Innovative approaches to initial and terminal value problems of fractional differential equations with two different derivative orders

This research investigates the properties of fractional differential equations characterized by variable orders, focusing on their existence, uniqueness, and Ulam-Hyers stability. By transforming the problem into an integral equation format, we established essential conditions and utilized well-known fixed-point theorems to derive our results. The application of these mathematical strategies enables us to confirm the existence, uniqueness, and stability of solutions for the equations under consideration. Additionally, to highlight the practical implications of our findings, we present a specific example that effectively illustrates the core outcomes of this study.