Fractional calculus is the important branch of mathematical analysis field. It covenants with the requests and exploration of derivatives and integrals of random order. Periodic motion and special phenomena are very important not only in natural science, but also in social science. In general, not all differential equations have periodic solutions. But there exist special differential equations which are interesting to get their periodic solutions, which are neutral and Rayleigh differential equations. Rayleigh differential equation has many applications in physics and biology. It deals with experimental results with periodicity. The existence of periodic and positive outcome is established in a new method. Some available work in the literature are developed and extended on the existence and uniqueness of a class of fractional differential equation. All the classes of fractional differential equations are interpreted under Riemann-Liouville fractional calculus. This thesis focuses on the development of fractional calculus to extend and generalize many classes of fractional differential equations. The main scope of this study is fractional neutral differential equation, fractional Rayleigh differential equation and fractional neutral Rayleigh differential equation. This study establishes new results on the existence and uniqueness of fractional neutral, fractional Rayleigh and fractional neutral Rayleigh differential equations. A mathematical model was studied by generalization of the neutral differential equation of the first order. A composition theorem for m-PΛΛ functions under appropriate conditions was proved based on interpolation theory and Banach’s fixed point theorem. Therefore, the solution, in this case, is unique. As the Riemann–Liouville fractional calculus was not periodic, a construction to get the periodicity of some classes of fractional differential equations was introduced. A Rayleigh-type equation with state-dependent delay was considered and the existence of periodic solutions to this equation was investigated. During the generalization, Riemann–Liouville fractional derivatives was utilized and sufficient conditions for the existence of periodic solutions were obtained.
