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E. Capelas Oliveira - One of the best experts on this subject based on the ideXlab platform.
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Leibniz type rule: $\Psi-$Hilfer Fractional Derivative
arXiv: Classical Analysis and ODEs, 2018Co-Authors: J. Vanterler Da C. Sousa, E. Capelas OliveiraAbstract:In this paper, we present the Leibniz rule for the $\Psi-$Hilfer ($\Psi-$H) Fractional Derivative in two versions, the first in relation to $\Psi-$RL Fractional Derivative and the second in relation to the $\Psi-$H Fractional Derivative. In this sense, we present some particular cases of Leibniz rules and Leibniz type rules from the investigated case.
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A New Truncated M-Fractional Derivative Type Unifying Some Fractional Derivative Types with Classical Properties
International Journal of Analysis and Applications, 2018Co-Authors: J. Vanterler Da C. Sousa, E. Capelas OliveiraAbstract:We introduce a truncated $M$-Fractional Derivative type for $\alpha$-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 $_{i}\mathscr{D}_{M}^{\alpha,\beta }$, where the parameter $\alpha$, associated with the order of the Derivative is such that $ 0 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.
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On the ψ-Hilfer Fractional Derivative
Communications in Nonlinear Science and Numerical Simulation, 2018Co-Authors: J. Vanterler Da C. Sousa, E. Capelas OliveiraAbstract:Abstract 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 results involving uniformly convergent sequence of function, uniformly continuous function and examples including 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.
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Mittag–Leffler Functions and the Truncated \({\mathcal {V}}\)-Fractional Derivative
Mediterranean Journal of Mathematics, 2017Co-Authors: J. Vanterler Da C. Sousa, E. Capelas OliveiraAbstract:In this paper, we introduce a new type of Fractional Derivative, which we called truncated \({\mathcal {V}}\)-Fractional Derivative, for \(\alpha \)-differentiable functions, by means of the six-parameter truncated Mittag–Leffler function. One remarkable characteristic of this new Derivative is that it generalizes several different Fractional Derivatives, recently introduced: conformable Fractional Derivative, alternative Fractional Derivative, truncated alternative Fractional Derivative, M-Fractional Derivative and truncated M-Fractional Derivative. This new truncated \({\mathcal {V}}\)-Fractional Derivative satisfies several important properties of the classical Derivatives of integer order calculus: linearity, product rule, quotient rule, function composition and the 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 were able to extend some of the classical results of the integer-order calculus, namely: Rolle’s theorem, the mean value theorem and its extension. In addition, we present a theorem on the law of exponents for Derivatives and as an application we calculate the truncated \({\mathcal {V}}\)-Fractional Derivative of the two-parameter 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 the inverse property, the fundamental theorem of calculus, a theorem associated with classical integration by parts, and the mean value theorem for integrals. We also calculate the \({\mathcal {V}}\)-Fractional integral of the two-parameter Mittag–Leffler function. Further, we were able to establish the relation between the truncated \({\mathcal {V}}\)-Fractional Derivative and the truncated \({\mathcal {V}}\)-Fractional integral and the Fractional Derivative and Fractional integral in the Riemann–Liouville sense when the order parameter \(\alpha \) lies between 0 and 1 (\(0
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On the $\psi$-Hilfer Fractional Derivative
arXiv: Classical Analysis and ODEs, 2017Co-Authors: J. Vanterler Da C. Sousa, E. Capelas OliveiraAbstract:In this paper we introduce a new Fractional Derivative with respect to another function the so-called $\psi$-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 $\psi$-Hilfer Fractional Derivative.
J. Vanterler Da C. Sousa - One of the best experts on this subject based on the ideXlab platform.
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Leibniz type rule: $\Psi-$Hilfer Fractional Derivative
arXiv: Classical Analysis and ODEs, 2018Co-Authors: J. Vanterler Da C. Sousa, E. Capelas OliveiraAbstract:In this paper, we present the Leibniz rule for the $\Psi-$Hilfer ($\Psi-$H) Fractional Derivative in two versions, the first in relation to $\Psi-$RL Fractional Derivative and the second in relation to the $\Psi-$H Fractional Derivative. In this sense, we present some particular cases of Leibniz rules and Leibniz type rules from the investigated case.
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A New Truncated M-Fractional Derivative Type Unifying Some Fractional Derivative Types with Classical Properties
International Journal of Analysis and Applications, 2018Co-Authors: J. Vanterler Da C. Sousa, E. Capelas OliveiraAbstract:We introduce a truncated $M$-Fractional Derivative type for $\alpha$-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 $_{i}\mathscr{D}_{M}^{\alpha,\beta }$, where the parameter $\alpha$, associated with the order of the Derivative is such that $ 0 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.
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On the ψ-Hilfer Fractional Derivative
Communications in Nonlinear Science and Numerical Simulation, 2018Co-Authors: J. Vanterler Da C. Sousa, E. Capelas OliveiraAbstract:Abstract 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 results involving uniformly convergent sequence of function, uniformly continuous function and examples including 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.
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Mittag–Leffler Functions and the Truncated \({\mathcal {V}}\)-Fractional Derivative
Mediterranean Journal of Mathematics, 2017Co-Authors: J. Vanterler Da C. Sousa, E. Capelas OliveiraAbstract:In this paper, we introduce a new type of Fractional Derivative, which we called truncated \({\mathcal {V}}\)-Fractional Derivative, for \(\alpha \)-differentiable functions, by means of the six-parameter truncated Mittag–Leffler function. One remarkable characteristic of this new Derivative is that it generalizes several different Fractional Derivatives, recently introduced: conformable Fractional Derivative, alternative Fractional Derivative, truncated alternative Fractional Derivative, M-Fractional Derivative and truncated M-Fractional Derivative. This new truncated \({\mathcal {V}}\)-Fractional Derivative satisfies several important properties of the classical Derivatives of integer order calculus: linearity, product rule, quotient rule, function composition and the 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 were able to extend some of the classical results of the integer-order calculus, namely: Rolle’s theorem, the mean value theorem and its extension. In addition, we present a theorem on the law of exponents for Derivatives and as an application we calculate the truncated \({\mathcal {V}}\)-Fractional Derivative of the two-parameter 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 the inverse property, the fundamental theorem of calculus, a theorem associated with classical integration by parts, and the mean value theorem for integrals. We also calculate the \({\mathcal {V}}\)-Fractional integral of the two-parameter Mittag–Leffler function. Further, we were able to establish the relation between the truncated \({\mathcal {V}}\)-Fractional Derivative and the truncated \({\mathcal {V}}\)-Fractional integral and the Fractional Derivative and Fractional integral in the Riemann–Liouville sense when the order parameter \(\alpha \) lies between 0 and 1 (\(0
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On the $\psi$-Hilfer Fractional Derivative
arXiv: Classical Analysis and ODEs, 2017Co-Authors: J. Vanterler Da C. Sousa, E. Capelas OliveiraAbstract:In this paper we introduce a new Fractional Derivative with respect to another function the so-called $\psi$-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 $\psi$-Hilfer Fractional Derivative.
Emanuel Guariglia - One of the best experts on this subject based on the ideXlab platform.
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Fractional Derivative of the complex gamma function
2018Co-Authors: Emanuel GuarigliaAbstract:In this paper the Fractional Derivative of the complex gamma function are explicitly computed. The convergence is studied. Comparison with the real case is given. In particular, the link with the Fractional Derivative of the Riemann zeta function is shown. Furthermore, other properties of this Fractional Derivative are presented and discussed.
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On the Critical Strip of the Riemann zeta Fractional Derivative
Fundamenta Informaticae, 2017Co-Authors: Carlo Cattani, Emanuel Guariglia, Shuihua WangAbstract:The Fractional Derivative of the Dirichlet eta function is computed in order to investigate the behavior of the Fractional Derivative of the Riemann zeta function on the critical strip. Its converg ...
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A functional equation for the Riemann zeta Fractional Derivative
2017Co-Authors: Emanuel Guariglia, Sergei SilvestrovAbstract:In this paper a functional equation for the Fractional Derivative of the Riemann zeta function is presented. The Fractional Derivative of the zeta function is computed by a generalization of the Gr ...
Melda Duman - One of the best experts on this subject based on the ideXlab platform.
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crank nicolson method for the Fractional diffusion equation with the riesz Fractional Derivative
Journal of Computational Physics, 2012Co-Authors: Cem Celik, Melda DumanAbstract:We examine a numerical method to approximate to a Fractional diffusion equation with the Riesz Fractional Derivative in a finite domain, which has second order accuracy in time and space level. In order to approximate the Riesz Fractional Derivative, we use the ''Fractional centered Derivative'' approach. We determine the error of the Riesz Fractional Derivative to the Fractional centered difference. We apply the Crank-Nicolson method to a Fractional diffusion equation which has the Riesz Fractional Derivative, and obtain that the method is unconditionally stable and convergent. Numerical results are given to demonstrate the accuracy of the Crank-Nicolson method for the Fractional diffusion equation with using Fractional centered difference approach.
Zivorad Tomovski - One of the best experts on this subject based on the ideXlab platform.
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a new extension of the riemann liouville Fractional Derivative operator
arXiv: Classical Analysis and ODEs, 2018Co-Authors: Gauhar Rahman, Kottakkaran Sooppy Nisar, Zivorad TomovskiAbstract:The main aim of this present paper is to present a new extension of the Fractional Derivative operator by using the extension of beta function recently defined by Shadab et al. Moreover, we establish some results related to the newly defined modified Fractional Derivative operator such as Mellin transform and relations to extended hypergeometric and Appell's function via generating functions.
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Fractional diffusion equation with a generalized riemann liouville time Fractional Derivative
Journal of Physics A, 2011Co-Authors: Trifce Sandev, Zivorad Tomovski, Ralf MetzlerAbstract:In this paper, the solution of a Fractional diffusion equation with a Hilfer-generalized Riemann–Liouville time Fractional Derivative is obtained in terms of Mittag–Leffler-type functions and Fox's H-function. The considered equation represents a quite general extension of the classical diffusion (heat conduction) equation. The methods of separation of variables, Laplace transform, and analysis of the Sturm–Liouville problem are used to solve the Fractional diffusion equation defined in a bounded domain. By using the Fourier–Laplace transform method, it is shown that the fundamental solution of the Fractional diffusion equation with a generalized Riemann–Liouville time Fractional Derivative defined in the infinite domain can be expressed via Fox's H-function. It is shown that the corresponding solutions of the diffusion equations with time Fractional Derivative in the Caputo and Riemann–Liouville sense are special cases of those diffusion equations with the Hilfer-generalized Riemann–Liouville time Fractional Derivative. The asymptotic behaviour of the solutions are found for large values of the spatial variable. The Fractional moments of the fundamental solution of the Fractional diffusion equation are obtained. The obtained results are relevant in the context of glass relaxation and aquifer problems.