arXiv subjects
Publications and source records attributed to E. Capelas de Oliveira.
Using Gronwall inequality we will investigate the Ulam-Hyers and generalized Ulam-Hyers-Rassias stabilities for the solution of a fractional order pseudoparabolic partial differential equation.
By means of the recent $ψ$-Hilfer fractional derivative and of the Banach fixed-point theorem, we investigate stabilities of Ulam-Hyers, Ulam-Hyers-Rassias and semi-Ulam-Hyers-Rassias on closed intervals $[a,b]$ and $[a,\infty)$ for a particular class of fractional integro-differential equations.
In this paper, using the Riemann-Liouville fractional integral with respect to another function and the $ψ-$Hilfer fractional derivative, we propose a fractional Volterra integral equation and the fractional Volterra integro-differential equation. In this sense, for this new fractional Volterra integro-differential equation, we study the Ulam-Hyers stability and, also, the fractional Volterra integral equation in the Banach space, by means of the Banach fixed-point theorem. As an application, we present the Ulam-Hyers stability using the $α$-resolvent operator in the Sobolev space $W^{1,1}(\mathbb{R}_{+},\mathbb{C})$.
We present the validation of a recent fractional mathematical model for erythrocyte sedimentation proposed by Sharma et al. \cite{GMR}. The model uses a Caputo fractional derivative to build a time fractional diffusion equation suitable to predict blood sedimentation rates. This validation was carried out by means of erythrocyte sedimentation tests in laboratory. Data on sedimentation rates (percentages) were analyzed and compared with the analytical solution of the time fractional diffusion equation. The behavior of the analytical solution related to each blood sample sedimentation data was described and analyzed.
Using the $ψ-$Hilfer fractional derivative, we present a study of the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of the fractional Volterra integral-differential equation by means of fixed-point method.
In this paper, we introduce two new non-singular kernel fractional derivatives and present a class of other fractional derivatives derived from the new formulations. We present some important results of uniformly convergent sequences of continuous functions, in particular the Comparison's principle, and others that allow, the study of the limitation of fractional nonlinear differential equations.
In this short note, using the variable-order differential operator introduced by means of the inverse Laplace transform \cite{coimbra}, we questioned the result obtained by Yang and Tenreiro Machado \cite{yang}.
We study the existence and uniqueness of solution of a nonlinear Cauchy problem involving the $ψ$-Hilfer fractional derivative. In addition, we discuss the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of its solutions. A few examples are presented in order to illustrate the possible applications of our main results.
We propose a new fractional derivative, the Hilfer-Katugampola fractional derivative. Motivated by the Hilfer derivative this formulation interpolates the well-known fractional derivatives of Hilfer, Hilfer-Hadamard, Riemann-Liouville, Hadamard, Caputo, Caputo-Hadamard, Liouville, Weyl, generalized and Caputo-type. As an application, we consider a nonlinear fractional differential equation with an initial condition using this new formulation. We show that this equation is equivalent to a Volterra integral equation and demonstrate the existence and uniqueness of solution to the nonlinear initial value problem.
In high-energy hadron-hadron collisions, the dependence of the total cross-section ($σ_{tot}$) with the energy still constitutes an open problem for QCD. Phenomenological analyses usually relies on analytic parameterizations provided by the Regge-Gribov formalism and fits to the experimental data. In this framework, the singularities of the scattering amplitude in the complex angular momentum plane determine the asymptotic behavior of $σ_{tot}$ in terms of the energy. Usual applications connect simple and triple pole singularities with asymptotic power and logarithmic-squared functions of the energy, respectively. More restrict applications have considered as a leading component for $σ_{tot}$ an empirical function consisting of a logarithmic raised to a real exponent, which is treated as a free fit parameter. With this function, data reductions lead to good descriptions of the experimental data and real (not integer) values of the exponent. In this paper, making use of two independent formalisms (fractional calculus and Mellin transform), we first show that the singularity associated with this empirical function is a branch point and then we explore the mathematical consequences of the result and possible physical interpretations. After reviewing the determination of the singularities from asymptotic forms of interest through the Mellin transform, we demonstrate that the same analytic results can be obtained by means of the Caputo fractional derivative, leading, therefore, to fractional calculus interpretations. Besides correlating Mellin transform, non-local fractional derivatives and exploring the generalization from integer to real exponents and derivative orders, this fractional calculus result may provide insights for physical interpretations on the asymptotic rise of $σ_{tot}$.
In this paper, we propose a generalized Gronwall inequality through the fractional integral with respect to another function. The Cauchy-type problem for a nonlinear differential equation involving the $ψ$-Hilfer fractional derivative and the existence and uniqueness of solutions are discussed. Finally, through generalized Gronwall inequality, we prove the continuous dependence of data on the Cauchy-type problem.
In this paper we introduce a new fractional derivative with respect to another function the so-called $ψ$-Hilfer fractional derivative. We discuss some properties and important results of the fractional calculus. In this sense, we present some uniformly convergent sequence of function results and examples involving the Mittag-Leffler function with one parameter. Finally, we present a wide class of integrals and fractional derivatives, by means of the fractional integral with respect to another function and the $ψ$-Hilfer fractional derivative.
We introduce a new fractional derivative that generalizes the so-called alternative fractional derivative recently proposed by Katugampola. We denote this new differential operator by $\mathscr{D}_{M}^{α,β}$, where the parameter $α$, associated with the order, is such that $0<α<1$, $β>0$ and $M$ is used to denote that the function to be derived involves a Mittag-Leffler function with one parameter. This new derivative satisfies some properties of integer-order calculus, e.g.\ linearity, product rule, quotient rule, function composition and the chain rule. Besides as in the case of the Caputo derivative, the derivative of a constant is zero. Because Mittag-Leffler function is a natural generalization of the exponential function, we can extend some of the classical results of integer-order calculus, namely: Rolle's theorem, the mean value theorem and its extension. Further, when the order of the derivative is $α=1$ and the parameter of the Mittag-Leffler function is also unitary, our definition is equivalent to the definition of the ordinary derivative of order one. Finally, we present the corresponding fractional integral from which, as a natural consequence, new results emerge which can be interpreted as applications. Specifically, we generalize the inversion property of the fundamental theorem of calculus and prove a theorem associated with the classical integration by parts.
We introduce a truncated $M$-fractional derivative type for $α$-differentiable functions that generalizes four other fractional derivatives types recently introduced by Khalil et al., Katugampola and Sousa et al., the so-called conformable fractional derivative, alternative fractional derivative, generalized alternative fractional derivative and $M$-fractional derivative, respectively. We denote this new differential operator by $_{i}\mathscr{D}_{M}^{α,β}$, where the parameter $α$, associated with the order of the derivative is such that $ 0 <α<1 $, $β>0$ and $ M $ is the notation to designate that the function to be derived involves the truncated Mittag-Leffler function with one parameter. The definition of this truncated $M$-fractional derivative type satisfies the properties of the integer-order calculus. We also present, the respective fractional integral from which emerges, as a natural consequence, the result, which can be interpreted as an inverse property. Finally, we obtain the analytical solution of the $M$-fractional heat equation and present a graphical analysis.
In this paper, we present and prove a new truncated $\mathcal{V}$-fractional Taylor's formula using the truncated $\mathcal{V}$-fractional variation of constants formula. In this sense, we present the truncated $\mathcal{V}$-fractional Taylor's remainder by means of $\mathcal{V}$-fractional integral, essential for analyzing and comparing the error, when approaching functions by polynomials. From these new results, some applications were made involving some inequalities, specifically, we generalize the Cauchy-Schwartz inequality.
In this paper, using a fractional integral as proposed by Katugampola we establish a generalization of integral inequalities of Gruss-type. We prove two theorems associated with these inequalities and then immediately we enunciate and prove others inequalities associated with these fractional operator.
Using the six parameters truncated Mittag-Leffler function, we introduce a convenient truncated function to define the so-called truncated $\mathcal{V}$-fractional derivative type. After a discussion involving some properties associated with this derivative, we propose the derivative of a vector valued function and define the $\mathcal{V}$-fractional Jacobian matrix whose properties allow us to say that: the multivariable truncated $\mathcal{V}$-fractional derivative type, as proposed here, generalizes the truncated $\mathcal{V}$-fractional derivative type and can bee extended to obtain a truncated $\mathcal{V}$-fractional partial derivative type. As applications we discuss and prove the change of order associated with two index i.e., the commutativity of two truncated $\mathcal{V}$-fractional partial derivative type and propose the truncated $\mathcal{V}$-fractional Green's theorem.
We introduce a new derivative, the so-called truncated $\mathcal{V}$-fractional derivative for $α$-differentiable functions through the six parameters truncated Mittag-Leffler function, which generalizes different fractional derivatives, recently introduced: conformable fractional derivatives, alternative fractional derivative, truncated alternative fractional derivative, $M$-fractional derivative and truncated $M$-fractional derivative. This new truncated $\mathcal{V}$-fractional derivative satisfies properties of the entire order calculus, among them: linearity, product rule, quotient rule, function composition, and chain rule. Also, as in the case of the Caputo derivative, the derivative of a constant is zero. Since the six parameters Mittag-Leffler function is a generalization of Mittag-Leffler functions of one, two, three, four, and five parameters, we can extend some of the classic results of the entire order calculus, namely: Rolle's theorem, the mean value theorem and its extension. In addition, we present the theorem involving the law of exponents for derivatives and we calculated the truncated $\mathcal{V}$-fractional derivative of the two parameters Mittag-Leffler function. Finally, we present the $\mathcal{V}$-fractional integral from which, as a natural consequence, new results appear as applications. Specifically, we generalize inverse property, the fundamental theorem of calculus, a theorem associated with classical integration by parts, and the mean value theorem for integrals. Also, we calculate the $\mathcal{V}$-fractional integral of the two parameters Mittag-Leffler function. Further, through the truncated $\mathcal{V}$-fractional derivative and the $\mathcal{V}$-fractional integral, we obtain a relation with the fractional derivative and integral in the Riemann-Liouville sense, in the case $0<α<1$.