Research Projects

The topic of fractional differential equations (FDEs) was initiated in 1695 by Newton and Leibnitz when L'Hospital pointed out the problem of the derivative 1/2. The investigation of derivatives of fractional orders is covered in the area of mathematics known as fractional calculus. Since it has developed into a potent instrument with more exact and effective outcomes in modelling a variety of complex problems in many apparently different and broad areas of science and engineering, fractional calculus has been a tool of many researchers throughout the past centuries. The primary benefit of FDE is that the memory and inherited characteristics of many different objects and activities may be described using its derivatives. The objective of the study is to obtain the existence and controllability outcomes for Caputo FDEs and inclusion systems. 1. Results on existence and uniqueness for deterministic and stochastic Caputo FDEs of order ? ? (1,2). 2. An analysis of controllability and asymptotic stability for deterministic and stochastic FDEs of order ? ? (1,2). 3. Asymptotic stability and optimal control outcomes for deterministic and stochastic fractional integrodifferential equations of order ? ? (1,2). 4. Null and boundary controllability results for deterministic and stochastic FDEs of order ? ? (1,2). 5. Trajectory and relative controllability results for deterministic and stochastic FDEs of order ? ? (1,2). 6. Exponential stability results for deterministic and stochastic FDEs of order ? ? (1,2). The semigroup theory, sectorial operators, cosine functions, stability analysis, and fixed-point theorem approach can all be used to study the existence and controllability results for fractional differential systems of order ? ? (1,2). The Banach contraction principle, Scauder's fixed point theorem, Dhage's fixed point technique, asymptotic stability, Krasnoselskii's fixed point theorem, Bohnenblust and Karlin's fixed point theorem, Leray-fixed Schauder's point theorem, and Martelli's fixed point theorem will all be used in particular. For numerical, graphical, and simulation, MATLAB will be used.