Research Projects

Delay differential equations (DDEs) are crucial mathematical structures used in biological sciences for analysis and forecasting. DDEs rely on memory and past data, making them challenging to interpret when time delays are involved. Recent developments in DDE theory have improved our understanding of the qualitative properties of solutions and have implications for biomedical sciences and other disciplines. Delay parameters are used in mathematics to characterize delay dynamical systems, which often have a temporal delay and involve noise. Deterministic delay differential systems were used in the past due to the lack of effective computing devices. However, recent advances in computer techniques have shown that stochastic delay differential systems can represent real events more accurately. Classical differential equations are becoming increasingly inadequate due to the complexity of memory and genetic factors in physical processes. Fractional order operators provide helpful tools for explaining nonlocal processes. Controllability is essential in systems defined by differential equations, as it guides the system's solution from its starting state to its terminal state. The project aims to define a mild solution to stochastic differential equations using semigroup and operator theories, using fixed-point theorems to show a solution exists, prove the system's controllability under certain conditions, and construct examples to illustrate the effectuality of the results.