Initial value problems for fractional differential equations involving Riemann–Liouville sequential fractional derivative

Abstract

In this paper, we shall discuss the properties of the well-known Mittag-Leffler function, and consider the existence and uniqueness of solution of the initial value problem for fractional differential equation involving Riemann-Liouville sequential fractional derivative by using monotone iterative method.

On the existence of solutions of fractional integro-differential equations

“…In [11], Clément et al proved the existence of Hölder continuous solutions for a partial fractional differential equation and in [18], Kilbas et al studied the existence of solutions of several classes of ordinary fractional differential equations. Also, Samko et al [35], Anguraj et al [4], Baleanu and Mustafa [9], Diethelm and Ford [15], Kilbas and Marzan [17], Kosmatov [21], Tian and Bai [39], Wei et al [40], Aghajani et al [3], Pilipović and Stojanović [30], Yuste and Acedo [41], Idczak and Kamocki [16], and Kostić [22], between so many more, have investigated the existence of solutions for various types of fractional differential and integral equations. Furthermore, several analytical and numerical methods have been proposed for approximate solutions of fractional differential equations, e.g.…”

The quasi-boundary value regularization method for identifying the initial value with discrete random noise

“…Recently, many researchers have presented results of the initial value problem and boundary value problem on fractional differential equations, such as [14][15][16]. In [17], the authors used the monotone iterative method to consider the existence and uniqueness of solution of the initial value problem for a fractional differential equation. In [18], the authors used quasi-reversible method to consider initial value problem for a time-fractional diffusion equation.…”

Monotone iterative technique for periodic problem involving Riemann–Liouville fractional derivatives in Banach spaces

“…For details, see the monographs of Miller and Ross [1], Kiryakova [2], Podlubny [3], and Kilbas et al [4] and the papers by Lakshmikantham and Vatsala [5], Agarwal et al [6], Darwish et al [7][8][9][10]. Some recent contributions to the theory of fractional differential equations can be seen in [11][12][13][14][15][16][17][18][19][20][21][22][23].…”