كلية العلوم الرياضية والاحصاء
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Item Theory of Lie algebra with applications to symmetry and conservation laws(Al-Neelain University, 2016) Ismail Mustafa Mohammed OthmanAbstract In this research we studied Lie groups and Lie algebras . We considered the classification theory of complex and semi-simple Lie algebras , provided with sufficient knowledge of the properties of Lie algebras . We then concentrated on the application of Lie algebra to the conservation laws , in particular Noether theorem . We also studied the application of Lie algebra to particle physics including the utility of the representation theory of the five dimensional special unitary group. We concluded our research program with the study of symmetry groups of differential equations . IV المستخلص في هذا البحث درسنا زمر لي و جبر لي. درسنا نظرية تصنيف جبر لي المركبة وشبه البسيطة، مع توفير قدر كاف من المعرفة عن خصائص جبر لي. ثم ركزنا على تطبيق جبر لي لقوانين الحفظ ، وال سيما نظرية نوثر . درسنا أيضا تطبيق جبر لي لفيزياء الجسيمات يضم استخدام نظرية تمثيل الزمر ذات البعد الخامس في دراسة زمرة الوحدة الخاصة. ختمنا بحثنا بدراسة زمر التماثل للمعادالت التفاضلية.Item Lie Algebra Treatment of Differential Equations(Al-Neelain University, 2016) Mahmoud Adam Aliي ھذا البحث ندرس تمـــــاثل المعادلات التفاضلیة و توضیح كیفیة إستخـــدام التمـــــــاثل فــــي تكــــــــامل المعــــادلات التفاضلیة . في الواقع التماثلات ھي زمر لي مع جبر لي التي یمكن إستخدامھا لإكتشاف اللامتغیرات للمعـــادلات التفاضــلیة التي تحتاج إلي إنشاء حلول. خلال ھذه الدراســـــة إعتبرنا زمرلي لإنشاء حلول المعادلات التفاضلیة. في نھایة ھذأ الدراســـــــــــــة اوجدنا حلولا ً لبعض الامثلـــــــــــــة لتوضیــــــــح تقنــــــــــــیات التمـــــاثل.Item Geometrical Reformulation of some Equations of Fluid Mechanics(Al-Neelain University, 2015) Mansour Hassan MansourAbstract The aim of this thesis is to explain some of the connections between fluid mechanics and differential geometry and to shed light on formulation of classical fluid mechanics in a differential geometric language. The thesis presents a reformulation of some of the most basic entities and equations of fluid mechanics, the continuity equation and the momentum equation of motion, in a modern differential geometric language using calculus of exterior differential forms on manifold (exterior calculus). Also, the study investigates the integrability of some fluid problems from geometrical perspective, with particular attention to the Euler equations of motion.Item Convective Heat Transfer in Micropolar Fluid in Presence of Radiation(Al-Neelain University, 2009) Mohamed Elhafiz MohamedAbstract In this thesis we study the developing theory of micropolar fluids and compare that with ordinary fluids, and the of heat transfer mechanism in boundary layer micropolar fluids, after reviewing of boundary layer in ordinary fluid as well as magnetohydrodynamics. Mainly the effects of radiation on the flows and heat transfer of an electrically conducting micropolar fluid past semi-infinite vertical plate in presence of uniform magnetic field are considered. The non-linear partial differential equations governing the problem under consideration are converted into a system of non-linear ordinary differential equations by means of similarity transformations, the resulting system of coupled non-linear ordinary differential equations, is solved numerically for a range of values of radiation, magnetic, microrotation and vortex viscosity parameters and also, Grashoff’s number, is obtained by applying an efficient numerical technique based on shooting method, using (mathematica pages). The effects of these parameters are examined on the velocity of the fluid, temperature distribution and angular velocity of microstructure, as well as the coefficient of heat flux and shearing stress at the plate. The velocity of fluid, angular velocity and temperature profiles are shown graphically and the numerical results are obtained. 4 الخلاصة نافشت الرسالة النظرية والتطبيق فى الموائع الميكروبولرية ( دقيقة القطبية ) و مقارنتها بالموائع التعتيادية . وبصسسورة رئيسسسية درست انتقال الحرارة جوار جدار راسسى نصسف لنهسائى وتساثيرات العشعاع فى وجود مجال مغناطيسى تعلى حركسة المسائع, وفسى هسذا حلت معادلت النظام تعدديا بعد تحويلها من معادلت تفاضلية جزئية غير خطية التماثلية الى معادلت تفاضلية تعادية غيسر خطيسة وامكسن رسسم منحنيسات توزيسع سسرتعات النمسوذج للمسائع, السسرتعة الزاويسة للجزئيات و توزيع الحرارة لقيم مختلقسة للعوامسل الستى تتحكسم فسى الحركة. كما امكن ايضا حسساب القيسم العدديسة لمركبسات الحتكساك .تعلى جدارومعامل انتقال الحرارةItem Applications of Homotopy Perturbation and Variational Iteration Methods in Fractional Calculsus(Al-Neelain University, 2016) Altayb Ahmed Mohammed AliAbstract This research aims to solve some problems of differential equations which contain fractional derivatives by using homotopy perturbation method (HPM) and variational iteration method (V IM). Chapter one contains Homotopy perturbation (HPM) and its use in solving differential equations. Chapter two contains some examples in fractional derivatives. Chapter three contains applications of the (HPM) to solve the equations that contain fractional derivatives. Chapter four contains applications of variational iteration method (V IM) to differential equations. Comparison with (HPM) is made through examples.Item Applications of Finite Element Method on Beams(Al-Neelain University, 2015) Eman Jamal-Aldeen Alkhair Altahirمستلخص طریقة العناصر المنتھیة ھي عرض بطریقة متغایره لحل المعادلھ التفاضلیھ . نضع فیھا المسألھ المستمره في cjj ،الدالھ التقریبیھ شكل معادلھ تفاضلیھ مكافئة لصیغھ التغایر، ویفترض الحل في صورة تركیب خطي، ھي j . البارمتیرات cj تحدد باستخدام صیغة التغایر . طریقة العناصر المنتھیھ تحسن تقنیھ النظام للدالھ التقریبیھ لمجال بسیط مركب ھندسیا . في طریقة العناصر المنتھیة، الدالھ التقریبیھ ھي كثیرة حدود (كثیره الحدود تعرف لكل مجال وتسمي بالعنصر) Abstract The finite element method is introduced as a variationaly based technique of solving differential equations. A continuous problem described by a differential equations is put into an equivalent variational from, and the approximate solution is assumed to be a linear combination , Pcjφj , of approximation function φj . The parameters cj a determined using the associated varitional form. The finite element method provides a systematic technique for deriving the approximation function for simple subregions by which a geometrically complex region can be represented. In the finite element method, the approximation function are piecewise polynomials (i.e, polynomials that are defined only on a subregion, called an element)Item Application of Spectral Adomian Decomposition Method on Heat and Mass Transfer of Boundary Layer(Al-Neelain University, 2015) Salma Hassan Mohammed HassanAbstract In this study, we proposed numerical method for solving non-linear differential equations. The method is hybired of spectral and Adomian decomposition method. The method will applied on the problem of convection heat and mass transfer, the convergence of the method tested and displayed on tables. The velocity, temperature and concentration have been obtained and discussed for various physical parameter.