Investigation of the solutions of some differential equations in mathematical physics with fractional calculus

In nature as well as human life, there exists many complex systems which develops stochastically. In this thesis, the insufficiency of the standard mathematical physics for describing the dynamics of complex physical systems are presented and the importance of the fractional calculus in describing stochastic systems realistically are outlined. With the aim of to reveal the physical mechanism underlying the success of fractional calculus to describe complex stochastic systems, cumulative method is developed in view of the Fibonacci approach which takes care of the development of the system in a compound manner. Thus, it is seen that complex processes which evolve in a fractal space with memory effects could be handled with cumulative method in parallel to the fractional calculus. In cumulative method, the physical property of the environment in which the complex physical system evolves is imposed with a parameter $\alpha$. In our derivation, the development of physical systems have been handled with the operator technique of the cumulative method. In our approach, the parameter $\alpha$ emerges as the power of the operator which corresponds to the order of the differintegration of the fractional calculus and parameter $\alpha$ is considered as a measure of the coupling of the physical system with its environment. For the purpose of application, the investigation of the Brownian movements, population dynamics and the life and the mortality probabilities of the living creatures are considered within the framework of fractional calculus and cumulative approaches. The effects of environment on the living population dynamics, life and mortality probabilities of individuals are illustrated via $\alpha$ on the sketched figures.