Certain classes of ordinary and partial differential equations solvable by means of fractional calculus

An Elementary and Introductory Approach to Fractional Calculus and Its Applications

“…has solutions of the form: 20) where K is an arbitrary constant and H(z; p, q) is given by (4.18), it being provided that the second member of (4.20) exists.…”

“…Next, for a function u = u (z, t) of two independent variables z and t, we find it to be convenient to use the notation: 24) provided that the second member of (4.22) exists in each case.…”

“…The main purpose of this section (and Section 7 below) is to follow rather closely and analogously the investigations in (for example) [16], [23], [53], [63] and [64] of solutions of some general families of second-order linear ordinary differential equations, which are associated with the familiar Bessel differential equation of general order ν (cf. [5], Vol.…”

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]).…”