Caputo Hadamard Research Articles

Revised and Generalized Results of Averaging Principles for the Fractional Case

The averaging principle involves approximating the original system with a simpler system whose behavior can be analyzed more easily. Recently, numerous scholars have begun exploring averaging principles for fractional stochastic differential equations. However, many previous studies incorrectly defined the standard form of these equations by placing ε in front of the drift term and ε in front of the diffusion term. This mistake results in incorrect estimates of the convergence rate. In this research work, we explain the correct process for determining the standard form for the fractional case, and we also generalize the result of the averaging principle and the existence and uniqueness of solutions to fractional stochastic delay differential equations in two significant ways. First, we establish the result in Lp space, generalizing the case of p=2. Second, we establish the result using the Caputo–Katugampola operator, which generalizes the results of the Caputo and Caputo–Hadamard derivatives.

Observer‐based control for nonlinear Hadamard fractional‐order systems via SOS approach

AbstractPractical stability refers to the notion that the origin is not an equilibrium point (EP) and that the system states tend to converge toward a sphere centered at the origin. The first goal of this paper is to analyze the concept of “practical stability” in Caputo–Hadamard fractional‐order derivative (CHFOD) systems. Then, using the Lyapunov approach, a polynomial fuzzy (PF) observer‐based controller for stabilizing CHFOD PF systems is created. The observer‐based control is innovative since it was created and proven using the sum‐of‐squares (SOS) method. In conclusion, a numerical illustration is provided to corroborate the theoretical findings.

Fractional logarithmic Chebyshev cardinal functions: Application for Caputo–Hadamard fractional optimal control problems

This study investigates a specific type of fractional optimization problems, subject to a dynamical system including the Caputo–Hadamard fractional differentiation. In this way, a novel class of basis functions known as the fractional logarithmic Chebyshev cardinal functions is introduced. An operational matrix regarding the Hadamard fractional integral of these fractional functions is derived and employed to develop an efficient numerical method for the provided problem. The established algorithm converts the solution of the primary fractional optimization problem into the solution of an algebraic system of equations by representing the state and control variables via the introduced fractional functions. Two test problems are considered to confirm the effectiveness of the suggested method. The derived outcomes of solving these examples highlight the satisfactory accuracy and efficiency of the designed method.

An Euler–Maruyama method for Caputo–Hadamard fractional stochastic differential equations on exponential meshes and its fast approximation

An Euler–Maruyama method for Caputo–Hadamard fractional stochastic differential equations on exponential meshes and its fast approximation

Spectral deferred correction method for fractional initial value problem with Caputo–Hadamard derivative

This paper considers an efficient and accurate spectral deferred correction (SDC) method for the initial value problem (IVP) with Caputo–Hadamard derivative. We first apply the basic idea of the SDC method to derive the numerical scheme. Then the iteration matrix which is the key to convergence of the proposed scheme can be obtained for the linear problem. Detailed computation of history term is presented using the spectral collocation method based on mapped Jacobi log orthogonal functions (MJLOFs). Finally, numerical simulations for both linear and nonlinear cases are shown to verify the feasibility and efficiency of the proposed method.

Ulam–Hyers stability of Caputo–Hadamard fractional stochastic differential equations with time-delays and impulses

Ulam–Hyers stability of Caputo–Hadamard fractional stochastic differential equations with time-delays and impulses

Logarithmic Chelyshkov functions for one- and two-dimensional nonlinear Caputo–Hadamard fractional Rosenau equation

In this work, the Caputo–Hadamard time fractional version of the Rosenau equation in one and two dimensions is introduced. The logarithmic Chelyshkov functions (LCFs), as a novel family of basis functions, are introduced and used together with the classical Chelyshkov polynomials (CCPs) to established a hybrid method to solve the expressed fractional equations. For this aim, the ordinary and fractional derivative matrices associated with these basis functions are determined. To establish the desired approach by considering a hybrid expansion of the fractional problem’s solution using the CCPs (for the spatial variables) and LCFs (for the temporal variable), and employing the derived derivative matrices, solving the fractional problem under consideration turns into solving an algebraic system of nonlinear equations. Some numerical examples are considered to investigate the capability and accuracy of this proposed scheme.

Existence and stability results for a Langevin system with Caputo–Hadamard fractional operators

This paper is devoted to analyzing a new model of a coupled Langevin system with fractional operators under nonlocal antiperiodic integral boundary conditions. This model involves nonlinear Langevin fractional equations with Caputo–Hadamard and Caputo fractional operators. Also, the existence and uniqueness of solutions to the suggested model have been investigated by the fixed-point technique. Moreover, the Hyers–Ulam stability of the solutions has been discussed. Finally, we provide an illustrative example to support the theoretical results.

Darbo’s Fixed-Point Theorem: Establishing Existence and Uniqueness Results for Hybrid Caputo–Hadamard Fractional Sequential Differential Equations

Darbo’s Fixed-Point Theorem: Establishing Existence and Uniqueness Results for Hybrid Caputo–Hadamard Fractional Sequential Differential Equations

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.

Non-Polynomial Collocation Spectral Scheme for Systems of Nonlinear Caputo–Hadamard Differential Equations

In this paper, we provide a collocation spectral scheme for systems of nonlinear Caputo–Hadamard differential equations. Since the Caputo–Hadamard operators contain logarithmic kernels, their solutions can not be well approximated using the usual spectral methods that are classical polynomial-based schemes. Hence, we construct a non-polynomial spectral collocation scheme, describe its effective implementation, and derive its convergence analysis in both L2 and L∞. In addition, we provide numerical results to support our theoretical analysis.

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.

Identifying the source term and the initial value simultaneously for Caputo–Hadamard fractional diffusion equation on spherically symmetric domain

Identifying the source term and the initial value simultaneously for Caputo–Hadamard fractional diffusion equation on spherically symmetric domain

Solvability of infinite systems of Caputo–Hadamard fractional differential equations in the triple sequence space $$c^3(\triangle )$$

Solvability of infinite systems of Caputo–Hadamard fractional differential equations in the triple sequence space $$c^3(\triangle )$$

Mathscr {S}mathscr {E}mathscr {I}mathscr {A}mathscr {R}mathscr {S} model for analyzing mathscr {C}mathscr {O}mathscr {V}mathscr {I}mathscr {D}-19 pandemic process via uppsi -Caputo fractional derivative and numerical simulation

The objective of this study is to develop the mathscr {S}mathscr {E}mathscr {I}mathscr {A}mathscr {R}mathscr {S} epidemic model for mathscr {C}mathscr {O}mathscr {V}mathscr {I}mathscr {D}-{textbf {19}} utilizing the uppsi -Caputo fractional derivative. The reproduction number (breve{mathscr {R}}_0) is calculated utilizing the next generation matrix method. The equilibrium points of the model are computed, and both the local and global stability of the disease-free equilibrium point are demonstrated. Sensitivity analysis is discussed to describe the importance of the parameters and to demonstrate the existence of a unique solution for the model by applying a fixed point theorem. Utilizing the fractional Euler procedure, an approximate solution to the model is obtained. To study the transmission dynamics of infection, numerical simulations are conducted by using MatLab. Both numerical methods and simulations can provide valuable insights into the behavior of the system and help in understanding the existence and properties of solutions. By placing the values texttt{t}, ln (texttt{t}) and sqrt{texttt{t}} instead of uppsi , the derivatives of the Caputo and Caputo–Hadamard and Katugampola appear, respectively, to compare the results of each with real data. Besides, these simulations specifically with different fractional orders to examine the transmission dynamics. At the end, we come to the conclusion that the simulation utilizing Caputo derivative with the order of 0.95 shows the prevalence of the disease better. Our results are new which provide a good contribution to the current research on this field of research.

Analysis of Local Discontinuous Galerkin Method for the Variable-order Subdiffusion Equation with the Caputo–Hadamard Derivative

Analysis of Local Discontinuous Galerkin Method for the Variable-order Subdiffusion Equation with the Caputo–Hadamard Derivative

Stability of a Solution for a Hybrid Fractional Differential Equation

This study focuses on examining the existence, uniqueness, and U-lam stability of a solution for a hybrid fractional equation by utilizing the derivative of Caputo-Hadamard (C-H). The primary tools used in our research are the Banach contraction mapping principle (BCMP) and Schaefer’s fixed point theorem. Additionally, we provide an example to demonstrate our results.

On Caputo-Hadamard fractional pantograph problem of two different orders with Dirichlet boundary conditions

This manuscript aims to study the effectiveness of two different levels of fractional orders in the frame of Caputo-Hadamard (CH)-derivatives on a special type class of delay problem supplemented by Dirichlet boundary conditions. The corresponding Hadamard fractional integral equation is derived for a proposed CH-fractional pantograph system. The Banach, Schaefer, and Krasnoselskii fixed point theorems (FPTs), are used to investigate sufficient conditions of the existence and uniqueness theorems for the proposed system. Furthermore, the Green functions properties are investigated and used to discuss the Ulam-Hyers (UH) stability and its generalized by utilizing nonlinear analysis topics. Finally, three mathematical examples are provided with numerical results and figures by using Matlab software to illustrate the validity of our findings.