Linear fractional differential equations and eigenfunctions of fractional differential operators

• Main Author: Grigoletto, Eliana Contharteze [UNESP]
• Publication Date: 2018
• Other Authors: de Oliveira, Edmundo Capelas, de Figueiredo Camargo, Rubens [UNESP]
• Format: Article
• Language: eng
• Source: Repositório Institucional da UNESP
• Download full: http://dx.doi.org/10.1007/s40314-016-0381-1; http://hdl.handle.net/11449/171031
• topic: Caputo derivatives; Linear fractional differential equations; Mittag-Leffler functions; Riemann–Liouville derivatives

Eigenfunctions associated with Riemann–Liouville and Caputo fractional differential operators are obtained by imposing a restriction on the fractional derivative parameter. Those eigenfunctions can be used to express the analytical solution of some linear sequential fractional differential equations. As a first application, we discuss analytical solutions for the so-called fractional Helmholtz equation with one variable, obtained from the standard equation in one dimension by replacing the integer order derivative by the Riemann–Liouville fractional derivative. A second application consists of an initial value problem for a fractional wave equation in two dimensions in which the integer order partial derivative with respect to the time variable is replaced by the Caputo fractional derivative. The classical Mittag-Leffler functions are important in the theory of fractional calculus because they emerge as solutions of fractional differential equations. Starting with the solution of a specific fractional differential equation in terms of these functions, we find a way to express the exponential function in terms of classical Mittag-Leffler functions. A remarkable characteristic of this relation is that it is true for any value of the parameter n appearing in the definition of the functions, i.e., we have an infinite family of different expressions for ex in terms of classical Mittag-Leffler functions.