INVESTIGATING HIGHER ORDER BOUNDARY VALUE PROBLEMS ON TIME SCALES
G.J. Muralikanth
Mathematics
December 2023
Higher-order boundary value problems (BVPs) for delay differential equations are the subject of this research, which is important to control theory, physics, and applied mathematics. Although there has been a lot of research on lower-order BVPs with delays, the domain of higher-order situations has been relatively less studied. This work fills that gap. The paper presents an explicit first-order system representation and develops multi-point higher-order BVPs for ordinary differential equations. The nuances of two-point and multipoint BVP are examined, with a focus on the importance of separate boundary conditions and periodic rewriting of the system for computing efficiency. While recognizing the inherent difficulties of these challenges, the study examines the complex questions of whether or not BVPs exist and whether or not their solutions are unique. A brief overview is given of various numerical techniques, such as collocation, shooting techniques, and finite differences. As the investigation progresses, it becomes clear that higher-order BVPs with delays present theoretical and computational difficulties, providing opportunities for additional study and improvement of numerical methods. This study advances our knowledge of boundary value issues on time scales and establishes the foundation for upcoming advancements in mathematical modeling and analysis
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