Second-Order Self-Adjoint Differential Equations Using a Conformable Proportional Derivative

TL;DR

This paper explores second-order conformable differential equations, demonstrating their self-adjoint properties, establishing identities, and connecting solutions to Riccati equations, with implications for boundary value problems and Green's functions.

Contribution

It introduces the formal self-adjointness of conformable second-order equations, develops related identities, and links solutions to Riccati equations, advancing the theoretical framework.

Findings

Established self-adjointness with respect to a specific inner product

Derived a conformable version of Abel's formula and Lagrange identity

Connected solutions to Riccati equations and boundary value problems

Abstract

In this study, linear second-order conformable differential equations using a proportional derivative are shown to be formally self-adjoint equations with respect to a certain inner product and the associated self-adjoint boundary conditions. Defining a Wronskian, we establish a Lagrange identity and Abel's formula. Several reduction-of-order theorems are given. Solutions of the conformable second-order self-adjoint equation are then shown to be related to corresponding solutions of a first-order Riccati equation and a related quadratic functional and a conformable Picone identity. The first part of the study is concluded with a comprehensive roundabout theorem relating key equivalences among all these results. Subsequently, we establish a Lyapunov inequality, factorizations of the second-order equation, and conclude with a section on boundary value problems and Green's functions.