Explicit solutions of differential equations with the help of fractional analysis

Fractional analysis, which is a branch of mathematical analysis, is an extension of derivative and integral to arbitrary orders. It has a wide application area in science and engineering. In this thesis, the history of fractional analysis and certain definitions and theorems are mentioned. Our aim is to handle the non-homogeneous linear second order fractional differential equation and solve it with the help the fractional derivate operator by using Leibniz rule. In addition, the time-independent Schrödinger differential equation was taken into account, the second-order Hermite differential equation was obtained with the help of appropriate transformations and its explicit solution was found with the help of fractional analysis.