Masters theses : Statistics

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APPLICATION OF LIE GRoUPs IN THE SOLUTION OF SoME ORDINARY DIFFERENTIAL EQUATIONS
(ALNEELAIN UNIVERSITY, 2003) MOHAMMED ABD AL—BAcI MOHAMMED
ABSTRACT 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). IV
Approximation by the Method of Least Square
(Al Neelain University, 2003-05) Sara Ahmed Ebrahim
Numerical 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.
STUDY OF SOME Problems In Bubble Dynamics
(Al Neelain University, 2003-08) Mohamed Saad El-Din Abdel Gafoor
Some problems of the bubble dynamics are discussed and the solutions are found. The ﬂuid in which the bubble is moving is asstuned to be incompressible, irrotational and inviscid. The stability of the ﬂuid ﬂows with spherical symmetry is studied and the conditions for its occurrence are deduced. The motion of the bubble produced during Taylor instability of superposed ﬂuids in cylindrical tubes is studied and the effect of the surface tension force and the curvature of the interface are taken into account. Both the steady and the unsteady state of motion are considered. i Spherical — cap models are presented, and the comparison between the theoretical studies and the experimental results is found. V The potential solution for the boundary value problem of the toroidal bubble in an inﬁnite ﬂuid is found by using the toroidal coordinate system. Botl1 the three dimensional and the axisymmetric case are considered.
Applications Of Symmetry Methods to Differential equation
(Al Neelain University, 2004) Sami H i gazi Mustafa
In 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 symmetry
Lubrication Theory
(Al-Neelain University, 2004) Jawahir Zakaria Adam
This thesis is a study of Lubrication theory .Thc consept of Lubrication its to types and practice and some deﬁnitions are introduced in chapter one . In chapter Two the Reyn0ld's equation is derived. Chapter three deals with Further theory of Lubrication. In chapter four the variational ,approach to Lubrication problems is studied boundary conditions uses are also discussed .
SOME APPLICATIONS OF GREEN'S FUNCTIONS METHOD
(Alneelain University, 2005) TARIQA. AZIME A. HALEEM
This thesis deals with solving the Boundary Value Problems by applying the Green's Function Method, on the Ordinary Differential Equations of the second order, and Partial Differential Equations of the ﬁrst and second order including higher orders. Using Green's Function Method to solve such equations by using simple ways such as: Laplace and Fourier transformations and their inverse transformations, including Bessel's functions, and F0urier's series. It also include the basic concepts, i.e. inner product, differential operators, adjoint operator, self—adjoint operator, unit-step function, the delta function, convolution and known integrals. .
symmetry condition of algebraic and differential equations
(Neelain University, 2005) mnahil mohammed bashier
This study deals with the application of symmetry concept of the solution of the differential equations. The study covers the meaning of symmetry for the differential equation and it gives some examples of it, for centain group especially the rotation group in theplane. »=~ - ~ ~-’ " " We examine if a centain group represents symmetry group of a differential equation, by considering the differential equation as algebraic equation. This is done by studding the partial derivatives of the dependent variables with respect to the independent variables. This leads to consider space called Jet space. The differential equation is a kernel of a map, whose kernel is in fact subspace of the Jet space, which is invariant under group prolongation G, the symmetry group for differential equation. The study introduced some basic concepts which we need to calculate the symmetry group for the differential equations. One of these is the inﬁnitesimal generator for the one parameter group. For this we consider the vector ﬁelds with some examples and we give the important features of it. We also provide the criterion of the group G to be a symmetry group for the differential equation. We calculate the symmetry group for the heat equation and we deﬁne all the one parameter groups which represent the symmetry of heat equation and we conclude-with the general form of the solution of the heat equation. i To prove this utility we studied the integration theory of ordinaiy differential equation through the symmetry concept. The theory shows that this concept uniﬁed several ways for solving the differential equation of ﬁrst order especially the ideas of separation of variables and exactness which represent the basis of solving the ﬁrst ODE
symmetry condition of algebraic differential equations
(Neelain University, 2005) mnahil mohammed bashier
This study deals with the application of symmetry concept of the solution of the differential equations. The study covers the meaning of symmetry for the differential equation and it gives some examples of it, for centain group especially the rotation group in theplane. »=~ - ~ ~-’ " " We examine if a centain group represents symmetry group of a differential equation, by considering the differential equation as algebraic equation. This is done by studding the partial derivatives of the dependent variables with respect to the independent variables. This leads to consider space called Jet space. The differential equation is a kernel of a map, whose kernel is in fact subspace of the Jet space, which is invariant under group prolongation G, the symmetry group for differential equation. The study introduced some basic concepts which we need to calculate the symmetry group for the differential equations. One of these is the inﬁnitesimal generator for the one parameter group. For this we consider the vector ﬁelds with some examples and we give the important features of it. We also provide the criterion of the group G to be a symmetry group for the differential equation. We calculate the symmetry group for the heat equation and we deﬁne all the one parameter groups which represent the symmetry of heat equation and we conclude-with the general form of the solution of the heat equation. i To prove this utility we studied the integration theory of ordinaiy differential equation through the symmetry concept. The theory shows that this concept uniﬁed several ways for solving the differential equation of ﬁrst order especially the ideas of separation of variables and exactness which represent the basis of solving the ﬁrst ODE.
SOLVING SOME PROBLEMS OF LINEAR ALGEBRA WITH COMPUTATIONS
(Al Neelain University, 2005-07) ALI MOHAMED ABU OAM
This thesis provides a computational solution for some problems in Linear Algebra, giving many examples and applications. I considered the computational solution that depends on maple soﬁware, version 9 which is distributed in June 2003. To deal with any math soltware, you have to use packages and commands to display results. A package is a collection of routines (and perhaps other data) that are collected together in some way. Typically a package provided a range of functionality for solving problems in some well-deﬁned problem domain. The scope of a package may be quite wide or very small. The package must be writ- ten aﬁer the " with " command, which is an interactive package management utilities. and is effective only at the top level. The packages which were used in this text are : O plots I is a graphics package. 0 linalg 1 is a linear algebra package based on array data structures. 0 Linear Algebra : is a linear algebra package based on rtable data structures. 6 The "rrab/e( )" function is the low level routine which is used by maple to build an array. a matrix and a vector. I student[LinearAlgebra]: is a subpackage from LinearAlgebra package which has three princ- ipals components : interactive, visualization and matrix computation Each package provided many commands to activate the functions that the package is based on. A command is a direct order for a certain problem to be solved. a command can be short or a long stat- ment. The command is in red ( input ), and the result is in blue ( output). To display the result, end the command's statement with a semicolon. otherwise end the statement with a colon. l explained each co- mmand in its location when it has been used.
ON THE LINEAR INTEGRAL I: EQUATIONS
(Neelain University, 2006) KHALID ABD ASSALAM ATEIA
The reader may be surprised about what the linear integral equations. And that for the little importance that had paid for this topic. The integral equations is a mathematics method which had used to solve many problems in physics arid chemistry, and also at mathematic. Our considerable mathematic method can be also used to solve the differential equations which can be also reduces to a one of the integral equation. Our research will consider the following points:- l\ the deﬁnition of the linear integral equations, and explain some of its usefulness. 2\ some method of solution for the linear integral equations. So in our ﬁrst section we had to deﬁne the linear integral equations and we obtained some of its uses in mathematics, physics and chemistry , also we take the numerical solution for the linear integral equations. But we had just explained the method of solution for some reason we had written. _ In the second section we explained a method to solve the integral equations of the ﬁrst kind , and also it uses to solve the integral equations of the second kind . At the third section we had the approximation solution, and we clear the co-theory for this method of solution. More over our fourth section is specialized on the algebraically solution to the linear integral equations with the assistance theorems for it. And at special case for the kernel of the linear integral equations we take our ﬁfth section and we joined our algebraically solution with Fourier series. _ The sixth section had taken as applications to some problems which had reduced to an integral equations. . At last we had a conclusion to our research, and we take some problems in the last section as applications for the methods of solutions which we had explained in the previous sections.
On The Propagation Of Wave Motion In An Incomparessible Fluids
(Al Neelain University, 2006-08)
ln this research the researcher discussed in chapter l, the fundamentals of ﬂuid mechanics and the principal equations that govern the motion ofideal and real ﬂuids. ln chapter 2, the researcher encountered to the general concepts in Wave motion in ﬂuids, and the mathematical description of waves. Chapter 3, was devoted for the theory of linear surface waves. lt discussed the wave velocity and group velocity and the energy contained in the waves , and standing and stationary waves. The researcher concluded the research by chapter 4, which discussed non linear waves ofsurface waves, and classifying them according to their waveforms.
FUNCTIQN L N SIS EA§E@ QN EXISTEiﬁ€E%ﬂ% %NlQUENES§ Féﬁ PA§ik NTiAL
(Neelain University, 2007) Um Kalthoum Suliman Kanona
A§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. \_
Functional Analysis Based Methods on Existence and Uniqueness Problems for Partial Differential Equations
(Al Neelain University, 2007-11) Um Kalthoum Suliman Kanona
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 me/thods, variation methods and energy integral method. We have illustrated the techniques with some examples.
FUNCTIQN L N SIS EA§E@ QN EXISTEiﬁ€E%ﬂ% %NlQUENES§ Féﬁ PA§ik NTiAL
(Alneelain University, 2007-11) Um Kalthoum Suliman Kanona
A§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.
the geometric formulation of electromagnetic field
(Neelain University, 2008) Tagreed Ahmed F adeel
Abstract 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.
INTEGRAL TRANSFORM METHODS SOLUTION OF BINGHAM FLUID FLOW
(Neelain University, 2008) fawzia mansour elniel dalam
ln this research we study the Bingham ﬂuid ﬂow by the method of integral transforms which are considered as the most efficient methods for solving physical and engineering problems. ' Investigation of ﬂow characteristics of Bingham ﬂuid and the differences between a Newtonian and non-Newtonian ﬂuid was given. The basic equations that describing shear stress and velocities of non-Newtonian ﬂuids were discussed. Finally we applied the Laplace and Fourier transforms for solving a problem of unsteady state unidirectional ﬂow of Bingham ﬂuid and We found that the integral transform methods is suitable for problems of this kind.
Indeﬁnite Quadratic Programming
(Neelain University, 2008) Hager Elhadi Gibrcel
ABSTRACT - 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.
numerical solution of poisson equation using pascal language
(Neelain University, 2008) Nada Osman Hassan
ABSTRACT The researcher targets to use the numerical solutions methods in solving the partial differential equations using the computer speed in the solution . The researcher has come to use the numerical solutions because the mathematical solutions needs great effort and an limited time. Moreover, there are some equations which can not be solved manually. Also the computer adds to the fact that it is very accurate in solving the equations, because of its great power in dealing with big numbers either integer or real. The research has included an example of aproblem that has been solved by the computer and compared with the manual solution. In additionto the above the research includes the decriesed of the number of the solution iterations.
INTEGRAL TRANSFORM METHODS SOLUTION OF BINGHAM FLUID FLOW
(Alneelain University, 2008) fawzia monsour dalam
abstract In this research We stu dy the Blngham ﬂuid ﬂow by the method of integral transforms which are considered as the most efficient methods for solving physical and engineering problems. Investigation of ﬂow characteristics of Bingham ﬂuid and the differences between a Newtonian and non-Newtonian ﬂuid Was given. The basic equations that describing shear stress and velocities of non-Newtonian ﬂuids were discussed. Finally We applied the Laplace and Fourier transforms for solving a problem of unsteady state unidirectional ﬂow of Bingham ﬂuid and we found that the integral transform methods is suitable for problems of this kind.
|NTEGRAL TRANSFORM METHODS SOLUTION OF ‘ BINGHAM FLUID FLOW ‘
(Al Neelain University, 2008-04) Fawzia Mansour Elniel Dalam