Transformation Mittag-Lefler function to an exponential function and its some applications to problems with a fractional derivative

TL;DR

This paper establishes a relation between the Mittag-Leffler function and the exponential function, applying it to solve fractional differential equations and demonstrating consistency with classical results as fractional order approaches integers.

Contribution

It introduces a transformation of the Mittag-Leffler function into an exponential form and applies this to solve fractional differential equations with constant coefficients.

Findings

The Mittag-Leffler function can be transformed into an exponential function.

Solutions to fractional differential equations converge to classical solutions as fractional order approaches integers.

The method simplifies solving linear operator differential equations with fractional derivatives.

Abstract

In this work at first the relation the Mittag-Lefler function to the exponential is given. The results are applied to the construction of the solution of Cauchy problem for ordinary linear operator differential equations with constant coefficients and fractional derivatives. On the example is shown that when the order of the derivatives (fractional) approaches to integers the results coincide with the classical.