Generalized Bernoulli Differential Equation

TL;DR

This paper introduces a generalized Bernoulli differential equation using fractional derivatives, proves related inequalities, analyzes stability, and offers a finite difference method for solutions, enhancing understanding of fractional differential systems.

Contribution

It generalizes the Bernoulli differential equation with fractional derivatives, extending solution techniques and stability analysis methods.

Findings

Proved a generalized Gronwall's inequality for fractional systems.

Established qualitative behavior of solutions for the generalized equation.

Validated a finite difference method for solving the generalized Bernoulli equation.

Abstract

In this paper we propose and solve a generalization of the Bernoulli Differential Equation, by means of a generalized fractional derivative. First we prove a generalization of Gronwall's inequality, which is useful for studying the stability of systems of fractional differential equations and we state results about the qualitative behavior of the trivial solution of the proposed equation. After that, we prove and state the main results about the solution of the generalized Bernoulli Differential Equation and also we give some examples that show the advantage of considering this fractional derivative approach. We also present a finite difference method as an alternative to the solution of the generalized Bernoulli equation and prove its validity by means of examples.