الفهرس 
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Abstract
Fractional calculus is a generalization of the ordinary differentiation and integration to any order. This thesis introduced a numerical method depends on shifted Gegenbauer polynomial and its orthogonality property for forming a new matrix called shifted Gegenbauer operation matrix of Riemann- Liouville fractional integral for solving some types of fractional linear differential equations, systems and optimal control problems. Also at the thesis, we are studying the main structure properties of fractional normal and singular control systems.