Analysis of fractional derivatives and integral operators and novel applications

Fractional calculus, unlike integer-order analysis, contains various derivative and integral definitions. For this reason, considering that each definition has some unique features, a confusion of meaning and usage difficulties may arise. In the last few decades, due to the fact that fractional calculus has become very common with its numerous applications in many different sciences, new definitions of fractional derivative and integral operators are added to the existing definitions and this leads to disagreements among researchers. One of the most important reasons for these differences of opinion is the lack of specific classification criteria that are universally accepted to distinguish different types of fractional operators. There are also different opinions about which derivative operators can be used as "fractional" or which fractional operator definitions presented in the literature are considered "new" and which are just a generalization or modification. On the other hand, there are hundreds of studies in which the comparison analysis of classical (integer-order) derivative and different fractional operators on mathematical models using real data, and models with arbitrary order derivatives give more advantageous results. In this thesis, first of all, after giving detailed information about Riemann Liouville fractional calculus, which is the basis of all other fractional operators, a reasonable classification and classification criteria are presented to cover the different perspectives available in the literature in order to distinguish the definitions of fractional derivative and integral according to their properties. Therefore, considering these classification criteria, it will be possible to predict which fractional derivative definitions encountered are useful for the problem under consideration by determining which class they belong to and their important properties. Throughout the current study, it introduces various local derivatives and non-local fractional derivative operators with singular kernels; Mathematical models in physics, engineering, and biology are examined in detail by making a comparative analysis between the operators utilized. On the other hand, the solution methods redefined using the proportional derivative in the class of local derivatives are given for the solution of differential equations. The proportional derivative used in control theory is a derivative with a strong theoretical background and has been frequently used throughout this thesis with its non-local versions. For the models investigated with both local and non-local derivatives, the behavior of the solution curves on the graphs obtained with MATLAB is interpreted by observing. In terms of the results obtained in this study, it has been observed that both local derivatives and non-local fractional derivatives have advantageous effects on linear differential equations and nonlinear differential equation systems. With this study, which allows many different types of arbitrary order derivative and integral operators to be seen together and presents both theoretical and numerical analysis, it has been tried to provide a broad perspective on fractional calculus.