The Fractional Integrating Factor Operator: A New Framework for Exactness in Boundary Value Problems

Abstract

The integrating factor is a classical tool in ordinary differential equations. This paper extends that idea to the fractional setting by introducing the Fractional Integrating Factor Operator (FIF operator), a new analytical construct tailored to fractional calculus. Rather than multiplying by a single function, the FIF operator is designed as a structured operator that blends fractional derivatives of different orders into a unified form. This framework opens a direct route for turning complex fractional differential equations into exact, integrable problems. We prove a general reduction theorem showing that boundary value problems with Caputo-type derivatives can be rewritten as first-order integral problems with explicit solution formulas. Beyond the theory, the operator yields constructive formulas for fractional Green’s functions, sharpens Lyapunov-type inequalities, and supports existence–uniqueness results that remain uniform across fractional orders. In this way, the FIF operator provides a foundational advance for the analysis of fractional boundary value problems, with implications for both rigorous theory and computational practice.