Explicit Solutions of Singular Differential Equation by Means of Fractional Calculus Operators

Abstract

Recently, several authors demonstrated the usefulness of fractional calculus operators in the derivation of particular solutions of a considerably large number of linear ordinary and partial differential equations of the second and higher orders. By means of fractional calculus techniques, we find explicit solutions of second-order linear ordinary differential equations.

A Different Solution Method for the Confluent Hypergeometric Equation

“…Definition 2.1. If the function ( ) is analytic (regular) inside and on , where 37 = { − , + }, − is a contour along the cut joining the points and −∞ + Im( ), 38 which starts from the point at −∞, encircles the point once counter-clockwise, and 39 returns to the point at −∞, and + is a contour along the cut joining the points and 40 ∞ + Im( ), which starts from the point at ∞, encircles the point once counter-41 clockwise, and returns to the point at ∞, 42 43 is Euler's function gamma (Oldham and Spanier, 1974;Miller and Ross, 1993;Podlubny, 1999;Yilmazer and Ozturk, 2013).…”

“…where ∈ ℕ and Γ is Euler's function gamma (Oldham and Spanier, 1974;Miller and 34 Ross, 1993;Podlubny, 1999;Yilmazer and Ozturk, 2013). 35 design, mechanics, optics, modelling and so on (Akgül, 2014;Akgül et al, 2015).…”

“…here''in sola dayalı yazılması ve bu ifadenin ''where ,'' olarak sonraki ''(Yilmazer and Ozturk, 2013). ''ün sola dayalı yazılması, '' and fractional order derivatives exist,'' yerine ''fractional order yazılması, an sonraki satırın sola dayalı yazılması ve ''and are constants and , '' tants and'' yazılması ve de enter yapılmadan bu ifadeden hemen sonra elmesi, ''( ) and ( ) fractional order derivatives exist,'' yerine ''fractional ) exist,'' yazılması, en sonraki satırın sola dayalı yazılması ve ''and .…”

N-Fractional Calculus Operator Method to the Euler Equation

“…In the recent years, by making use of the following definition, properties, and characteristics of a fractional differintegral operator of order ν ∈ R, many scientists have explicitly obtained particular solutions of a number of families of homogeneous (as well as non-homogeneous) linear ordinary and partial fractional differintegral equations [6,[11][12][13][14].…”

Fractional Solutions of the Associated Legendre Equation

“…where Γ stands for Euler's function gamma [10]. contour along the cut joining the points and −∞ + Im( ), which starts from the point at −∞, encircles the point once counter-clockwise, and returns to the point at −∞, and + is a contour along the cut joining the points and ∞ + Im( ), which starts from the point at ∞, encircles the point once counterclockwise, and returns to the point at ∞,…”