Linear fractional differential equations and eigenfunctions of fractional differential operators

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Grigoletto, Eliana Contharteze [UNESP]
de Oliveira, Edmundo Capelas
de Figueiredo Camargo, Rubens [UNESP]

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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.



Caputo derivatives, Linear fractional differential equations, Mittag-Leffler functions, Riemann–Liouville derivatives

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Computational and Applied Mathematics, v. 37, n. 2, p. 1012-1026, 2018.