Analysis and applications of mathematical models using general integral transformations for different fractional derivatives

In this doctoral dissertation, a new analytical framework based on general integral transforms integrated with different definitions of fractional derivatives is developed by going beyond classical approaches for solving fractional differential equations. In addition to classical fractional derivatives such as Riemann-Liouville and Caputo, modern fractional derivatives including Caputo-Fabrizio, Atangana-Baleanu, and constant proportional fractional derivatives are considered, demonstrating their capability to model different memory effects through their kernel function structures. First, a general integral transform theory encompassing the Laplace, Sumudu, and Elzaki transforms is presented, and the compatibility of these transforms with various fractional derivative operators is systematically investigated. This transform, known in the literature as the \emph{New General Integral Transform}, is shown to cover a wide range of special cases through its parametric structure and to exhibit high compatibility with modern fractional derivatives such as Caputo-Fabrizio and Atangana-Baleanu. In the applied part of the thesis, original mathematical models are developed using fractional derivatives in three different fields: respiratory mechanics, economic growth (GDP), and epidemic spread, and these models are analytically solved using general integral transforms. Moreover, a new integral transform called the SUM integral transform is introduced, and its fundamental properties are examined in detail. It is shown that the SUM transform can be reduced to several integral transforms available in the literature as special cases through appropriate parameter selections, and pole analyses and transfer functions corresponding to different fractional derivative definitions are obtained using this transform. In addition, a new transform named the Compatible Fractional General Integral Transform is defined, and some of its properties and applications are presented. In conclusion, this dissertation provides significant theoretical contributions by combining fractional calculus with general integral transform techniques and presents a powerful modeling approach for interdisciplinary applications.