M.Sc.

The Fractional Order Bounded Delay Differential Equations (FOBDDE’s) has been studied in this work. The Existence and Uniqueness theorems of such type of differential equation have been proved, by using the successive approximation techniques. Also, the analytic solution of (FOBDDE’s) are presented, using Laplace Transformation, and the numerical solutions are discussed, using general one-step methods and linear multi-step methods. The comparison, among these methods and the exact solutions are presented.

   In this thesis, some properties and basic definitions of fractional integral and derivatives of Riemann-Liouvill  are presented , to construct the optimality conditions of  mixed order  unconstrained and constrained variational problems with continuous and discontinuous functional, on fixed and moving boundaries ,based on the classical product  rule for Riemann-Liouvill , Several tested example are presented to demonstrate the implementation of the optimality necessary conditions.
    

Fractional (or non-integer) differentiation is an important concept both from theoretical and applicational points of view. The study of problems of the calculus of variations with fractional derivatives is a rather recent subject. In this work, some properties and basic definitions of fractional integral and derivatives of Riemann-Liouvill are presented.