This paper discuss the longstanding problems of fractional calculus such as too many definitions while lacking physical or geometrical meanings, and try to extend fractional calculus to any dimension. First, some different definitions of fractional derivatives, such as the Riemann-Liouville derivative, the Caputo derivative, Kolwankar’s local derivative and Jumarie’s modified Riemann-Liouville ...

The aim of this tutorial survey is to revisit the basic theory of relaxation processes governed by linear differential equations of fractional order. The fractional derivatives are intended both in the Rieamann-Liouville sense and in the Caputo sense. After giving a necessary outline of the classical theory of linear viscoelasticity, we contrast these two types of fractional derivatives in thei...

In this note, we establish the sectorial property of the Caputo fractional derivative operator of order α ∈ (1, 2) with a zero Dirichlet boundary condition.

In this paper, existence and uniqueness of solution to two-point boundary value for two-sided fractional differential equations involving Caputo fractional derivative is discussed, by means of the Min-Max Theorem.

We study an initial value problem for an implicit fractional differential equation with the Liouville–Caputo fractional derivative. By using fixed point theory and an approximation method, we obtain some existence and uniqueness results.

In this article, we establish sufficient conditions for the existence of solutions for a class of initial value problem for impulsive fractional differential equations involving the Caputo fractional derivative.

This paper contains a variety of new integral inequalities for (s,m)-convex functions using Caputo fractional derivatives and Caputo–Fabrizio operators. Various generalizations Hermite–Hadamard-type containing operators are derived those whose (s,m)-convex. Inequalities involving the digamma function special means deduced as applications.