Fractional Calculus: Integral and Differential Equations of Fractional Order

TL;DR

This paper introduces fractional calculus operators within the Riemann-Liouville framework, emphasizing Laplace transform techniques to solve simple fractional integral and differential equations, highlighting the Mittag-Leffler function's role.

Contribution

It provides accessible methods for solving fractional equations using Laplace transforms and explores fundamental properties of fractional calculus and the Mittag-Leffler function.

Findings

Analytical solutions for fractional integral and differential equations derived.

Laplace transform techniques effectively handle fractional operators.

The Mittag-Leffler function is fundamental in fractional calculus solutions.

Abstract

We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. By applying this technique we shall derive the analytical solutions of the most simple linear integral and differential equations of fractional order. We show the fundamental role of the Mittag-Leffler function, whose properties are reported in an ad hoc Appendix. The topics discussed here will be: (a) essentials of Riemann-Liouville fractional calculus with basic formulas of Laplace transforms, (b) Abel type integral equations of first and second kind, (c) relaxation and oscillation type differential equations of fractional order.