Analytic solutions of fractional differential equations by operational methods

TL;DR

This paper introduces an operational method for solving fractional differential equations, providing analytic solutions for physics models and fractional Poisson processes, highlighting the relation between ordinary and fractional derivatives.

Contribution

It presents a general operational approach for fractional differential equations, including derivation of analytic solutions and analysis of relations between derivatives, applicable to physics and stochastic processes.

Findings

Derived analytic solutions for fractional differential equations in physics

Established operational relations between ordinary and fractional derivatives

Applied method to fractional Poisson process analysis

Abstract

We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation of differential equations of mathematical physics. Fractionality is obtained by substituting the ordinary integer-order derivative with the Caputo fractional derivative. Furthermore, operational relations between ordinary and fractional differentiation are shown and discussed in detail. Finally, a last example concerns the application of the method to the study of a fractional Poisson process.