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We consider the Cauchy problem $(mathbb D_{(k)} u)(t)=lambda u(t)$, $u(0)=1$, where $mathbb D_{(k)}$ is the general convolutional derivative introduced in the paper (A. N. Kochubei, Integral Equations Oper. Theory {bf 71} (2011), 583--600), $lambda >0$. The solution is a generalization of the function $tmapsto E_alpha (lambda t^alpha)$ where $0<alpha <1$, $E_alpha$ is the Mittag-Leffler function. The asymptotics of this solution, as $tto infty$, is studied.
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Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann-Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel functions. We demonstrate, under some assumptions, how all of these modifications can be considered as special cases of a single, unifying, model of fractional calculus. We provide a fundamental connection with classical fractional calculus by writing these general fractional operators in terms of the original Riemann-Liouville fractional integral operator. We also consider inversion properties of the new operators, prove analogues of the Leibniz and chain rules in this model of fractional calculus, and solve some fractional differential equations using the new operators.
A demonstration of how the point symmetries of the Chazy Equation become nonlocal symmetries for the reduced equation is discussed. Moreover we construct an equivalent third-order differential equation which is related to the Chazy Equation under a generalized transformation, and find the point symmetries of the Chazy Equation are generalized symmetries for the new equation. With the use of singularity analysis and a simple coordinate transformation we construct a solution for the Chazy Equation which is given by a Right Painleve Series. The singularity analysis is applied to the new third-order equation and we find that it admits two solutions, one given by a Left Painleve Series and one given by a Right Painleve Series where the leading-order behaviors and the resonances are explicitly those of the Chazy Equation.
This paper deals with the solution of unified fractional reaction-diffusion systems. The results are obtained in compact and elegant forms in terms of Mittag-Leffler functions and generalized Mittag-Leffler functions, which are suitable for numerical computation. On account of the most general character of the derived results, numerous results on fractional reaction, fractional diffusion, and fractional reaction-diffusion problems scattered in the literature, including the recently derived results by the authors for reaction-diffusion models, follow as special cases.
Several approaches to the formulation of a fractional theory of calculus of variable order have appeared in the literature over the years. Unfortunately, most of these proposals lack a rigorous mathematical framework. We consider an alternative view on the problem, originally proposed by G. Scarpi in the early seventies, based on a naive modification of the representation in the Laplace domain of standard kernels functions involved in (constant-order) fractional calculus. We frame Scarpis ideas within recent theory of General Fractional Derivatives and Integrals, that mostly rely on the Sonine condition, and investigate the main properties of the emerging variable-order operators. Then, taking advantage of powerful and easy-to-use numerical methods for the inversion of Laplace transforms of functions defined in the Laplace domain, we discuss some practical applications of the variable-order Scarpi integral and derivative.
We aim to introduce the generalized multiindex Bessel function $J_{left( beta _{j}right) _{m},kappa ,b}^{left( alpha _{j}right)_{m},gamma ,c}left[ zright] $ and to present some formulas of the Riemann-Liouville fractional integration and differentiation operators. Further, we also derive certain integral formulas involving the newly defined generalized multiindex Bessel function $J_{left( beta _{j}right) _{m},kappa ,b}^{left( alpha _{j}right)_{m},gamma ,c}left[ zright] $. We prove that such integrals are expressed in terms of the Fox-Wright function $_{p}Psi_{q}(z)$. The results presented here are of general in nature and easily reducible to new and known results.
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