Masters theses : Statistics
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Item Application of la place transform on orddinary and fractional ordianry differential equations(Al-Neelain University, 2014) Samah Abd Elazim Ali\ i Abstract In this thesis we focused on the concept of Ordinary Differential Equations which appears to be important in physical systems, electri- cal mechanical, Fractional Ordinary Differential Equations and other fields, in particular the basic theory of Fractional Ordinary Differential Equations involving Riernman-Liouville operators and some examples explain their applications. The Laplace transform method has been successfully applied to solve the Ordinary and Fractional Ordinary Differential Equati0ns.The method is very powerful and eflicient in finding solution.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 Applications of Caputo derivatives and Homotopy Perturbation Method in Fractional Calculus(Al Neelain University, 2018-03) Nabil Sulaiman Ebrahim AbdulazizThe aim of this study is to study fractional derivatives by Caputo and Mainardi. We found the exact solution for the linear and nonlinear Reaction-Diffusion equation by using a Homotopy perturbation method, modified Homotopy perturbation method, Laplace transform and Fourier transform.Item Applications Of Symmetry Methods to Differential equation(Al Neelain University, 2004) Sami H i gazi MustafaIn this research we utilize the symmetry condition of differential equations to solve some of these equations . Also we show how Lie algebra methods are much advanced for understanding and unifying all classical methods of solving differential equations . At the end of our research we discuss the variational methods and Euler — Lagrange equations on the light of symmetryItem Approximation by the Method of Least Square(Al Neelain University, 2003-05) Sara Ahmed EbrahimNumerical analysis is concerned with developing methods to find approximate solutions for scientific problems, these methods are used extensively to treat commercial and industrial problems. The problem is first put in a mathematical model, in a form that describes the behavior of the variables, after applying particular relations between them. In this thesis we study some methods in approximation theory. There are several reasons for studying approximation theory and methods, ranging from a need to represent functions in computer calculations to an interest in the mathematics of the subject. Although approximation algorithms are used through out the science and in many industrial and commercial fields, some of the theory has became highly specialized and abstract. Work in numerical analysis and in mathematical s oflware is the o ne of the main links between these two extremes, for it’s purpose is to provide computer users with the efficient programs for general approximation calculations, in order that useful advances in the subject can be applied. This work presents methods using polynomials to approximate a given function or a given discrete data. The work shows many approximation methods, begining with graphical methods, and the methods of average briefly, then the method of least square is given in more details and it is the main topic of this thesis. Some programs are designed to solve general problems in discrete least square problems, and special problems in continuous one.Item Comparison between Elzaki and Laplace Transforms for Solving some Differential Equation(Al Neelain University, 2018-04) Roa Alhussin Saad GebrilIn this study, we solve differential equations by using Laplace transformation and we compare it with Elzaki transformation and we also solve system of ordinary differential equations by using combined Laplace transform - Adomian decomposition method compared with combined Elzaki transform - Adomian decomposition method .Item Comparison between Laplace-Elzaki Decomposition Methods for Solving Some Linear and Nonlinear Partial Differential Equations Rua(Al Neelain University, 2018-04) Ruaa Faisal Musa Abd AlgleelIn this study we presented two kinds of Integral Transforms Laplace Transform, and New Integral Transform "Elzaki Transform". We using this transformation to solve differential equations. We applied Laplace decomposition method and Elzaki decomposition method to solution for nonlinear partial differential equations. We solved some problems(Partial, System of DES and Integral Equations) using Laplace transform and Elzaki transform, and then compared the solutionsItem Comparison Between Riemann-Liouville with Caputo and a New Modified Fractional Derivatives(Al-Neelain University, 2022) Islam Mohammed Arbab AliABSTRACT In this research, we dealt with fractional derivatives and fractional integrals. Also we show the comparison between Riemann-Liouvelle, Caputo and the new modification using the vibration equation in one dimension with homogenous and nonhomogeneous initial conditions. We introduced Fourier and Laplace transform in solution. المستخلص في هذا البحث تناولنا الاشتقاق الكسري والتكامل الكسري. وايضا المقارنة بين ريمان –لوفيل، كابوتو والتعديل الجديد باستحدام معادلة الاهتزاز في بعد واحد مع الشروط أولية اذا كانت متجانسة اوغير متجانسة. وادخلنا تحويلات فوريير ولابلاس.Item Convective heat and mass transfer from a Vertical wall in a non-Darcy porous medium(Al-Neelain University, 2022-01) Ekram Hashim Atta Elmanan HassanAbstract The problem of convective heat transfer from vertical wall in a non-Darcy porous Medium has been investigated numerically by using Spectral quasi-linearization method, the governing equations ruled by Boussinesq approximation together with the thermal boundary layer approximations have been reduced to a nonlinear ordinary differential equation then solved by using quasi-linearization method, our results have been analized and presented in tabular and graphical form الخلاصة مسالة انتقال الحرارة بالحمل الحرارى من جدار عمودى فى وسط مسامى غير دارسي درست عدديا باستخدام طريقة الطيف شبه الخطي . المعادلات الحاكمة في هذا النموذج محكومة بتقريب بوذينيسك وتقريب الطبقة الحديه معاً وتم تقليصها من معادلات تفاضلية جزئيه غير خطية الي معادلات تفاضلية عادية غير خطية عن طريق التحويلات المتشابهه , ومن ثم تم حلها عددياً باستخدام طريقة الطيف شبه الخطى , والنتائج التي تم التوصل لها عرضت في شكل جداول ورسوماتItem Convective Heat Transfer Over Inverted Cone Saturated in Porous Medium(Al-Neelain University, 2019-07) Sahar Abd Alla Abd-Elrahman MohammedAbstract This thesis presents a numerical investigation of convective heat transfer over inverted cone saturated in porous meduim. Using the boundary layer approximations together with the Boussinesq approximation, the set of steady partial differential equations governing the fluid flow is transformed into nonlinear dimensional PDEs form then transfored again into ordinary differential equations using a suitable transformation. To obtain clear insights into the physics of the fluid flow, we used the theory of boundary layer approximation to transform the physical quantities into physical pa- rameters. The resulting highly nonlinear ordinary differential equations were solved numerically by using the spectral quasi-linearastion method. A comparison between SQLM and published results is made to test the accuracy and convergence analysis of the method. The method is found to be convergent and give very accurate results with very few grid points in the numerical discretization procedure.Item The effects of closure (full and partial) on the Covid-19 pandemic using mathematical modeling approach: Case study of India(Al-Neelain University, 2022) Rogaia Mohamed Hassan Alnaiemالخـلاصـة درسنا في هذا البحث ديناميكيات وباء مرض كورونا المستجد (كوفيـد - 19) لدراسة إمكانية السيطرة على انتشار المرض من خلال الإغلاق الكامل أو الجزئي في دولة الهند باستخدام النمذجة الرياضية. لقد حددنا شروط الإيجابية وحدود الحلول. قدمنا شرطًا لوجود نقاط التوازن. تم الحصول على شروط الاستقرار المحلي عن طريق رقم التكاثر الأساسي باستخدام طريقة ليابونوف، وتحققنا من الاستقرار العالمي للتوازن. واستخدمنا البيانات الفعلية لتقدير معلمات النموذج. تم إجراء تحليل الحساسية والمحاكاة العددية لقيم المعلمات المختلفة. Abstracts In this research, we studied the dynamics of the emerging corona epidemic disease (Covid-19) to study the possibility of controlling the spread of the disease through complete or partial closure in the country of India using mathematical modelling approach. We determined conditions for positivity and boundedness of solutions. We provided condition for the existence of equilibrium points. Conditions for the local stability was obtained by the means of the basic reproduction number. Using Lyapunov functions we proved the global stability of the equilibria. We used actual data to estimate the parameters of the model. Sensitivity analysis and numerical simulations for different parameter values were present.Item Fractional Adomian Decomposition Method Compared by Jumarie Fractional Derivative(Al-Neelain University, 2021) Mohammed Ali Mohammed ShaeldinAbstract This research aims to study the Fractional derivatives by means of Jumarie and we found the solution to the equations for wave and heat (linear and nonlinear) respectively, using two methods: the Jumarie Fractinal Derivative Via Tanh-Method. الخلاصة يهدف هذا البحث الي دراسة المشتقات الكسرية عن طريق جوميري واوجدنا الحل لمعادلتي الحرارة والموجة الخطية وغير الخطية علي التوالي باستخدام طريقتان : الطريقة الاولى هي ادميان الكسرية ومقارنتهما مع المشتقة الكشرية لجيوميري بواسطة طريقة الظل الزائدية .Item Functional Analysis Based Methods on Existence and Uniqueness Problems for Partial Differential Equations(Al Neelain University, 2007-11) Um Kalthoum Suliman KanonaIn this study, we considered existence and uniqueness problems for partial differential . equations. . We used functional analysis techniques, where a partial differential equation is regarded as an operator on an appropriate Hilbert space. This Hilbert space is in fact a Sobolev space. In particular, we have dealt with two me/thods, variation methods and energy integral method. We have illustrated the techniques with some examples.Item FUNCTIQN L N SIS EA§E@ QN EXISTEifi€E%fl% %NlQUENES§ Féfi PA§ik NTiAL(Neelain University, 2007) Um Kalthoum Suliman KanonaA§STfiA€T" In this study, we considered existence and uniqueness problems for partial differential equations. . We used functional analysis techniques, where a partial differential equation is regarded as an operator on an appropriate Hilbert space. This Hilbert space is in fact a Sobolev space. In particular, we have dealt with two methods, variation methods and energy integral method. \ We have illustrated the techniques with some examples. \_Item FUNCTIQN L N SIS EA§E@ QN EXISTEifi€E%fl% %NlQUENES§ Féfi PA§ik NTiAL(Alneelain University, 2007-11) Um Kalthoum Suliman KanonaA§STfiA€T" In this study, we considered existence and uniqueness problems for partial differential equations. . We used functional analysis techniques, where a partial differential equation is regarded as an operator on an appropriate Hilbert space. This Hilbert space is in fact a Sobolev space. In particular, we have dealt with two methods, variation methods and energy integral method. We have illustrated the techniques with some examples.Item The geometric formulation of electromagnetic fied(Alneelain University, 2008-08) Tagreed Ahmed F adeel; Tagreed Ahmed F adeelAbstract In this research we studied Maxwell's field equations. The treatment is different from the classical approach. It is both global and fi'ee of coordinates. So we used the language of differential forms and fiber bundle whose base space is a general manifold. The geometrical description is neat and short. Moreover the gravitational force is incorporated in the electromagnetic field, being written in curved space —time.Item the geometric formulation of electromagnetic field(Neelain University, 2008) Tagreed Ahmed F adeelAbstract In this research we studied Maxwell's field equations. The treatment is different from the classical approach. It is both global and free of coordinates. So we used the language of difl’erential forms and fiber bundle whose base space is a general manifold. The geometrical description is neat and short. Moreover the gravitational force is incorporated in the electromagnetic field, being written in curved space —time.Item Homotopy and Homology Theory with Some Applications(AL-Neelain University, 2014) ELHAM DAUOED HAMDOUN ADAMThe purpose of this project is to study topological spaces in terms of certain groups as~ sociated with them, these groups are topological invariants in the sense that isomorphic groups are associated with hornomeomorphic spaces. We discuss two types of groups. The homotopy group for the lower~dimension spaces and homology groups for higher-dimension spaces. As an application of this study we present some computation of homotopy and homology group and prove several theorems in algebra and analysis.Item Homotopy Perturbation Method and Adomian Decomposition Method for solving the non-liner fractional Fisher’s equation(Neelain University, 2018) Hamda Talat Fozy SorialAbstract The aim of this research is to solve the nonlinear fractional Fisher’s equation by using a new modification of Homotopy Perturbation Method (HPM) with Caputo and Reimman derivative and using by the Adomian Decomposition Method (ADM) then we change the period in Fisher’s equation and solved it by new modification of HPM with Caputo and Reimman derivative .Item Indefinite Quadratic Programming(Neelain University, 2008) Hager Elhadi GibrcelABSTRACT - By ageneral quadratic programming QP problem we mean aQP problem with ageneral indefinite Hessian matrix . In this research an algorithm to 'solve general QP problems is designed .It is based on the extended Dantzig-Wolfe method. The main idea of the method is to use stable factorizations of the Lagrangian matrix ,where QR-factorization of the active set matrix is used. The algorithm is tested using different problems.