M.Sc.

The objective of this work may be oriented toward two objectives.The first objective is to study fuzzy set theory, as well as some of its basic algebraic properties and theoretical results.The second objective is to study D-metric spaces and M-fuzzy metric spaces, and some of their properties. Also, this objective includes the study of complete fuzzy metric spaces using M-fuzzy distance function. In addition, some additional results are presented and proved in this work.

In this thesis, we introduce a modified approach for solving fractional order boundary value problems. This approach is given by applying the Riesz-Feller operator to obtain a modified finite difference equation, which is symmetric to the equation of fractional boundary value problems. Also, the main objective of this work is to study the existence and uniqueness theorem of solutions of the fractional boundary value problems, and to present their proof depending on Schauder fixed point theorem for fractional order integral operator.

In this work, we consider two Monte Carlo methods for evaluating the n-dimensional integrals for bounded integrand. Statistical properties of these methods are illustrated and unified. The supported number of trials to estimate the integrals, confidence interval and the efficiency for each method were derived theoretically and assessed practically.