Masters theses : Statistics
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Item APPLICATION OF LIE GRoUPs IN THE SOLUTION OF SoME ORDINARY DIFFERENTIAL EQUATIONS(ALNEELAIN UNIVERSITY, 2003) MOHAMMED ABD AL—BAcI MOHAMMEDABSTRACT Lie's group theory of diflerential equations was initiated by the Norwegian mathematician Marius Sophus Lie (1842-1899). Today, this area of research is actively engaged. In chapter one of this thesis, we give the fundamental concepts of the one-parameter Lie group of transformations, and it also contains the main theorems and definitions. In chapter two, we apply the Lie's theory to the following second order ODE's x:_v3+xy1-1=0, (1) xiv; —y,: —l = 0. (2) _v: —y1 —£=O. (3) y -t:(1”:+,t:_v]: — Zxyl + 2 = 0 . (4) k d y h . = i . k = 1.2. w ere y_ LN In this chapter. we obtain the following results (i) The symmetry groups of(1), (2). (3). and (4). (ii) Reduce the order of (1). (2), (3), and (4) to the first order ordinary differential equations. (111) The general solution of (1). (2), (3), and (4). In chapter three, we considered the non-linear third order ODE y;+Zyy:-_v|2=O. (5)where _vk which is k In this cha (i) (ii) Red equ (m) The (iv) Ne The symmetry groups of (0) w __-Q _ ,___ _ k — d f k -1 v 3 dx nown as Goldstein equation [7]. pter, we obtain the following results uce the order of (5) to the first order ordinary diflerential ation. invariant solutions of (5). solutions from known solutions for (5). IVItem The THEORY OF FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS AND APPLICATIONS(Neelain University, 2017) OSMAN ALI OSMAN MOHMAMEDالخلاصة المعادلات التفاضلية الجزئية مهمة جداً في نمذجة الظواهر الفيزيائية وتصف الإختلافات التي تكون في سلسلة وتعبر عنها الدالة التفاضلية ومشتقاتها الجزئية. من الدرجة الأولى وتصنيفها وتفردها. وقمنا بمعالجة مسائل كوشي للمعادلات الزائدية وحل بعض المسائل وثم طبقنا على المعادلات التي تتعلق بمعادلات الموجة. وقمنا بتنفيذ الحل العددي للمعادلات التفاضلية من الدرجة الأولى. Abstract Partial differential equation are very important in modeling physical phenomena. They describe variation, which are smooth and expressed by differentiable functions and their partial derivatives. In this work we considered first order partial differential equations, their classifications and the uniqueness and existence problems. In particular, we have treated Cauchy problems for hyperbolic equation and solved some of their problems. This has been also applied to wave propagation in addition to this numerical solution of first order differential equations have been investigated for the characteristics formulation.