In this paper, we deal with a Cauchy problem for nonlinear fractional differential equation the Caputo derivative of order $$\alpha \in (0, 1)$$ . As initial data, consider pair consisting an point, which does not necessarily coincide inferior limit derivative, and function that determines values solution on interval from to point. We study differentiability properties functional associating da...

This paper studies a new class of instantaneous and non-instantaneous impulsive boundary value problem involving the generalized ?-Caputo fractional derivative with weight. Depending on critical point theorems some properties ?-Caputo-type integration differentiation, variational construction multiplicity result solutions are established.

The formulations of Riemann-Liouville and Caputo derivatives in the complex plane are presented. Two versions corresponding to the whole or half plane. It is shown that they can be obtained from the Grünwald-Letnikov derivative.

The present study aims at indicating the existence and uniqueness result of system in extended colombeau algebra. The Caputo fractional derivative is used for solving the system of ODEs. In addition, Riesz fractional derivative of  Colombeau generalized algebra is considered. The purpose of introducing Riesz fractional derivative is regularizing it in Colombeau sense. We also give a solution to...

In this paper, images of the tempered Ψ-Hilfer fractional integral and Ψ-Caputo derivative under generalized Laplace transform are derived. The results applied to find a solution an initial value problem for nonhomogeneous linear differential equation with order α n− 1 <α<n N. An illustrative example is given 0 <α<1 comparing solutions same but different tempering Ψ.