Stability of Integral Caputo-Type Boundary Value Problem with Noninstantaneous Impulses

Zada, Akbar; Ali, Sartaj

doi:10.1007/s40819-019-0640-0

Cited by 15 publications

(11 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The same approach has been applied, and is incorrect for the same reasons, for many other more complicated NIFDEs. 23,26,29,31,33 For example, let us consider the problem posed in Agarwal et al 26 :…”

Section: Approach A1: Fixed Lower Limitmentioning

confidence: 99%

On non‐instantaneous impulsive fractional differential equations and their equivalent integral equations

Fernandez

Ali²,

Zada

2021

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

Real-world processes that display non-local behaviours or interactions, and that are subject to external impulses over non-zero periods, can potentially be modelled using non-instantaneous impulsive fractional differential equations or systems. These have been the subject of many recent papers, which rely on re-formulating fractional differential equations in terms of integral equations, in order to prove results such as existence, uniqueness, and stability. However, specifically in the non-instantaneous impulsive case, some of the existing papers contain invalid re-formulations of the problem, based on a misunderstanding of how fractional operators behave. In this work, we highlight the correct ways of writing non-instantaneous impulsive fractional differential equations as equivalent integral equations, considering several different cases according to the lower limits of the integro-differential operators involved.

show abstract

Section: Approach A1: Fixed Lower Limitmentioning

confidence: 99%

On non‐instantaneous impulsive fractional differential equations and their equivalent integral equations

Fernandez

Ali²,

Zada

2021

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

show abstract

“…Fractional calculus has much applications in many engineering and scientific disciplines, such as the modeling of processes and systems in the fields of signal and image processing, 2 mathematical biology, 3 electronics, economics, control theory, 4 chemistry, biophysics, and blood flow phenomena. For more applications of fractional differential equations, we refer the reader to other studies 5‐13 . Since then, in order to describe and imitate the pragmatic phenomena with mathematical tools more accurately, the fractional q ‐integrals and q ‐derivatives make an appearance at the historic moment.…”

Section: Introductionmentioning

confidence: 99%

“…For more applications of fractional differential equations, we refer the reader to other studies. [5][6][7][8][9][10][11][12][13] Since then, in order to describe and imitate the pragmatic phenomena with mathematical tools more accurately, the fractional q-integrals and q-derivatives make an appearance at the historic moment.…”

Section: Introductionmentioning

confidence: 99%

Analysis ofq‐fractional implicit boundary value problems having Stieltjes integral conditions

Zada

Alam

Riaz

2020

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

In this article, we make analysis of the implicit q‐fractional differential equations involving integral boundary conditions associated with Stieltjes integral and its corresponding coupled system. We are using some sufficient conditions to achieve the existence and uniqueness results for the given problems by applying the Banach contraction principle, Schaefer's fixed point theorem, and Leray–Schauder result of cone type. Moreover, we present different kinds of stability such as Hyers–Ulam stability, generalized Hyers–Ulam stability, Hyers–Ulam–Rassias stability, and generalized Hyers–Ulam–Rassias stability using the classical technique of functional analysis. At the end, the results are verified with the help of examples.

show abstract

“…Such problems arise due to abrupt changes in the state of systems like earth quack, fluctuation of pendulum, etc. Here, we refer to some recent papers on impulsive problems [11][12][13][14][15][16]. e important class of FODEs known as IFODEs has been given much devotion by researchers.…”

Section: Introductionmentioning

confidence: 99%

Mathematical Analysis of Nonlocal Implicit Impulsive Problem under Caputo Fractional Boundary Conditions

Ali

Gupta

Abdeljawad

et al. 2020

Mathematical Problems in Engineering

View full text Add to dashboard Cite

This paper is related to frame a mathematical analysis of impulsive fractional order differential equations (IFODEs) under nonlocal Caputo fractional boundary conditions (NCFBCs). By using fixed point theorems of Schaefer and Banach, we analyze the existence and uniqueness results for the considered problem. Furthermore, we utilize the theory of stability for presenting Hyers-Ulam, generalized Hyers-Ulam, Hyers-Ulam-Rassias, and generalized Hyers-Ulam-Rassias stability results of the proposed scheme. Finally, some applications are offered to demonstrate the concept and results. The whole analysis is carried out by using Caputo fractional derivatives (CFDs).

show abstract

Stability of Integral Caputo-Type Boundary Value Problem with Noninstantaneous Impulses

Cited by 15 publications

References 38 publications

On non‐instantaneous impulsive fractional differential equations and their equivalent integral equations

On non‐instantaneous impulsive fractional differential equations and their equivalent integral equations

Analysis ofq‐fractional implicit boundary value problems having Stieltjes integral conditions

Mathematical Analysis of Nonlocal Implicit Impulsive Problem under Caputo Fractional Boundary Conditions

Contact Info

Product

Resources

About