Existence of Mild Solutions of Abstract Fractional Differential Equations with Fractional Non-Instantaneous Impulsive Conditions

Anguraj, A.; Kanjanadevi, Subramaniam; Trujillo, Juan J.

doi:10.5890/dnc.2017.06.005

Cited by 4 publications

(3 citation statements)

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“…This is the correct way of formulating a NIFDE to be solved using the approach A2, as seen in several papers, 28,30,37 but unfortunately it has often been confused with the other type of NIFDE (fixed lower bound for the Caputo derivatives) discussed in the previous subsection. In some papers, 38,39 it is even unclear which problem is being considered, that of our § 3.2.1 or § 3.2.2, since the lower limit of the Caputo derivative used is not clearly stated in these papers.…”

Section: Analysing the Approachesmentioning

confidence: 96%

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

Fernandez

Ali²,

Zada

2021

Math Methods in App Sciences

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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.

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Section: Analysing the Approachesmentioning

confidence: 96%

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

Fernandez

Ali²,

Zada

2021

Math Methods in App Sciences

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“…In this regard, we will assume the following hypothesis for all the remaind of paper: These types of problem have been attracting many researchers. For instant, results in [10,11,12,13] showed the well-posedness of certain classes of nonlinear and noninstantaneous impulsive differential equations. In the next section, we will discuss the result for the linear case.…”

Section: Abstract Formulation Of the Problemmentioning

confidence: 99%

Approximate Controllability of the Burgers Equation With Impulses and Delay

Leiva¹,

Sundar²

2017

FJMS

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The semilinear heat equation with non instantaneous impulses (NII), memory and delay is considered and its approximate controllability is obtained. This is done by employing a technique that avoids fixed point theorems and pulls back the control solution to a fixed curve in a short time interval. We demonstrate, once again, that the controllability of a system is robust under the influence of non instantaneous impulses, memory and delays. In support, a numerical example with simulation for a linear heat equation is given to validate the obtained controllability result for the linear part. Finally we present some open problems and a possible general framework to study the controllability of non instantaneous impulsive (NII) semilinear equations. Contents 11 6.3. Open Problem 12 References 12

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“…Ramadan et al (2020) improved the method by combining the homotopy perturbation approach with Taylor’s original power series formula to find the exact solution of integral equation. The results related to uniqueness (and existence) were mentioned in Agarwal and O’Regan (2000), Mishra and Sen (2016), Pathak and Mishra (2022), Pathak and Mishra (2023) and Wang et al (2014), resolvent methods (Becker, 2016; Farid et al , 2022; Paul et al , 2023; Paul et al , 2023a, 2023b), etc. In He (2000), the author provided a novel iteration method to solve autonomous ordinary differential systems.…”

Section: Introductionmentioning

confidence: 99%

Precision and efficiency of an interpolation approach to weakly singular integral equations

Bhat,

Mishra,

Mishra

et al. 2024

HFF

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Purpose This study aims to discuss the numerical solutions of weakly singular Volterra and Fredholm integral equations, which are used to model the problems like heat conduction in engineering and the electrostatic potential theory, using the modified Lagrange polynomial interpolation technique combined with the biconjugate gradient stabilized method (BiCGSTAB). The framework for the existence of the unique solutions of the integral equations is provided in the context of the Banach contraction principle and Bielecki norm. Design/methodology/approach The authors have applied the modified Lagrange polynomial method to approximate the numerical solutions of the second kind of weakly singular Volterra and Fredholm integral equations. Findings Approaching the interpolation of the unknown function using the aforementioned method generates an algebraic system of equations that is solved by an appropriate classical technique. Furthermore, some theorems concerning the convergence of the method and error estimation are proved. Some numerical examples are provided which attest to the application, effectiveness and reliability of the method. Compared to the Fredholm integral equations of weakly singular type, the current technique works better for the Volterra integral equations of weakly singular type. Furthermore, illustrative examples and comparisons are provided to show the approach’s validity and practicality, which demonstrates that the present method works well in contrast to the referenced method. The computations were performed by MATLAB software. Research limitations/implications The convergence of these methods is dependent on the smoothness of the solution, it is challenging to find the solution and approximate it computationally in various applications modelled by integral equations of non-smooth kernels. Traditional analytical techniques, such as projection methods, do not work well in these cases since the produced linear system is unconditioned and hard to address. Also, proving the convergence and estimating error might be difficult. They are frequently also expensive to implement. Practical implications There is a great need for fast, user-friendly numerical techniques for these types of equations. In addition, polynomials are the most frequently used mathematical tools because of their ease of expression, quick computation on modern computers and simple to define. As a result, they made substantial contributions for many years to the theories and analysis like approximation and numerical, respectively. Social implications This work presents a useful method for handling weakly singular integral equations without involving any process of change of variables to eliminate the singularity of the solution. Originality/value To the best of the authors’ knowledge, the authors claim the originality and effectiveness of their work, highlighting its successful application in addressing weakly singular Volterra and Fredholm integral equations for the first time. Importantly, the approach acknowledges and preserves the possible singularity of the solution, a novel aspect yet to be explored by researchers in the field.

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Existence of Mild Solutions of Abstract Fractional Differential Equations with Fractional Non-Instantaneous Impulsive Conditions

Cited by 4 publications

References 0 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

Approximate Controllability of the Burgers Equation With Impulses and Delay

Precision and efficiency of an interpolation approach to weakly singular integral equations

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