Hadamard Fractional Differential Equations Research Articles

On Ulam–Hyers–Rassias stability of the boundary value problem of Hadamard fractional differential equations of variable order

On Ulam–Hyers–Rassias stability of the boundary value problem of Hadamard fractional differential equations of variable order

Finite-time stability of Hadamard fractional differential equations in weighted Banach spaces

The main purpose of this paper is to investigate the finite-time stability of Hadamard fractional differential equations (HFDEs). Firstly, the standard definitions of finite-time stability of HFDEs in compatible Banach spaces are proposed. In light of the method of successive approximation and Beesack inequality with weakly singular kernel, the criteria of finite-time stability for linear and nonlinear HFDEs are established, respectively. Then with regard to linear HFDEs with pure delay, a novel fractional delayed matrix function (also called delayed Mittag-Leffler matrix function) is given. Specific to nonlinear HFDEs with constant time delay, both Beesack inequality and Hölder inequality are utilized in the framework of the generalized Lipschitz condition. Finally, several indispensable simulations are implemented to verify the effectiveness and practicability of the main results.

On solvability of BVP for a coupled Hadamard fractional systems involving fractional derivative impulses

<abstract><p>Hadamard fractional calculus is one of the most important fractional calculus theories. Compared with a single Hadamard fractional order equation, Hadamard fractional differential equations have a more complex structure and a wide range of applications. It is difficult and challenging to study the dynamic behavior of Hadamard fractional differential equations. This manuscript mainly deals with the boundary value problem (BVP) of a nonlinear coupled Hadamard fractional system involving fractional derivative impulses. By applying nonlinear alternative of Leray-Schauder, we find some new conditions for the existence of solutions to this nonlinear coupled Hadamard fractional system. Our findings reveal that the impulsive function and its impulsive point have a great influence on the existence of the solution. As an application, we discuss an interesting example to verify the correctness and validity of our results.</p></abstract>

Existence of positive solutions of Hadamard fractional defferential equations with integral boundary conditions

In this paper, We study the existence of positive solutions for Hadamard fractional differential equations with integral conditions. We employ Avery-Peterson fixed point theorem and properties of Green's function to show the existence of positive solutions of our problem. Furthermore, we present an example to illustrate our main result.

Ulam-Hyers-Rassias stability for a class of nonlinear implicit Hadamard fractional integral boundary value problem with impulses

<abstract><p>This paper considers a class of nonlinear implicit Hadamard fractional differential equations with impulses. By using Banach's contraction mapping principle, we establish some sufficient criteria to ensure the existence and uniqueness of solution. Furthermore, the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of this system are obtained by applying nonlinear functional analysis technique. As applications, an interesting example is provided to illustrate the effectiveness of main results.</p></abstract>

Some results on the study of Caputo–Hadamard fractional stochastic differential equations

In this article, we show the existence and uniqueness of solution of Caputo–Hadamardfractional stochastic differential equations (CHFSDE) via the Banach fixed point method (BFPM). We analyze the Ulam–Hyers stability (UHS) of CHFSDE by the generalized and the classical Gronwall inequalities. Two examples are presented to illustrate our results.

Existence and uniqueness results for Riemann–Stieltjes integral boundary value problems of nonlinear implicit Hadamard fractional differential equations

In this paper, we study the existence and uniqueness of solutions for Riemann–Stieltjes integral boundary value problems of nonlinear implicit Hadamard fractional differential equations. The investigation of the main results depends on Schauder’s fixed point theorem and Banach’s contraction principle. An illustrative example is given to show the applicability of theoretical results.

Hilfer–Hadamard Fractional Boundary Value Problems with Nonlocal Mixed Boundary Conditions

This paper is concerned with the existence and uniqueness of solutions for a Hilfer–Hadamard fractional differential equation, supplemented with mixed nonlocal (multi-point, fractional integral multi-order and fractional derivative multi-order) boundary conditions. The existence of a unique solution is obtained via Banach contraction mapping principle, while the existence results are established by applying the fixed point theorems due to Krasnoselskiĭ and Schaefer and Leray–Schauder nonlinear alternatives. We demonstrate the application of the main results by presenting numerical examples. We also derive the existence results for the cases of convex and non-convex multifunctions involved in the multi-valued analogue of the problem at hand.

Numerical Methods for Caputo–Hadamard Fractional Differential Equations with Graded and Non-Uniform Meshes

We consider the predictor-corrector numerical methods for solving Caputo–Hadamard fractional differential equations with the graded meshes logtj=loga+logtNajNr,j=0,1,2,…,N with a≥1 and r≥1, where loga=logt0<logt1<⋯<logtN=logT is a partition of [logt0,logT]. We also consider the rectangular and trapezoidal methods for solving Caputo–Hadamard fractional differential equations with the non-uniform meshes logtj=loga+logtNaj(j+1)N(N+1),j=0,1,2,…,N. Under the weak smoothness assumptions of the Caputo–Hadamard fractional derivative, e.g., DCHa,tαy(t)∉C1[a,T] with α∈(0,2), the optimal convergence orders of the proposed numerical methods are obtained by choosing the suitable graded mesh ratio r≥1. The numerical examples are given to show that the numerical results are consistent with the theoretical findings.

Multiple positive solutions for a system of (p_{1}, p_{2}, p_{3})-Laplacian Hadamard fractional order BVP with parameters

In this article, we are pleased to investigate multiple positive solutions for a system of Hadamard fractional differential equations with (p_{1}, p_{2}, p_{3})-Laplacian operator. The main results rely on the standard tools of different fixed point theorems. Finally, we demonstrate the application of the obtained results with the aid of examples.

Existence results for a multipoint boundary value problem of nonlinear sequential Hadamard fractional differential equations

In this paper, existence and uniqueness results are established for a nonlinear sequential Hadamard fractional differential equation with multi-point boundary conditions, via Banach and Krasnosel'skiǐ's fixed point theorems and Leray-Schauder nonlinear alternative. An example illustrating the existence of a unique solution is also constructed.

A wavelet method for solving Caputo–Hadamard fractional differential equation

PurposeThe purpose of the present work is to propose a wavelet method for the numerical solutions of Caputo–Hadamard fractional differential equations on any arbitrary interval.Design/methodology/approachThe author has modified the CAS wavelets (mCAS) and utilized it for the solution of Caputo–Hadamard fractional linear/nonlinear initial and boundary value problems. The author has derived and constructed the new operational matrices for the mCAS wavelets. Furthermore, The author has also proposed a method which is the combination of mCAS wavelets and quasilinearization technique for the solution of nonlinear Caputo–Hadamard fractional differential equations.FindingsThe author has proved the orthonormality of the mCAS wavelets. The author has constructed the mCAS wavelets matrix, mCAS wavelets operational matrix of Hadamard fractional integration of arbitrary order and mCAS wavelets operational matrix of Hadamard fractional integration for Caputo–Hadamard fractional boundary value problems. These operational matrices are used to make the calculations fast. Furthermore, the author works out on the error analysis for the method. The author presented the procedure of implementation for both Caputo–Hadamard fractional initial and boundary value problems. Numerical simulation is provided to illustrate the reliability and accuracy of the method.Originality/valueMany scientist, physician and engineers can take the benefit of the presented method for the simulation of their linear/nonlinear Caputo–Hadamard fractional differential models. To the best of the author’s knowledge, the present work has never been proposed and implemented for linear/nonlinear Caputo–Hadamard fractional differential equations.

Well-posedness and regularity of Caputo–Hadamard fractional stochastic differential equations

Well-posedness and regularity of Caputo–Hadamard fractional stochastic differential equations

Boundary Value Problems of Hadamard Fractional Differential Equations of Variable Order

A boundary value problem for Hadamard fractional differential equations of variable order is studied. Note the symmetry of a transformation of a system of differential equations is connected with the locally solvability which is the same as the existence of solutions. It leads to the necessity of obtaining existence criteria for a boundary value problem for Hadamard fractional differential equations of variable order. Also, the stability in the sense of Ulam–Hyers–Rassias is investigated. The results are obtained based on the Kuratowski measure of noncompactness. An example illustrates the validity of the observed results.

On the Boundary Value Problems of Hadamard Fractional Differential Equations of Variable Order via Kuratowski MNC Technique

In this manuscript, we examine both the existence and the stability of solutions of the boundary value problems of Hadamard-type fractional differential equations of variable order. New outcomes are obtained in this paper based on the Darbo’s fixed point theorem (DFPT) combined with Kuratowski measure of noncompactness (KMNC). We construct an example to illustrate the validity of the observed results.

A study of a coupled system of Hadamard fractional differential equations with nonlocal coupled initial-multipoint conditions

In this paper, we obtain the existence results for a coupled system of Hadamard fractional differential equations supplemented with nonlocal coupled initial-multipoint conditions via fixed point theorems. An example is constructed for the illustration of the uniqueness result.

Maximal and minimal iterative positive solutions for $ p $-Laplacian Hadamard fractional differential equations with the derivative term contained in the nonlinear term

<abstract><p>In this paper, the maximal and minimal iterative positive solutions are investigated for a singular Hadamard fractional differential equation boundary value problem with a boundary condition involving values at infinite number of points. Green's function is deduced and some properties of Green's function are given. Based upon these properties, iterative schemes are established for approximating the maximal and minimal positive solutions.</p></abstract>

Existence and Uniqueness of Solutions for Nonlinear Implicit Hadamard Fractional Differential Equations with Nonlocal Conditions in a Weighted Banach Space

Existence and Uniqueness of Solutions for Nonlinear Implicit Hadamard Fractional Differential Equations with Nonlocal Conditions in a Weighted Banach Space

Lyapunov-type inequalities for Hadamard fractional differential equation under Sturm-Liouville boundary conditions

<abstract><p>In this paper, we establish new Lyapunov-type inequalities for a Hadamard fractional differential equation under Sturm-Liouville boundary conditions. Our conclusions cover many results in the literature.</p></abstract>

Existence of solutions for Hadamard fractional differential equations in Banach spaces

In this work, we investigate the existence of solutions for Hadamard fractional differential equations with integral boundary conditions in a Banach space. We will make use the measure of noncompactness and the Monch fixed point theorem to prove the ¨ main results. An example is given to illustrate our results.