The Experts below are selected from a list of 90 Experts worldwide ranked by ideXlab platform
E. Capelas De Oliveira - One of the best experts on this subject based on the ideXlab platform.
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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 De 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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Mittag-Leffler functions and the truncated $\mathcal{V}$-fractional derivative
arXiv: Classical Analysis and ODEs, 2017Co-Authors: J. Vanterler Da C. Sousa, E. Capelas De OliveiraAbstract:We introduce a new derivative, the so-called truncated $\mathcal{V}$-fractional derivative for $\alpha$-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
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On the local $M$-derivative
arXiv: Classical Analysis and ODEs, 2017Co-Authors: J. Vanterler Da C. Sousa, E. Capelas De OliveiraAbstract: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 $\mathscr{D}_{M}^{\alpha,\beta }$, where the parameter $\alpha$, associated with the order, is such that $0 0$ 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 $\alpha=1$ 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.
J. Vanterler Da C. Sousa - One of the best experts on this subject based on the ideXlab platform.
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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 De 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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Mittag-Leffler functions and the truncated $\mathcal{V}$-fractional derivative
arXiv: Classical Analysis and ODEs, 2017Co-Authors: J. Vanterler Da C. Sousa, E. Capelas De OliveiraAbstract:We introduce a new derivative, the so-called truncated $\mathcal{V}$-fractional derivative for $\alpha$-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
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On the local $M$-derivative
arXiv: Classical Analysis and ODEs, 2017Co-Authors: J. Vanterler Da C. Sousa, E. Capelas De OliveiraAbstract: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 $\mathscr{D}_{M}^{\alpha,\beta }$, where the parameter $\alpha$, associated with the order, is such that $0 0$ 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 $\alpha=1$ 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.
S. Morley - One of the best experts on this subject based on the ideXlab platform.
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The chain Rule for F-differentiation
2016Co-Authors: T. Chaobankoh, Joel Feinstein, S. MorleyAbstract:Let X be a perfect, compact subset of the complex plane, and let D (1)(X) denote the (complex) algebra of continuously complex-differentiable functions on X. Then D(1)(X) is a normed algebra of functions but, in some cases, fails to be a Banach function algebra. Bland and the second author investigated the completion of the algebra D(1)(X), for certain sets X and collections F of paths in X, by considering F-differentiable functions on X. In this paper, we investigate composition, the chain Rule, and the Quotient Rule for this notion of differentiability. We give an example where the chain Rule fails, and give a number of sufficient conditions for the chain Rule to hold. Where the chain Rule holds, we observe that the Fa a di Bruno formula for higher derivatives is valid, and this allows us to give some results on homomorphisms between certain algebras of F-differentiable functions.
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Homomorphisms between algebras of $\mathcal F$-differentiable functions
arXiv: Functional Analysis, 2015Co-Authors: T. Chaobankoh, Joel Feinstein, S. MorleyAbstract:Let $X$ be a perfect, compact subset of the complex plane, and let $D^{(1)}(X)$ denote the (complex) algebra of continuously complex-differentiable functions on $X$. Then $D^{(1)}(X)$ is a normed algebra of functions but, in some cases, fails to be a Banach function algebra. Bland and the second author investigated the completion of the algebra $D^{(1)}(X)$, for certain sets $X$ and collections $\mathcal{F}$ of paths in $X$, by considering $\mathcal{F}$-differentiable functions on $X$. In this paper, we investigate composition, the chain Rule, and the Quotient Rule for this notion of differentiability. We also investigate homomorphisms between certain algebras of $\mathcal{F}$-differentiable functions.
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The chain Rule for $\mathcal F$-differentiation
arXiv: Functional Analysis, 2015Co-Authors: T. Chaobankoh, Joel Feinstein, S. MorleyAbstract:Let $X$ be a perfect, compact subset of the complex plane, and let $D^{(1)}(X)$ denote the (complex) algebra of continuously complex-differentiable functions on $X$. Then $D^{(1)}(X)$ is a normed algebra of functions but, in some cases, fails to be a Banach function algebra. Bland and the second author investigated the completion of the algebra $D^{(1)}(X)$, for certain sets $X$ and collections $\mathcal{F}$ of paths in $X$, by considering $\mathcal{F}$-differentiable functions on $X$. In this paper, we investigate composition, the chain Rule, and the Quotient Rule for this notion of differentiability. We give an example where the chain Rule fails, and give a number of sufficient conditions for the chain Rule to hold. Where the chain Rule holds, we observe that the Fa\'a di Bruno formula for higher derivatives is valid, and this allows us to give some results on homomorphisms between certain algebras of $\mathcal{F}$-differentiable functions.
Koji Hasegawa - One of the best experts on this subject based on the ideXlab platform.
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Perfectly Matched Layers for Elastic Waves in Piezoelectric Solids
Japanese Journal of Applied Physics, 2013Co-Authors: Koji Hasegawa, Shingo SatoAbstract:The material constants of perfectly matched layers (PMLs) for elastic waves in piezoelectric solids in orthogonal coordinates, such as the cylindrical and spherical coordinates, in the frequency domain were derived from the differential form. Using the coordinate transformation laws of tensors, the Quotient Rule, and complex coordinate stretching, we obtained the material parameters of PMLs in the real coordinate. Our results on stress and piezoelectric stress constants are different from the parameters determined by the analytic continuation because we include or exclude the transformation of the contravariant components in the differential form or the analytic continuation, respectively. The presented results are extensions of our results for anisotropic solids without piezoelectricity
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Perfectly Matched Layers in the Cylindrical and Spherical Coordinates for Elastic Waves in Solids
Japanese Journal of Applied Physics, 2010Co-Authors: Takao Shimada, Koji HasegawaAbstract:The material constants of perfectly matched layers (PMLs) in the cylindrical and spherical coordinates in the frequency domain are presented. Using the coordinate transformation laws of tensors on manifolds, the Quotient Rule, and complex coordinate stretching, we obtain the material parameters of PMLs in the real coordinate. Our results show that PML parameters for elastic waves may be determined by the same procedure in the Cartesian coordinates. However, this Rule has been determined for PML material constants derived from the analytic continuation in the cylindrical and spherical coordinates by Zheng and Huang in 2002. Our derivation based on differential forms shows that this Rule holds for PML parameters in any orthogonal coordinate system.
T. Chaobankoh - One of the best experts on this subject based on the ideXlab platform.
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The chain Rule for F-differentiation
2016Co-Authors: T. Chaobankoh, Joel Feinstein, S. MorleyAbstract:Let X be a perfect, compact subset of the complex plane, and let D (1)(X) denote the (complex) algebra of continuously complex-differentiable functions on X. Then D(1)(X) is a normed algebra of functions but, in some cases, fails to be a Banach function algebra. Bland and the second author investigated the completion of the algebra D(1)(X), for certain sets X and collections F of paths in X, by considering F-differentiable functions on X. In this paper, we investigate composition, the chain Rule, and the Quotient Rule for this notion of differentiability. We give an example where the chain Rule fails, and give a number of sufficient conditions for the chain Rule to hold. Where the chain Rule holds, we observe that the Fa a di Bruno formula for higher derivatives is valid, and this allows us to give some results on homomorphisms between certain algebras of F-differentiable functions.
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Homomorphisms between algebras of $\mathcal F$-differentiable functions
arXiv: Functional Analysis, 2015Co-Authors: T. Chaobankoh, Joel Feinstein, S. MorleyAbstract:Let $X$ be a perfect, compact subset of the complex plane, and let $D^{(1)}(X)$ denote the (complex) algebra of continuously complex-differentiable functions on $X$. Then $D^{(1)}(X)$ is a normed algebra of functions but, in some cases, fails to be a Banach function algebra. Bland and the second author investigated the completion of the algebra $D^{(1)}(X)$, for certain sets $X$ and collections $\mathcal{F}$ of paths in $X$, by considering $\mathcal{F}$-differentiable functions on $X$. In this paper, we investigate composition, the chain Rule, and the Quotient Rule for this notion of differentiability. We also investigate homomorphisms between certain algebras of $\mathcal{F}$-differentiable functions.
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The chain Rule for $\mathcal F$-differentiation
arXiv: Functional Analysis, 2015Co-Authors: T. Chaobankoh, Joel Feinstein, S. MorleyAbstract:Let $X$ be a perfect, compact subset of the complex plane, and let $D^{(1)}(X)$ denote the (complex) algebra of continuously complex-differentiable functions on $X$. Then $D^{(1)}(X)$ is a normed algebra of functions but, in some cases, fails to be a Banach function algebra. Bland and the second author investigated the completion of the algebra $D^{(1)}(X)$, for certain sets $X$ and collections $\mathcal{F}$ of paths in $X$, by considering $\mathcal{F}$-differentiable functions on $X$. In this paper, we investigate composition, the chain Rule, and the Quotient Rule for this notion of differentiability. We give an example where the chain Rule fails, and give a number of sufficient conditions for the chain Rule to hold. Where the chain Rule holds, we observe that the Fa\'a di Bruno formula for higher derivatives is valid, and this allows us to give some results on homomorphisms between certain algebras of $\mathcal{F}$-differentiable functions.
