A unified presentation of certain families of non-Fuchsian differential equations via fractional calculus operators

A simple fractional-calculus approach to the solutions of the Bessel differential equation of general order and some of its applications

“…For various interesting applications of Theorem 1, one may refer to the earlier works [5][6][7][8][9][10][11][12][13][14][15][16][17][18], in each of which numerous further references on this subject can be found. The main object of the present paper is to investigate solutions of some general families of second-order linear ordinary differential equations, which are associated with the familiar Bessel differential equation of general order u (cf.…”

“…Recently, by applying the following definition of a fractional differintegral (that is, fractional derivative and fractional integral) of order u E R, many authors have explicitly obtained particular solutions of a number of families of homogeneous (as well as nonhomogeneous) linear ordinary and partial fractional differintegral equations (see, for details, [5][6][7][8][9][10][11][12][13][14][15][16][17][18], and the references cited in each of these earlier works). [19][20][21] for c-, (1.4) for c +, (1.5) then f~(z) ( First of all, we find it to be worthwhile to recall here the following potentially useful lemmas and properties associated with the fractional differintegration which is defined above (cf., e.g., [19,20]).…”

On Discrete Fractional Solutions of Non-Fuchsian Differential Equations

“…A similar theory was started for discrete fractional analysis and the definition and properties of fractional sums and differences theory were developed. Many articles related to this topic have appeared lately [5][6][7][8][9][10][11][12][13][14][15][16][17][18].…”

Integral inequalities of systems and the estimate for solutions of certain nonlinear two-dimensional fractional differential systems

“…The amount of literature on such differential equations and their applications is vast; see the monographs of Kilbas et al [1], Miller and Ross [2], Podlubny [3] and the papers of Daftardar-Gejji and Jafari [4], Diethelm [5], Lakshmikantham [6], Lakshmikantham and Vatsala [7,8], Lin et al [9], Hsieh et al [10], Wang et al [11], Zhou et al [12,13] and the references given therein.…”