المعادلات التفاضلية الكلاسيكية لهيرميت ولجاندر وتشيبي شيف مشهورة بحلولهم المتعددة الحدود تلك الحدوديات تساهم في حلول بعض المسائل في الرياضيات التطبيقية والفيزياء والهندسة .وحيث أن تلك المعادلات من الرتبة الثانية فإن لكل منها أيضاً حل ثاني مستقل خطيا ليس متعددة حدود هذه الحلول غالباً لا نستطيع وضعها في صورة دوال أولية بمفردها في هذا البحث سوف ندرس المعادلات الكلاسيكية لهيرميت ولجاندر وتشيبي شيف عندما يكون لهاالحد() على الطرف الأيمن والذي يعرف أحياناً بالحد المُجْبِر في المعادلة وسوف نثبت بأنه لكل معادلة وباختيار شرط ابتدائي محدد يكون ضروري وكافي نضمن الحل المتعدد الحدود . عندما يكون هذا الشرط الأبتدائي محدد فإن الشكل التام المضبوط للحل في صورة متعددات حدود يكون موجوداً. Abstract The classical differential equations of Hermite, Legendre, and Chebyshev are well known for their polynomial solutions. These polynomials occur in the solutions to numerous problems in applied mathematics, physics and engineering. However, since these equations are of second order, they also have second linearly independent solutions that are not polynomials. These solutions usually cannot be expressed in terms of elementary functions alone. In this thesis, the classical differential equations of Hermite, Legendre, and Chebyshev are studied when they have a forcing term on the right-hand side. It was shown that for each equation, choosing a certain initial condition is a necessary and sufficient condition for ensuring a polynomial solution. Once this initial condition is determined, the exact form of the polynomial solution is presented.
نجاة علي أحمد الجلالي (2014)

Abstract Laser surface treatment of materials is one of the most effective methods of improving properties and performance of metallic articles. This technique proved its capabilities and advantages in comparison with other methods of surface treatment As an example, it has been shown that the distortion caused by this type of treament is at least one order of magnitude smaller than by that due to other techniques. Moreeover, laser beam is capable of reaching practically any point of the surface whatever it is narrow or difficult to reach by other techniques. The current work investigates the possibility of improving certain mechanical properties of both technically pure aluminum and A1-4o/oSi alloy, using ruby laser of 3 xl 05 W/cm2 power density. The surface of the investigated materials has been impregnated by SiC particles of varying size ranges, with or without firrther alloying of the surface with Titanium powder addition to the impregnated mixture. The laser treatment included coating one surface of each specimen of SiC particles by means of a binder added at a certain ratio (with or without Ti additions to the mixture). The coated surface was then subjected to the laser beam, which melted a layer I OO-150Jilll thick. Due to hydrodynamic effects the melt mixes with SiC particles and dissolves some or all of the added Ti and rapidly solidifies forming a thin layer distinct in properties from the original surface. Two groups of specimens were used in this study. The first group was treated by impregnating SiC particles of one size range with or without Ti-additions. Various numbers of laser pulses have been applied increasmg in a geometrical progression (1-2-4-8 pulses). These specimens were used for the study of the effect of varying: 1) the chemical composition of the substrate materials ;2) the number of laser pulses :and 3) the effect of Ti additions to the impregnating mixture on the microstracture and microhardness of the treated zones. The specimens of the 2nd group were coated by a square network of laser treated zones each 1.5mm in diameter. These specimens have been used to study :1) the effect of the chemical composition of the substrate material ;2) the size of the SiC particles and 3) the effect of Ti addition to the mixture on the wear perfomance of the treated specimens. Identical specimens have been used to determine the solubility of Si into the aluminum phase during the test by means of x-ray diffraction. This dissolved Si is partly responsible for the increase ID microhardness and wear resistance of the investigated specimens. The results show the positive effect of the applied treatment on the properties and preformence of technically pure AI and A-4%Si alloy. The materials microhardness within the treated zones has been increased by a factor of 2.7 and 2.4 for pure Al and Al-4%Si substrates, respectively. The microstructure of the treated zones became more homogeneous and the penetration of SiC particles into them has been increased with increasing the number of the applied laser pulses. The wear resistance of pure Al has been increased by a factor of 73.6 due to laser assisted impregnation, while that of AI-4%Si by afactor of 42.5. Wear resistance was found to increase with I) Si addition to the base metal; 2) Ti addition to the impregnated mixture, and 3) increasing the SiC particle size. This thesis is a part of a wider study that deals with surface treatment of Al and AI-Si alloys undertaken in our laboratry. Such study proved how effective the laser alloying and impregnation of such alloys in improving some of their mechanical properties which are important in industrial application, especially in internal combustion engine design.
أسماء عبد الرزاق وهرة (2002)

لقد تم في هذه الدراسة تطوير منظومةحاسوبية لدراسة تفاعل سلوك نبضة ليزر ية ذات قدرة عالية مع البلازما. ولقد إستندت هذه الدراسة علي منظومة أولية قام بإعدادها كلا من جليرت وهاندكي (Gellert and Handke) وكذلك بعض الإضافات والتعديلات التي أجريت عليها لتأخد في عين الاغتبار التغير في الأطوال الموجية، وإحتمال تغير مادةالبلازما والمرحلة الأيونية المحتملة علاوة علي ذلك الحد الفيضي (flux limiter) الذي تم حسابه من دالة تجربية وتبلور هذا البحث حول دراسة نظرية علي مجموعة من العمليات اللاخطية مثل تشتت برلين(Brillouin scattering) وتشتت رامان (Raman scattering) وفي الأخير أخذ في الاعتبار تخامد لانداو اللاخطي (non-linear Landau damping) وهو الذي يؤدي لإنتاج إلكترونات ذات درجة حرارةعالية جداً نتيجة لإنهيار الموجة الإلكترونية وفي نفس الوقت يعمل علي التقليل من سرعة نمو تشتت رامان كما تم تعديل المنظومة لتأخد في عين الاعتبار إحتمالية حدوث تشتت رامان في تجارب الإندماج النووي الليزري. من هذه الدراسة إتضح حدوث زيادة في الطاقة الممتصة والمترسبة من نواتج عملية تسخين البلازما وذلك بزيادة عرض وقدرة وترردد النبضة الليزرية الساقطة، كما لوحظ أن المعلمات الأساسية مثل درجة الحرارة الإلكترورنية وكمية الطاقة الممتصة والمفقودة على هيئة إشعاع كهرومغنطيسي أنها تصل لأقصي قيمة لها بعد منتصف زمن النبضة الليزرية في حين أن درجة الحرارة الايونية تصل لأقصي قيمة لها بالقرب من نهاية النبضة وقبل الوصول لمرحلة موت اليلازما كما لوحظ أن عملية تحول الطاقة من الإلكترونات إلى الأيونات. كما تشير نتائج الدراسة إلي أن موت إمتصاص برمسترالنج العكسي لا يؤدي انقص كمية الطاقة الممتصة وذلك عند إستخدام نبضة ذات قدرة عاليةبسبب ظهورتستت رامان كواحد من العمليات اللاخطية الهامة التي تسيطر علي مجريات التفاعل بين النبضة الساقطة ومادة الهدف وذلك بسبب زيادة التهيجات الحاثة في كثافة اليلازما وأخيراَ لوحظ عدم حدوث إنحدار حرارى عند سطح البلازما، حيث لميتجاوز الحد الفيضي المقدار 0.1 علية فأن تشتت رامان يجب أن يؤخد في الحسبان عند دراسة الإندماج النووي الليزري ومادة البلازما وذلك عند إستعمال ليزرات ذات طاقة أعلي من .109 watt Abstract In the current work a modification of an algorithm to study the interaction between a high power laser beam and the plasma was undertaken. It was based on a model introduced by Gellert and Handke that was modified by K. K. Mohammed and A. A. Alsghir to take into account the changes due to laser and plasma parameters. The role of Raman scattering including non-linear Landau damping was investigated along with theoretical preliminary investigation of Brillouin scattering. The study shows that there is an increase in the amount of energy being absorbed, lost, and deposited in the plasma. The increase in the laser power, frequency and duration time lead to an increase in such quantities. The electron temperature and energy being absorbed and lost reach their maximum value at the middle of the laser pulse. However, ion temperature reaches that value at the end of the laser pulse and just before the death of the plasma. The transfer of energy from the electrons to the ions takes it's minimum value in the middle of the laser pulse, this results from the increase of the electron temperature. As the laser power increases, inverse Bremsstrahhmg role diminishes, however, the amount of energy being absorbed does not decrease due to the appearanc of Raman scattering as a major contributing process in the laser-plasma interaction. This is due to the fluctuations in the plasma density. Finally, it appears that there are no strong temperature gradients at the plasma surface where the flux limiter does not exceed the value 0.1 Hence Raman scattering should be taken into account in the analysis of the laser fusion experiment esp-ecially when high-power laser pulses are involved.
محمد أحمد خميس الحديدي (1996)