Mathematics and Computer Applications - Sciene

   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.

This work is oriented towards two objectives:

The first objective is to classify and study the generalized one-dimensional integral equations that contain n one-dimensional integral operators. This study includes the existence of a unique solution for special types of these integral equations and their solutions by using some quadrature methods, namely the trapezoidal rule, the modified trapezoidal rule and Simpson's rule.