Numerical methods for solving fractional differential equations

Fractional differential equations play an important role in modeling physical and engineering systems that exhibit memory and persistence properties. These equations are expressed using fractional derivatives, which are a generalized form of the classical derivative concept. In this thesis, the numerical solution of fractional differential equations formulated according to the Caputo definition is addressed. This study focuses on the application of the Euler method for such equations and analyzes its effectiveness. The accuracy, stability, and convergence properties of the method have been examined both theoretically and supported by various example problems. The results obtained demonstrate that the Euler method, under certain conditions, can be applied to solve Caputo fractional derivative systems and provide results with sufficient accuracy. This thesis offers simple and effective numerical solutions in the field of fractional calculus. Most fractional differential equations do not have an analytical solution. Therefore, approximation techniques and numerical methods are used. One of these methods is the Euler method. The aim of this study is to investigate the effectiveness and applicability of the Euler method for first-order ordinary and fractional differential equations.