A note on the mild solutions of Hilfer impulsive fractional differential equations

In this paper, the ‐Hilfer fractional derivative, the most generalized fractional derivative operator, is used to analyze nonlinear impulsive fractional differential equations. The initial goal of this work is to provide a suitable representation formula for the solutions of the ‐Hilfer nonlinear impulsive fractional differential equations. The second goal is to use the representation formula we came up with to show that there is a solution to the ‐Hilfer nonlinear impulsive fractional differential equations. Analysis of the work done is based on the calculus of the ‐Hilfer fractional derivative, the fixed point theorems, and the measure of noncompactness. We illustrate the outcomes of our work through examples.

Section: Introductionmentioning

confidence: 99%

On the (k,Ψ)‐Hilfer nonlinear impulsive fractional differential equations

Kharade

Kucche

2023

“…Fractional order calculus has become a powerful tool for the study of fractional order differential equations, fractal functions, and so forth, and is widely used in the study of fractal geometry, fractal functions, fractal partial differential equations, function spaces, and other fields. [1][2][3][4] Also fractional order has a wide range of applications in many fields such as quantum mechanics, chemical physics, dynamics, power networks, viscoelasticity, and medicine. [5][6][7][8][9] In the article, 10 the authors studied the dynamics of tuberculosis using a fractional order model.…”

Section: Introductionmentioning

confidence: 99%

“…Manderbrot, a professor at Yale University, points out that there are a large number of fractional dimensions in nature and in science. Fractional order calculus has become a powerful tool for the study of fractional order differential equations, fractal functions, and so forth, and is widely used in the study of fractal geometry, fractal functions, fractal partial differential equations, function spaces, and other fields 1‐4 . Also fractional order has a wide range of applications in many fields such as quantum mechanics, chemical physics, dynamics, power networks, viscoelasticity, and medicine 5–9 …”

Section: Introductionmentioning

confidence: 99%

Convergence analysis of Riemann‐Liouville fractional neural network

Dong

Liao

et al. 2022

Many fractional order calculus researchers believe that fractional order calculus is a good way to solve information processing as well as certain physical system modeling problems. In the training of neural networks, there is the problem of long convergence time. In order to shorten the convergence time of the network, an R-L gradient descent method is proposed in this study. The article begins with a theoretical proof of the convergence of fractional order derivatives using function approximation and interpolation inequality theorems. Finally, through multiple simulations, it can be obtained that the fractional-order neural network can maintain a higher accuracy rate compared with the integer-order neural network, and also can well solve the problem of longer convergence time of the neural network. The convergence time can be reduced by nearly 10% compared to the integer order.

“…Sousa et al 10 discussed the mild solutions of Hilfer impulsive fractional differential equations. Zhang and Zhou 11 studied with almost sectorial operators of the form

\begin{matrix} {frakturD}^{χ} ℘ false(t false) = & 𝒜 ℘ false(t false) + ℱ false(t, ℘ false(t false) false) t \in false(0, T false], \\ I_{0 +}^{false(1 - χ false)} ℘ false(0 false) = & ℘_{0}, \end{matrix}

where

{frakturD}^{χ}

is the R‐L fractional derivative of order χ and I (1 − χ ) is the R‐L fractional integral of order 1 − χ , 0 < χ < 1.…”

Section: Introductionmentioning

confidence: 99%

Almost sectorial operators on Ψ‐Hilfer derivative fractional impulsive integro‐differential equations

Karthikeyan

Karthikeyan²,

Baskonus

et al. 2021

135

This paper is concerned with the existence results of Ψ-Hilfer fractional impulsive integro-differential equations involving almost sectorial operators. The mild solutions of the problems are proved by using Schauder fixed-point theorem along with measure of noncompactness. We have discussed the two cases of operators associated semigroup. Also, we consider an abstract application via Hilfer fractional derivative system to verify the results.