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In this article, a new definition of fractional Hilfer difference operator is introduced. Definition based properties are developed and utilized to construct fixed point operator for fractional order Hilfer difference equations with initial condition. We acquire some conditions for existence, uniqueness, Ulam-Hyers and Ulam-Hyers-Rassias stability. Modified Gronwalls inequality is presented for discrete calculus with the delta difference operator.
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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 {xi}, with 0<{xi}<1 and type {eta}, with 0<{eta}<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.
In this article, we propose new proportional fractional operators generated from local proportional derivatives of a function with respect to another function. We present some properties of these fractional operators which can be also called proportional fractional operators of a function with respect to another function or proportional fractional operators with dependence on a kernel function.
Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. This paper investigates two typical applications: Lebiniz rule and Laplace transform. It is analytically shown that the commonly used Leibniz rule cannot be applied for Caputo derivative. Similarly, the well-known Laplace transform of Riemann-Liouville derivative is doubtful for n-th continuously differentiable function. By the aid of this series representation, the exact formula of Caputo Leibniz rule and the explanation of Riemann-Liouville Laplace transform are presented. Finally, three illustrative examples are revisited to confirm the obtained results.
For the following semilinear equation with Hilfer- Hadamard fractional derivative begin{equation*} mathcal{D}^{alpha_1,beta}_{a^+} u-Deltamathcal{D}^{alpha_2,beta}_{a^+} u-Delta u =vert uvert^p, qquad t>a>0, qquad xinOmega, end{equation*} where $Omegasubset mathbb{R}^N$ $(Ngeqslant 1)$, $p>1$, $0<alpha _{2}<alpha _{1}<1$ and $0<beta <1$. $mathcal{D}^{alpha_i,beta}_{a^+}$ $(i=1,2)$ is the Hilfer- Hadamard fractional derivative of order $alpha_i$ and of type $beta$, we establish the necessary conditions for the existence of global solutions.
This manuscript investigates the existence and uniqueness of solutions to the first order fractional anti-periodic boundary value problem involving Caputo-Katugampola (CK) derivative. A variety of tools for analysis this paper through the integral equivalent equation of the given problem, fixed point theorems of Leray--Schauder, Krasnoselskiis, and Banach are used. Examples of the obtained results are also presented.
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