General overview of fractional derivatives with applications

The fractional derivative account is more than 300 years old, one of the most interesting topics in the mathematical world. The basis of the idea of the fractional derivative account dates back to 1695. Many mathematicians have been involved in this subject since the time the idea of fractional derivation emerged. For a long time only theoretical mathematicians have been involved in this topic, but it is possible to come across the work of applied mathematicians and scientists from various branches of science recently. Scientists working on fractional derivative calculations came out of the classical definition of the derivative and developed it. Riemann, Grünwald, Letnikov, Liouville, Caputo, Euler, Abel, Fourier, Kobel, Erdelyi, Hadamard, Riesz and Laplace are major mathematicians contributing fractional derivatives. In this thesis study, it is aimed to investigate the general view of the fractional order and its usage areas in applied sciences. In the first part of this work, the introduction part is given and the appearance of the concept of fractional derivative is mentioned. In the second part, important functions and transformations used in fractional derivative calculations are given. In the third part, some fractional derivative definitions, fractional derivative properties and the derivation of Grünwald-Letnikov and Riemann-Liouville fractional derivatives are discussed. In the fourth chapter, practical fractional derivation methods of some special functions are mentioned. In addition, examples of fractional derivative definitions are given. In the fifth section, differential equations of fractional order are given. The solution methods of these equations are mentioned. Three different solution methods, including the Rezidual power series, Laplace Transformation and Adomian decomposition, are discussed and examples of these methods are given. In the sixth chapter, fractional derivatives are applied to other fields of science such as biology, physics, engineering. In the seventh section, the results of this thesis study are given. Keywords: Fractional derivatives, Fractional order differential equations, Fractional calculations, Grünwald-Letnikov Riemann-Liouville, Caputo fractional derivatives