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.

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.

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 $(p_{1}, p_{2}, p_{3})$ ( 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.

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 definition of finite-time stability of HFDEs in compatible Banach space are proposed. In light of the method of successive approximation and Beesack inequality with weakly singular kernel, the criteria for finite-time stability of 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.

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.

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.