Abstarct: Aabstract Using fractional derivatives in differential equations creates transitive properties for dynamical systems. Unlike the integer-order derivative, there are numerous definitions for the fractional derivative. These definitions, while attractive, also have shortcomings. This thesis first introduces and reviews some definitions of fractional derivatives, then their important shortcomings are analyzed. After that, generalized derivatives are introduced that partially cover the existing shortcomings. Using these new derivatives, various types of differential equations have been rewritten and the existence and uniqueness of their solutions have been investigated. Investigation of the existence of the solution and, more importantly, the conditions for the existence of the solution play a fundamental role in the efficiency of the equation, the definability space of the equation, the conditions for constant values, etc. Also, the uniqueness of the equation solution is considered an important feature of the equation in applied problems. Also, new numerical methods for generalized derivatives have been introduced and numerical simulations of the equations in question have been performed.