Fractional integral equations tell us how to impose initial values in fractional differential equations

TL;DR

This paper explores how to impose initial conditions in fractional differential equations, revealing that fewer initial values may be needed for unique solutions than traditionally assumed, especially in the Riemann-Liouville case.

Contribution

It establishes a connection between fractional differential equations and their integral analogues, showing that fewer initial conditions are often sufficient for uniqueness.

Findings

Fractional differential equations may require fewer initial conditions than the order suggests.

The number of initial conditions depends on the fractional orders and their decimal parts.

Results are derived specifically for Riemann-Liouville derivatives with constant coefficients.

Abstract

The goal of this work is to discuss how should we impose initial values in fractional problems to ensure that they have exactly one smooth unique solution, where smooth simply means that the solution lies in a certain suitable space of fractional differentiability. For the sake of simplicity and to show the fundamental ideas behind our arguments, we will do this only for the Riemann-Liouville case of linear equations with constant coefficients. In a few words, we study the natural consequences in fractional differential equations of the already existing results involving existence and uniqueness for their integral analogues, in terms of the Riemann-Liouville fractional integral. Under this scope, we derive naturally several interesting results. One of the most astonishing ones is that a fractional differential equation of order $\beta>0$ with Riemann-Liouville derivatives can demand, in principle, less initial values than $\lceil \beta \rceil$ to have a uniquely determined solution. In fact, if not all the involved derivatives have the same decimal part, the amount of conditions is given by $\lceil \beta - \beta_* \rceil$ where $\beta_*$ is the highest order in the differential equation such that $\beta-\beta_*$ is not an integer.