Abstract
Abstract This paper is devoted to the study of Caputo modification of the Hadamard fractional derivatives. From here and after, by Caputo-Hadamard derivative, we refer to this modified fractional derivative (Jarad et al. in Adv. Differ. Equ. 2012:142, 2012, p.7). We present the generalization of the fundamental theorem of fractional calculus (FTFC) in the Caputo-Hadamard setting. Also, several new related results are presented.
Highlights
Fractional calculus started to be considered deeply as a powerful tool to reveal the hidden aspects of the dynamics of the complex or hypercomplex systems [ – ].Finding new generalization of the existing fractional derivatives was always the main direction of research within this field
In [ – ], the authors recovered the concepts of fractional integrals and fractional derivatives in different forms and introduced a new version of fundamental theorem of fractional calculus (FTFC) in Caputo settings, which is regarded as a generalization of the classical fundamental theorem of calculus
4 Semigroup properties of Caputo-Hadamard operators We present the first proof of the semigroup properties of Caputo-Hadamard fractional derivatives
Summary
Fractional calculus started to be considered deeply as a powerful tool to reveal the hidden aspects of the dynamics of the complex or hypercomplex systems [ – ].Finding new generalization of the existing fractional derivatives was always the main direction of research within this field. D dx is located outside the integral in the case of Riemann-Liouville, which makes the fractional derivative of a constant of these two types not equal to zero in general. In [ – ], the authors recovered the concepts of fractional integrals and fractional derivatives in different forms and introduced a new version of FTFC in Caputo settings, which is regarded as a generalization of the classical fundamental theorem of calculus.
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