The application of the L2 method to some fractional derivatives with different kernels

This thesis consists of five chapters. In the first chapter, general information regarding fractional derivatives and the L1-2 formula is provided. In the second chapter, the definitions and properties of the Gamma function, Beta function, and Mittag-Leffler function are discussed. In the third chapter, the Riemann-Liouville fractional derivative, Riemann-Liouville fractional integral, Caputo fractional derivative, Caputo fractional integral, Caputo-Fabrizio fractional derivative, Caputo-Fabrizio fractional integral, Atangana-Baleanu fractional derivative in the sense of Caputo, constant proportional fractional derivative, constant proportional Caputo derivative, constant proportional Caputo-Fabrizio derivative, and constant proportional Atangana-Baleanu derivative are introduced, and some of their properties discussed. The fourth chapter presents the introduction of the L1-2 method and examines its application to various types of fractional derivatives. The applications include the following; applying the L1-2 method to the Caputo-Fabrizio derivative, to the constant proportional Caputo derivative, to the constant proportional Caputo-Fabrizio derivative, and to the constant proportional Atangana-Baleanu derivative. The fifth chapter presents the obtained conusions.