Caputo-type modification of the Hadamard fractional derivatives

Abstract

Generalization of fractional differential operators was subjected to an intense debate in the last few years in order to contribute to a deep understanding of the behavior of complex systems with memory effect. In this article, a Caputo-type modification of Hadamard fractional derivatives is introduced. The properties of the modified derivatives are studied.

Existence of solutions for nonlinear Caputo-Hadamard fractional differential equations via the method of upper and lower solutions

“…However, Hadamard fractional derivative operator was introduced and studied in order to consider the problems, which is different completely to the ones in the sense of Riemann-Liouville, Caputo and Grünwald-Letnikov (see [2,7,9,10,16,21]). In the present paper, we shall consider the following fractional nonlinear differential system involving Hadamard fractional integral term C D α a + x(t) = f(t, x(t), I α a + x(t)), t ∈ [a, b], x(a) = x a , (1.1) where C D α a + and I α a + stand for the Caputo-Hadamard derivative and Hadamard integral operators (see Definitions 2.1 and 2.4, below), respectively, f : [a, b] × R × R → R, x a ∈ R, and 0 < a < b < ∞.…”

Derivada fracionária no sentido de Caputo-Hadamard.

“…Introduzimos as integrais e derivadas fracionárias segundo Hadamard [4] e uma modificação para estas derivadas dando assim origemàs derivadas de Caputo-Hadamard [3].…”

“…2 Derivada fracionária no sentido de Caputo-Hadamard A seguir, apresentamos a definição da derivada de ordem não inteira segundo CaputoHadamard, a qual surgiu através de uma modificação na derivada fracionária conforme proposta por Hadamard [2,3]. No decorrer deste trabalho, consideraremos o conjunto…”

“…Após apresentar os operadores de integração e diferenciação fracionários no sentido de Caputo-Hadamard, vamos enunciar e demonstrar o teorema fundamental do cálculo fracionário associado a estes operadores [2,3]. Para tanto, consideremos o operador de integração segundo Hadamard e o operador de diferenciação segundo Caputo-Hadamard.…”

On the concept of general solution for impulsive differential equations of fractional-orderq∈ (2,3)

“…Fractional differential equations play an important part in modeling of many phenomena in various fields of science and engineering, and the subject of fractional differential equations is extensively researched (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the references therein).…”