Numerical Solutions for Solving Delay Differential Equations of Fractional Order

      The main theme of this thesis is to study and find the numerical solution of fractional order delay differential equations, and may be divided into two sub objectives, as follows:
  The first objective is to prove the existences, uniqueness and the stability of the solutions of fractional order delay differential equations. The second objective is to find the numerical solutions of fractional order delay differential equations by using the operational matrices of the generalized Hat functions.In this thesis, a modified technique by combining the method of steps and generalized Hat functions for solving fractional order delay differential equations will be proposed. This technique converts the fractional order delay differential equations on a given interval to a fractional order non-delay differential equations over that
interval, by using the function depend on previous interval.  Then apply the operational matrix for generalized Hat function on the obtained fractional order non-delay differential equations to transform linear and nonlinear the fractional order non-delay differential equations into a system of algebraic equations and find the solution. Some illustrative examples are presented and the results of these examples are compared with the existing methods such as Chebyshev wavelet method and the exact solution in order to illustrate the accuracy and efficiency of the proposed method.