relatively slow increase in the detected cases of the newly infected at the beginning of the pandemic. To characterize the specific dynamics, the Caputo-Hadamard fractional derivative is adopted to refine the classical SEIR model. Based on the reported data, we infer the fractional order and time-dependent parameters as well as unobserved dynamics of the fractional SEIR model via fractional physics-informed neural networks. Then, we make short-time predictions using the learned fractional SEIR model.

Existence and Ulam stability of solutions for Caputo-Hadamard fractional differential equations

In this paper, we study the existence of solutions for fractional differential equations with the Caputo-Hadamard fractional derivative of order 2 (1, 2]. The uniqueness result is proved via Banach’s contraction mapping principle and the existence results are established by using the Schauder’s fixed point theorem. Furthermore, the Ulam-Hyers and Ulam-Hyers-Rassias stability of the proposed equation is employed. Some examples are given to illustrate the results.

A second-order scheme with nonuniform time grids for Caputo–Hadamard fractional sub-diffusion equations

In this paper, a second-order scheme with nonuniform time meshes for Caputo–Hadamard fractional sub-diffusion equations with initial singularity is investigated. Firstly, a Taylor-like formula with integral remainder is proposed, which is crucial to studying the discrete convolution kernels and error estimate. Secondly, we come up with an error convolution structure (ECS) analysis for Llog,2−1σ interpolation approximation to the Caputo–Hadamard fractional derivative. The core result in this paper is an ECS bound and a global consistency analysis established at an offset point. By virtue of this result, we obtain a sharp L2-norm error estimate of a second-order Crank–Nicolson-like scheme for Caputo–Hadamard fractional differential equations. Ultimately, an example is presented to show the sharpness of our analysis.

Existence and uniqueness of solutions for generalized Sturm–Liouville and Langevin equations via Caputo–Hadamard fractional-order operator

PurposeThis paper aims to investigate the existence and uniqueness of solution for generalized Sturm–Liouville and Langevin equations formulated using Caputo–Hadamard fractional derivative operator in accordance with three nonlocal Hadamard fractional integral boundary conditions. With regard to this nonlinear boundary value problem, three popular fixed point theorems, namely, Krasnoselskii’s theorem, Leray–Schauder’s theorem and Banach contraction principle, are employed to theoretically prove and guarantee three novel theorems. The main outcomes of this work are verified and confirmed via several numerical examples.Design/methodology/approachIn order to accomplish our purpose, three fixed point theorems are applied to the problem under consideration according to some conditions that have been established to this end. These theorems are Krasnoselskii's theorem, Leray Schauder's theorem and Banach contraction principle.FindingsIn accordance to the applied fixed point theorems on our main problem, three corresponding theoretical results are stated, proved, and then verified via several numerical examples.Originality/valueThe existence and uniqueness of solution for generalized Sturm–Liouville and Langevin equations formulated using Caputo–Hadamard fractional derivative operator in accordance with three nonlocal Hadamard fractional integral boundary conditions are studied. To the best of the authors’ knowledge, this work is original and has not been published elsewhere.

Existence of Solutions for a Coupled System of p-Laplacian Caputo–Hadamard Fractional Sturm–Liouville–Langevin Equations with Antiperiodic Boundary Conditions

Fractional Sturm–Liouville and Langevin equations have recently attracted much attention. In this paper, we investigate a coupled system of fractional Sturm–Liouville–Langevin equations with antiperiodic boundary conditions in the framework of Caputo–Hadamard fractional derivative. Comparing with the existing literature, we study fractional differential equations with a p-Laplacian operator, which will enrich and generalize the previous work. Based on the Leray–Schauder nonlinear alternative and Krasnoselskii’s fixed point theorem, some interesting existence results are obtained. Finally, an example is constructed to illustrate our main results.

Lyapunov-type inequalities for fractional Langevin-type equations involving Caputo-Hadamard fractional derivative

In this study, some new Lyapunov-type inequalities are presented for Caputo-Hadamard fractional Langevin-type equations of the forms Da+βHC(HCDa+α+p(t))x(t)+q(t)x(t)=0,0<a<t<b,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\begin{aligned} &{}_{H}^{C}D_{a + }^{\\beta } \\bigl({}_{H}^{C}D_{a + }^{\\alpha }+ p(t)\\bigr)x(t) + q(t)x(t) = 0,\\quad 0 < a < t < b, \\end{aligned} $$\\end{document} and Da+ηHCϕp[(HCDa+γ+u(t))x(t)]+v(t)ϕp(x(t))=0,0<a<t<b,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\begin{aligned} &{}_{H}^{C}D_{a + }^{\\eta }{ \\phi _{p}}\\bigl[\\bigl({}_{H}^{C}D_{a + }^{\\gamma }+ u(t)\\bigr)x(t)\\bigr] + v(t){\\phi _{p}}\\bigl(x(t)\\bigr) = 0,\\quad 0 < a < t < b, \\end{aligned} $$\\end{document} subject to mixed boundary conditions, respectively, where p(t), q(t), u(t), v(t) are real-valued functions and 0 < beta < 1 < alpha < 2, 1 < gamma , eta < 2, {phi _{p}}(s) = |s{|^{p - 2}}s, p > 1. The boundary value problems of fractional Langevin-type equations were firstly converted into the equivalent integral equations with corresponding kernel functions, and then the Lyapunov-type inequalities were derived by the analytical method. Noteworthy, the Langevin-type equations are multi-term differential equations, creating significant challenges and difficulties in investigating the problems. Consequently, this study provides new results that can enrich the existing literature on the topic.

Halanay inequality involving Caputo-Hadamard fractional derivative and application

Abstract A Halanay inequality with distributed delay of non-convolution type is considered. We establish a decay of solutions as a Mittag-Leffler function composed with a logarithmic function. A general sufficient condition is found and a large class of admissible retardation kernels is provided. This needs the preparation of several lemmas on properties of the Hadamard derivative and some basic fractional differential problems with this kind of derivative. The obtained result is then applied to a Hopfield neural network system to discuss its stability.

New exploration of operators of fractional neutral integro-differential equations in Banach spaces through the application of the topological degree concept

&lt;abstract&gt;&lt;p&gt;In this paper, we analyze the behavior of the neutral integro-differential equations of fractional order including the Caputo-Hadamard fractional derivative. The results and solutions are obtained using the topological degree method. Furthermore, some specific numerical examples are given to ascertain the wide applicability and high efficiency of the suggested fixed point technique.&lt;/p&gt;&lt;/abstract&gt;

BOUNDARY VALUE PROBLEM FOR NONLINEAR CAPUTO-HADAMARD FRACTIONAL DIFFERENTIAL EQUATION WITH HADAMARD FRACTIONAL INTEGRAL AND ANTI-PERIODIC CONDITIONS

The aim of this work is to study a class of boundary value problem including a fractional order differential equation involving the Caputo-Hadamard fractional derivative. Suffcient conditions will be presented to guarantee the existence and uniqueness of solution of this fractional boundary value problem. The boundary conditions introduced in this work are of quite general nature and reduce to many special cases by fixing the parameters involved in the conditions.

Compact and Noncompact Solutions to Generalized Sturm–Liouville and Langevin Equation with Caputo–Hadamard Fractional Derivative

In this work, through using the Caputo–Hadamard fractional derivative operator with three nonlocal Hadamard fractional integral boundary conditions, a new type of the fractional-order Sturm–Liouville and Langevin problem is introduced. The existence of solutions for this nonlinear boundary value problem is theoretically investigated based on the Krasnoselskii in the compact case and Darbo fixed point theorems in the noncompact case with aiding the Kuratowski’s measure of noncompactness. To demonstrate the applicability and validity of the main gained findings, some numerical examples are included.

Numerical approaches to Caputo–Hadamard fractional derivatives with applications to long-term integration of fractional differential systems

In this paper, three kinds of numerical formulas are proposed for approximating the Caputo–Hadamard fractional derivatives, which are called L1-2 formula, L2-1σ formula, and H2N2 formula, respectively. Among them, the numerical formulas L1-2 and L2-1σ are for order α∈(0,1) with (3−α)-th order convergence, and H2N2 formula is for order α∈(1,2) with (3−α)-th order convergence too, where the theoretical convergence order has been verified by some illustrative examples. Finally, these three new formulas are applied to long-term integration of fractional differential systems.