Solutions of fractional differential equations with Mittag-Leffler kernel by Laplace and sumudu transforms

By looking at the relationship between Riemann-Liouville and Caputo operators, which are important operators of fractional analysis, and the non-singular Mittag-Leffler kernel of Atangana-Baleanu fractional operator, the solutions of fractional differential equations containing these derivatives with the help of Laplace and Sumudu transformations are given. These transformations are also applied to partial differential equations. Similar and superior aspects of the transformations to each other are mentioned. This thesis consists of 8 chapters. In the second chapter, the basic definitions and theorems used in fractional analysis are mentioned. In the third chapter, Gamma, Beta and Mittag-Leffler functions are introduced. In the fourth chapter, fractional derivatives of Riemann-Liouville, Caputo, Caputo-Fabrizio and Atangana-Baleanu are introduced. In the fifth chapter, Laplace and Sumudu transformations are mentioned. In the sixth chapter, examples of applications of Laplace and Sumudu transforms to differential equations containing ordinary, partial and fractional derivatives are given. In the seventh chapter, Sumudu transform is applied to some linear fractional differential equations and their graphs are interpreted. In the eighth chapter, the results are mentioned. Keywords: Atangana-Baleanu fractional derivative, Fractional Differential Equations, Laplace transform, Mittag-Leffler kernel, Non-singular kernel, Sumudu transform