Solution of Fractional Order Equations in the Domain of the Mellin Transform

This paper presents the Mellin transform for the solution of the fractional order equations. The Mellin transform approach occurs in many areas of applied mathematics and technology. The Mellin transform of fractional calculus of different flavours; namely the Riemann-Liouville fractional derivative, Riemann-Liouville fractional integral, Caputo fractional derivative and the Miller-Ross sequential fractional derivative were obtained. Three illustrative examples were considered to discuss the applications of the Mellin transform and its fundamental properties. The results show that the Mellin transform is a good analytical method for the solution of fractional order equations.

On the Nonlinear Fractional Differential Equations with Caputo Sequential Fractional Derivative

The purpose of this paper is to investigate the existence of solutions to the following initial value problem for nonlinear fractional differential equation involving Caputo sequential fractional derivativeDc0α2Dc0α1yxp-2Dc0α1yx=fx,yx,x>0,y(0)=b0,Dc0α1y(0)=b1, whereDc0α1,Dc0α2are Caputo fractional derivatives,0<α1,α2≤1,p>1, andb0,b1∈R. Local existence of solutions is established by employing Schauder fixed point theorem. Then a growth condition imposed tofguarantees not only the global existence of solutions on the interval[0,+∞), but also the fact that the intervals of existence of solutions with any fixed initial value can be extended to[0,+∞). Three illustrative examples are also presented. Existence results for initial value problems of ordinary differential equations withp-Laplacian on the half-axis follow as a special case of our results.

Bounds on the solution of a Cauchy-type problem involving a weighted sequential fractional derivative

In this paper we establish some bounds for the solution of a Cauchytype problem for a class of fractional differential equations with a weighted sequential fractional derivative. The bounds are based on a Bihari-type inequality and a bound on the Gauss hypergeometric function.