Searcharxiv⌕ Search

arXiv subjects

E. Capelas de Oliveira

Publications and source records attributed to E. Capelas de Oliveira.

At least 19 recordsLinked to original sources
arXiv2025

On fractional differential equations, dimensional analysis, and the double gamma function

In this paper we discuss some issues that arise in the process of writing a fractional differential equation (FDE) by replacing an integer order derivative by a fractional order derivative in a given differential equation. To address these issues, we propose a dimensional regularization of the Caputo fractional derivative, ensuring consistency in physical dimensions. Then we solve some FDEs using this proposed dimensional regularization. We show that the solutions of these FDEs are most conveniently written using the double gamma function. We also compare these solutions with those from equations involving the standard Caputo fractional derivative.

math.GM↗
arXiv2023

On the $\varepsilon$-regular mild solution for fractional abstract integro-differential equations

In this present paper, we first obtained some estimates involving parts of $\varepsilon$-regular mild solutions of the fractional integro-differential equation. In this sense, through these preliminary results, we investigate the main results of this paper, i.e., the existence, regularity and continuous dependence of $\varepsilon$-regular mild solutions for fractional abstract integro-differential equations in Banach space.

math.GM↗
arXiv2020

Analytical solutions for Navier-Stokes equations with $ψ$-Caputo fractional derivative

This work aims to use the homotopy analysis method to obtain analytical solutions of linear time-fractional Navier-Stokes equations with cylindrical coordinates and of a system of nonlinear time-fractional Navier-Stokes equations with Cartesian coordinates. These equations are described in the $ψ$-Caputo time-fractional derivative. The solutions obtained for time-fractional Navier-Stokes equations are graphically presented.

math.AP↗
arXiv2020

Solutions of time-fractional differential equations using homotopy analysis method

We have used the homotopy analysis method to obtain solutions of linear and nonlinear fractional partial differential differential equations with initial conditions. We replace the first order time derivative by $ψ$-Caputo fractional derivative, and also we compare the results obtained by the homotopy analysis method with the exact solutions.

math.AP↗
arXiv2020

Anomalous relaxation in dielectrics with Hilfer fractional derivative

We introduce a new relaxation function depending on an arbitrary parameter as solution of a kinetic equation in the same way as the relaxation function introduced empirically by Debye, Cole-Cole, Davidson-Cole and Havriliak-Negami, anomalous relaxation in dielectrics, which are recovered as particular cases. We propose a differential equation introducing a fractional operator written in terms of the Hilfer fractional derivative of order ξ, with 0<ξ<1 and type η, with 0<η<1. To discuss the solution of the fractional differential equation, the methodology of Laplace transform is required. As a by product we mention particular cases where the solution is completely monotone. Finally, the empirical models are recovered as particular cases.

cond-mat.stat-mech↗
arXiv2019

On the uniqueness of mild solutions to the time-fractional Navier-Stokes equations in $L^{N} \left( \mathbb{R} ^{N}\right) ^{N} $

In this paper, we present the result of maximum regularity of the mild solution of the fractional Cauchy problem. As our main result, we investigate the uniqueness of mild solutions for time-fractional Navier-Stokes equations in class $C\left([0,\infty);L^{N}\left( \mathbb{R}^{N}\right)^{N}\right)$ by means of the estimates $L^{p}-L^{q}$ of Giga-Shor inequality and the Gronwall inequality.

math.AP↗
arXiv2019

Ulam-Hyers stabilities of mild solutions of the fractional nonlinear abstract Cauchy problem

Since the main work on Ulam-Hyers dependable stabilities of differential equations to date, numerous significant and applicable papers have been published, both in the sense of integer order and fractional order differential equations. However, when we enter the field of fractional differential equations, the path that is still long to be traveled, although there is a range of published works. In this sense, in this paper, we will investigate the Ulam--Hyers and Ulam--Hyers--Rassias stabilities of mild solutions of the fractional nonlinear abstract Cauchy problem on the intervals $[0,T]$ and $[0,\infty)$, by means of Banach fixed point theorem.

math.CA↗
arXiv2019

Mild and strong solutions for Hilfer evolution equation

In this paper, we investigate the existence and uniqueness of mild and strong solutions of fractional semilinear evolution equations in the Hilfer sense, by means of Banach fixed point theorem and the Gronwall inequality.

math.CA↗
arXiv2018

A new class of mild and strong solutions of integro-differential equation of arbitrary order in Banach space

The motivation that the field of differential equations provide to several researchers for the challenges that have been challenging them over the decades has contributed to the strengthening of the area within mathematics. In this sense, investigating important properties of solutions of differential equations, in particular fractional, has been object of study due to the exponential growth of the fractional calculus. In this paper, we investigate the existence and uniqueness of a new class of mild and strong solution of the fractional integro-differential equations in the Hilfer fractional derivative sense in Banach space, by means of the continuously $C_{0}$-semigroup, Banach fixed point theorem and Gronwall inequality.

math.CA↗
arXiv2018

Leibniz type rule: $Ψ-$Hilfer fractional derivative

In this paper, we present the Leibniz rule for the $Ψ-$Hilfer ($Ψ-$H) fractional derivative in two versions, the first in relation to $Ψ-$RL fractional derivative and the second in relation to the $Ψ-$H fractional derivative. In this sense, we present some particular cases of Leibniz rules and Leibniz type rules from the investigated case.

math.CA↗
arXiv2018

On the $\mathbf{\rmΨ}-$fractional integral and applications

Motivated by the ${\rm Ψ}$-Riemann-Liouville $({\rm Ψ-RL})$ fractional derivative and by the ${\rm Ψ}$-Hilfer $({\rm Ψ-H})$ fractional derivative, we introduced a new fractional operator the so-called $\rmΨ-$fractional integral. We present some important results by means of theorems and in particular, that the $\rmΨ-$fractional integration operator is limited. In this sense, we discuss some examples, in particular, involving the Mittag-Leffler $({\rm M-L})$ function, of paramount importance in the solution of population growth problem, as approached. On the other hand, we realize a brief discussion on the uniqueness of nonlinear $Ψ$-fractional Volterra integral equation (${\rm VIE}$) using $β-$distance functions.

math.CA↗
arXiv2018

On the fractional functional differential equation with abstract Volterra operator

The present paper plans to examine the existence, uniqueness and data dependence of the solution of the fractional functional differential equation with the abstract operator of Volterra, in the context of the Picard operators. We present an example, in order to illustrate the results obtained. We present an application, with the end goal to illustrate the results obtained.

math.CA↗
arXiv2018

On the Ulam-Hyers stabilities of Ψ-Hilfer fractional differential equation by means of abstract Volterra operator

In this paper, we consider the new class of the fractional differential equation involving the abstract Volterra operator in the Banach space and investigate existence, uniqueness and stabilities of Ulam--Hyers on the compact interval $Δ=[a,b]$ and on the infinite interval $I=[a,\infty )$. Our analysis is based on the application of the Banach fixed point theorem and the Gronwall inequality involving generalized $Ψ$-fractional integral. At last, we performed out an application to elucidate the outcomes got.

math.CA↗
arXiv2018

Stability of $ψ$-Hilfer Impulsive Fractional Differential Equations

In this paper, we investigate the sufficient conditions for existence and uniqueness of solutions and δ-Ulam-Hyers-Rassias stability of an impulsive fractional differential equation involving $ψ$-Hilfer fractional derivative. Fixed point approach is used to obtain our main results.

math.CA↗
arXiv2018

Existence, uniqueness, estimation and continuous dependence of the solutions of a nonlinear integral and an integrodifferential equations of fractional order

By means of two fractional order integral inequalities we investigate the existence and uniqueness of the solutions of the fractional nonlinear Volterra integral equation and a fractional nonlinear integrodifferential equation in Banach space $C_ξ$, using an adequate norm, $||\cdot||_{ξ,\infty}$. We estimate the solutions and investigate their continuous dependence.

math.CA↗