A Study of Caputo-Hadamard-Type Fractional Differential Equations with Nonlocal Boundary Conditions

Shammakh, Wafa

doi:10.1155/2016/7057910

Cited by 17 publications

(13 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Although the Hadamard-type fractional calculus is an old topic, this type of fractional calculus has not yet been well studied and there is still much to explore. Let us only note some recent results in this direction ( [1,7,8,25,36] or [24]). Related work on such operators in abstract spaces can be found in [32].…”

Section: Definitionmentioning

confidence: 99%

On positive solutions of a system of equations generated by Hadamard fractional operators

2020

View full text Add to dashboard Cite

This paper is devoted to studying some systems of quadratic differential and integral equations with Hadamard-type fractional order integral operators. We concentrate on general growth conditions for functions generating right-hand side of considered systems, which leads to the study of Hadamard-type fractional operators on Orlicz spaces. Thus we need to prove some properties of such type of operators. In contrast to the case of Caputo or Riemann-Liouville type of fractional operators, it is not a convolution-type operator, so we need to study some of their new properties. Some more general problems than systems of quadratic integral equations are also studied, and the results are new even in the context of a single integral equation with the Hadamard fractional operator. The paper concludes with illustrative examples.

show abstract

Section: Definitionmentioning

confidence: 99%

On positive solutions of a system of equations generated by Hadamard fractional operators

2020

View full text Add to dashboard Cite

show abstract

“…In literature, 39 authors investigated the existence of three-point Caputo-Hadamard fractional boundary value problem. In literature, 40 authors investigated the existence and uniqueness of positive solutions for m-point Caputo-Hadamard fractional boundary value problem. We refer the reader to previous works [41][42][43][44][45] for detailed study of Hadamard-type fractional calculus.…”

Section: Introductionmentioning

confidence: 99%

Generalized sine–cosine wavelet method for Caputo–Hadamard fractional differential equations

Idrees

Saeed

2022

Math Methods in App Sciences

View full text Add to dashboard Cite

In this article, a wavelet method is introduced for solving Caputo-Hadamard fractional differential equations on an arbitrary interval. The proposed method is the fractional-order generalization of sine-cosine wavelets (FGSCWs). The operational matrices of fractional-order integration are constructed for solving initial value problem as well as boundary value problem. Furthermore, numerical solution of nonlinear Caputo-Hadamard fractional differential equation is obtained with the conjunction of proposed method with quasilinearization technique. We have constructed the FGSCW operational matrix, FGSCW operational matrix of Hadamard fractional integration of arbitrary order, and FGSCW operational matrix of Hadamard fractional integration for Caputo-Hadamard fractional boundary value problems. Convergence analysis of the proposed method is investigated. Numerical procedure is given for both Caputo-Hadamard initial and boundary value problems. Illustrative examples show the reliability and efficiency of the proposed method and give solution with less error.

show abstract

“…For more details, see [13,14,16,17,19,20] and reference therein. Many studies on differential equations of fractional order, involving different fractional operators such as Riemann-Liouville fractional derivative [6,9], Caputo fractional derivative [8,24], Hadamard fractional derivative [1,25],Caputo-Hadamard fractional derivative [15,22] and Atangana-Baleanu-Caputo fractional derivative [18] have appeared during the past several years. Moreover, by using many classical fixed-point theorems, several authors presented the existence and stability results for various classes of fractional differential equations, see for example [4,7,8,11,12,23].…”

Section: Introductionmentioning

confidence: 99%

Existence and Ulam Stability of Solutions for Nonlinear Caputo-Hadamard Fractional Differential Equations Involving Two Fractional Orders

Saadi

Houas

2022

FU Math Inform

View full text Add to dashboard Cite

In this paper, we study existence, uniqueness and Ulam-Hyers stability of solutions for integro-differential equations involving two fractional orders. By using Banach's fixed point theorem, we obtain some sufficient conditions for the existence and uniqueness of solution for the mentioned problem. Furthermore, we derive the Ulam-Hyers stability and the generalized Ulam-Hyers stability of solution. At the end, an illustrative example is discussed.

show abstract

A Study of Caputo-Hadamard-Type Fractional Differential Equations with Nonlocal Boundary Conditions

Cited by 17 publications

References 10 publications

On positive solutions of a system of equations generated by Hadamard fractional operators

On positive solutions of a system of equations generated by Hadamard fractional operators

Generalized sine–cosine wavelet method for Caputo–Hadamard fractional differential equations

Existence and Ulam Stability of Solutions for Nonlinear Caputo-Hadamard Fractional Differential Equations Involving Two Fractional Orders

Contact Info

Product

Resources

About