Mathematical modelling of real-life problems usually results in functional equations, like ordinary or partial differential equations, integral and integro- differential equations, stochastic equations. Many mathematical formulation of physical phenomena contain integro-differential equations, these equations arises in many fields like fluid dynamics, biological models and chemical kinetics integro-differential equations are usually difficult to solve analytically so it is required to obtain an efficient approximate solution.

In this paper we establish the existence of solution for two boundary value problems of Fredholm functional integro-differential equations with nonlocal boundary conditions.

The effect of the rare earth (RE) element Y on the microstructure and hardness of (Mg–0.5Zn–0.5Zr–2.8Nd –1.5Gd) wt% Mg alloy investigated. 1 wt. % Y was added and compared with the base alloy. The microstructure results show the refinement of the grain by the addition of Y and the grains became smaller about 31.8 % and the volume fraction was increases 11.1% %, which led to the increment of hardness from 48.33 HV (as-cast EV31A) to 53.71 HV (as-cast EV31A +1 Y). Energy dispersive spectroscopy (EDS) and X-ray diffraction (XRD) results showed that the base alloy mainly contained α-Mg matrix and Mg-(Nd, Gd) as a second phase crystallized along the grain boundaries and addition of (Y) yttrium resulted in the formation of the new phase, Mg-Zn-Y-phase was found as a new ternary phase, where Y combined with the original second phase.

The influence of holmium on the microstructure and hardness of Mg-Nd-Gd-Zn-Zr alloys 

were investigated. Conventional casting methods are used to produce the alloys. All the results were 

characterized by optical microscopy, scanning electron microscope (SEM) and the Vickers hardness 

test to highlight the influence of holmium addition. The addition of 2.0 wt.% holmium leads to the 

combination of rare earth elements which formed Mg-Zn-Nd-Ho phase. The results have shown the 

addition of Ho improved the microstructure and hardness of Mg-Nd-Gd-Zn-Zr alloys. By adding 

grain size of 2.0 wt.% holmium had reduced by 18.43%, while the volume fraction increased by 

7.34%. The Vickers hardness value improved 6.18% due to the grain refine and volume fraction 

precipitates. The 2.0 wt.% holmium addition showed a positive result in microstructure and 

hardness value.

رسالة ماجستير باللغة الفرنسية في القانون العام

تهدف هذه الدراسة إلى التعرف على مصادر استخدام علاقات التعلم التعاوني بين الطلاب الذين

يدرسون أي موضوع في الفصل .هذا قد يساعد المعلمين من أي موضوع في أي مستوى، سواء

كان ذلك اللغات الأجنبية والرياضيات أو العلوم الاجتماعية. الأداة الرئيسية التي من خلالها

الباحث يجمع البيانات هو استبيان لمعرفة كيفية استخدام المعلمين نصائح لتعليم التعلم التعاوني،

لذلك يجب على المدرسين اتباع النصائح لجعل التعلم نشط والمشاركة المستاوية في الفصول

الدراسية .

This study aims to investigate the sources of using cooperative learning relationships between learners studying any subject in a class .This study may help teachers of any subject at any level, whether that be foreign Languages, math's or social studies . The main tool through which the researcher collected data is questionnaire to find out how teachers use the tips of teaching cooperative learning, so teachers must follow these tips to make active learning, equal participation in the classrooms