The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses

Abstract

Based on some recent works about the general solution of fractional differential equations with instantaneous impulses, a Caputo-Hadamard fractional differential equation with noninstantaneous impulses is studied in this paper. An equivalent integral equation with some undetermined constants is obtained for this fractional order system with noninstantaneous impulses, which means that there is general solution for the impulsive systems. Next, an example is given to illustrate the obtained result.

A Coupled System of Caputo–Hadamard Fractional Hybrid Differential Equations with Three-Point Boundary Conditions

“…For the details of Hadamard fractional calculus, we mention the reader to the articles [4][5][6]. Fractional di erential equations involving Hadamard derivatives attracted remarkable interest in the latest years; for example, see [7][8][9][10][11][12][13][14][15][16][17][18][19][20] and [21,22].…”

Non-Instantaneous Impulsive Boundary Value Problems Containing Caputo Fractional Derivative of a Function with Respect to Another Function and Riemann–Stieltjes Fractional Integral Boundary Conditions

“…Therefore, the problem (1) generates many types and also mixed types of impulsive fractional differential equations with boundary conditions. There are some papers that have studied either Hadamard or Caputo fractional derivatives containing in noninstantaneous impulsive equations, see [32][33][34].…”