Hadamard Fractional Derivatives 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.

A new strategy based on the logarithmic Chebyshev cardinal functions for Hadamard time fractional coupled nonlinear Schrödinger–Hirota equations

In this research, the Hadamard fractional derivative is used to define the time fractional coupled nonlinear Schrödinger–Hirota equations. The logarithmic Chebyshev cardinal functions, as a new category of cardinal functions, are introduced to build a numerical method to solve this system. To do this, the Hadamard fractional differentiation matrix of these functions is obtained. In the developed method, by considering a hybrid approximation of the problem’s solution using the logarithmic Chebyshev cardinal functions (for the temporal variable) and classical Chebyshev cardinal polynomials (for the spatial variable), and employing the interpolation property of these basis functions, along with the expressed derivative matrix, solving the fractional system turns into obtaining the solution of an algebraic system of equations. Two numerical examples are investigated to acknowledge the high accuracy of the introduced procedure.

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.

Hadamard fractional discrete-time relaxation equation’s solutions and asymptotic stability

This paper suggests two numerical schemes for the famous fractional logistic equations by use of the time scale theory. First, the Hadamard discrete fractional calculus is introduced. Leibniz integral law is used to obtain the fractional sum equation. Then, Hadamard fractional derivative and integral’s numerical schemes are obtained and employed for numerical solutions. It can be concluded the fractional differential equation on time scales is very suitable for both continuous and discrete–time modeling.

Existence of Solutions for the Initial Value Problem with Hadamard Fractional Derivatives in Locally Convex Spaces

In this paper, we investigate an initial value problem for a nonlinear fractional differential equation on an infinite interval. The differential operator is taken in the Hadamard sense and the nonlinear term involves two lower-order fractional derivatives of the unknown function. In order to establish the global existence criteria, we first verify that there exists a unique positive solution to an integral equation based on a class of new integral inequality. Next, we construct a locally convex space, which is metrizable and complete. On this space, applying Schäuder’s fixed point theorem, we obtain the existence of at least one solution to the initial value problem.

New discretization of [formula omitted]-Caputo fractional derivative and applications

In the present paper, two approximations to evaluate the ψ-Caputo fractional derivative are developed using the linear and the quadratic polynomial interpolations. We present a study of the pointwise error for each approximation and illustrate some particular cases that correspond to approximations of the well known fractional derivatives, such as: Caputo, Katugampola and Hadamard fractional derivatives. In order to elucidate the investigated results, we present some examples for each approximation. For concreteness, we show some applications where we solve initial value problems and problems involving fractional sub-diffusion equations. Finally, some concluding remark are presented.

Study of Uniqueness and Ulam-Type Stability of Abstract Hadamard Fractional Differential Equations of Sobolev Type via Resolvent Operators

This paper focuses on studying the uniqueness of the mild solution for an abstract fractional differential equation. We use Banach’s fixed point theorem to prove this uniqueness. Additionally, we examine the stability properties of the equation using Ulam’s stability. To analyze these properties, we consider the involvement of Hadamard fractional derivatives. Throughout this study, we put significant emphasis on the role and properties of resolvent operators. Furthermore, we investigate Ulam-type stability by providing examples of partial fractional differential equations that incorporate Hadamard derivatives.

On a class of Lyapunov's inequality involving $ \lambda $-Hilfer Hadamard fractional derivative

<abstract><p>In this paper, we presented and proved a general Lyapunov's inequality for a class of fractional boundary problems (FBPs) involving a new fractional derivative, named $ \lambda $-Hilfer. We proved a criterion of existence which extended that of Lyapunov concerning the ordinary case. We used this criterion to solve the fractional differential equation (FDE) subject to the Dirichlet boundary conditions. In order to do so, we invoked some properties and essential results of $ \lambda $-Hilfer fractional boundary value problem (HFBVP). This result also retrieved all previous Lyapunov-type inequalities for different types of boundary conditions as mixed. The order that we considered here only focused on $ 1 < r\leq 2 $. General Hartman-Wintner-type inequalities were also investigated. We presented an example in order to provide an application of this result.</p></abstract>

Convergence to logarithmic-type functions of solutions of fractional systems with Caputo-Hadamard and Hadamard fractional derivatives

Convergence to logarithmic-type functions of solutions of fractional systems with Caputo-Hadamard and Hadamard fractional derivatives

Systems of Hilfer–Hadamard Fractional Differential Equations with Nonlocal Coupled Boundary Conditions

We study the existence and uniqueness of solutions for a system of Hilfer–Hadamard fractional differential equations. These equations are subject to coupled nonlocal boundary conditions that incorporate Riemann–Stieltjes integrals and a range of Hadamard fractional derivatives. To establish our key findings, we apply various fixed point theorems, notably including the Banach contraction mapping principle, the Krasnosel’skii fixed point theorem applied to the sum of two operators, the Schaefer fixed point theorem, and the Leray–Schauder nonlinear alternative.

Existence and Uniqueness Results for a Pantograph Boundary Value Problem Involving a Variable-Order Hadamard Fractional Derivative

This paper discusses the problem of the existence and uniqueness of solutions to the boundary value problem for the nonlinear fractional-order pantograph equation, using the fractional derivative of variable order of Hadamard type. The main results are proved through the application of fractional calculus and Krasnoselskii’s fixed-point theorem. Moreover, the Ulam–Hyers–Rassias stability of the nonlinear fractional pantograph equation is analyzed. To conclude this paper, we provide an example illustrating our findings and approach.

Positive Solutions for a System of Hadamard Fractional Boundary Value Problems on an Infinite Interval

Our investigation is devoted to examining the existence, uniqueness, and multiplicity of positive solutions for a system of Hadamard fractional differential equations. This system is defined on an infinite interval and is subject to coupled nonlocal boundary conditions. These boundary conditions encompass both Hadamard fractional derivatives and Riemann–Stieltjes integrals, and the nonlinearities within the system are non-negative functions that may not be bounded. To establish the main results, we rely on the utilization of mathematical theorems such as the Schauder fixed-point theorem, the Banach contraction mapping principle, and the Avery–Peterson fixed-point theorem.

A Generalized Lyapunov Inequality for a Pantograph Boundary Value Problem Involving a Variable Order Hadamard Fractional Derivative

The authors obtain existence and uniqueness results for a nonlinear fractional pantograph boundary value problem containing a variable order Hadamard fractional derivative. This type of model is appropriate for applications involving processes that occur in strongly anomalous media. They also derive a generalized Lyapunov-type inequality for the problem considered. Their results are obtained by the fractional calculus and Krasnosel’skii’s fixed point theorem. An example is given to illustrate their approach.

On a System of Hadamard Fractional Differential Equations with Nonlocal Boundary Conditions on an Infinite Interval

Our research focuses on investigating the existence of positive solutions for a system of nonlinear Hadamard fractional differential equations. These equations are defined on an infinite interval and involve non-negative nonlinear terms. Additionally, they are subject to nonlocal coupled boundary conditions, incorporating Riemann–Stieltjes integrals and Hadamard fractional derivatives. To establish the main theorems, we employ the Guo–Krasnosel’skii fixed point theorem and the Leggett–Williams fixed point theorem.

Use of Mixed Operator Method to a fractional Hadamard Dirichlet boundary value problem

The purpose of this paper is to deal with the following nonlinear Hadamard fractional boundary value problem HDα1+ u(t) + f(t, u(t), u(t)) + g(t, u(t)) = 0,1 < t < e, 1 < α ≤ 2,u(1) = u(e) = 0, where HDα 1+ is the Hadamard fractional derivative operator. Using the mixed monotone operator method, we prove an existence and uniqueness result for this mixed fractional Hadamard boundary value problem. As an application of this result, we give one example to establish an existence and uniqueness of a positive solution.

Analysis of p‐Laplacian Hadamard fractional boundary value problems with the derivative term involved in the nonlinear term

This paper investigates a new class of p‐Laplacian boundary value problems of Hadamard type fractional differential equations. We emphasize that few works on Hadamard fractional boundary value problems include p‐Laplacian Hadamard type fractional differential equations with nonlinear terms involving Hadamard fractional derivatives of unknown functions. By means of the five functional fixed point theorem, multiple positive solutions are established for the p‐Laplacian Hadamard fractional boundary value problem. An example is constructed for illustration of the main result.

Existence and stability results for a coupled system of impulsive fractional differential equations with Hadamard fractional derivatives

<abstract><p>The purpose of this study is to give some findings on the existence, uniqueness, and Hyers-Ulam stability of the solution of an implicit coupled system of impulsive fractional differential equations possessing a fractional derivative of the Hadamard type. The existence and uniqueness findings are obtained using a fixed point theorem of the type of Kransnoselskii. In keeping with this, many forms of Hyers-Ulam stability are examined. Ultimately, to support main results, an example is provided.</p></abstract>

The new general solution for a class of fractional-order impulsive differential equations involving the Riemann-Liouville type Hadamard fractional derivative

<abstract><p>In this paper, the new general solution for a class of higher-order impulsive fractional differential equations (IFDEs) involving the Riemann-Liouville (R-L) type Hadamard fractional derivative (FD) is presented. Specifically, the necessary and sufficient conditions of the solution are obtained by converting boundary value problems (BVPs) into integral equations and applying analytical techniques. The results in the paper provide a new method for converting BVPs or initial value problems (IVPs) for IFDEs to integral equations. Finally, some examples are devoted to explaining the application of the theorem.</p></abstract>