Browsing by Author "Mohammed, Seid (PhD)"
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Item Analysis of Fourier Transform in _L1 Space and its Inversion(Addis Ababa University, 2012-01) Jenber, Dagnachew; Mohammed, Seid (PhD)This project discusses the concept of Fourier transform of a function in Space with its properties theorem, inversion theorem, Fourier sine and cosine transforms theorem, Plancherel’s and Parseval’s identities theorem and the applications of Fourier transform in partial differential equations, Shannon’s sampling theorem and Heisenberg’s inequality. Therefore the purpose of this project is to solving certain problems in partial differential equations like for example Heat equation, Wave equation , and Laplace equation, to solve some complicated integrals shortly and simply, and it works in Shannon’s sampling theorem and Heisenberg’s inequality. This project uses some definitions and theorems as a preliminary from some real analysis and Fourier analysis booksItem The Basic theory of Sturm-Liouville Boundary Value Problems(Addis Ababa University, 2010-06-06) Assefa, Woldegebriel; Mohammed, Seid (PhD)In this paper we shall present the basic theory of Sturm-Liouville boundary value problem (BVP) together with a discussion of some of the powerful methods that are used to solve the Sturm-Liouville boundary value problem (SLBVP). Especially the paper focuses on Eigen values and Eigen functions of SLBVP. We devoted much time to the SLBVP due to their particular importance in solving many physical problems in physics, engineering, and other many sciences such as surface heat transfer, vibrations of a vibrating string, transverse displacements in a stretched string, a vertically hung elastic bar, heat flow in a non uniform rod, circularly symmetric heat flow, and the like. The main objective of this paper is to investigate the existence of Eigen values and Eigen functions of a Sturm-Liouville boundary value problem. The paper consists of four main sections: The first section, the introductory part, deals about general introduction and some preliminary considerations. In the second section, the SLBVP will be defined and discussed; Eigen values and Eigen functions of SLBVP will be defined & elaborated through examples. And then some basic properties of SLBVP (theorems on SLBVP) are stated and clarified where the proofs of most of these properties are also included. Having familiarized with SLBVP in the second section, in the third section,(the core part of the paper), We will come across with Eigen function expansion and existence of Eigen values; not only this more additional properties of the Eigen values and Eigen function of the SLBVP such as the completeness of Eigen functions, the Rayleigh quotient are presented in this section. In the last section, a method of solving non homogeneous SL type BVPs is presented where the existence of solution of non homogeneous second order SL type BVPs will be compared with analogous SLBVPs. It should be pointed out that these topics are far more in depth than what will be covering here. In fact we can do a whole course on these topics.Item Compa(ct Linear Operators on Blilbert Space and their App[ication to Integral Equations(Addis Ababa,, 2011-01) Getachew, Anteneh; Mohammed, Seid (PhD)Perhap the most theory developed about eigenvalue, eigenve t rand exi ten f oluti on of a certain operator equ ation i for matrix operator , but th fir t part t b con idered apart from matrices are integral operators, since the cia ical form ul ation from phy ics, chemistry, engineerin.g, stati stics are of thi nature and the tudy of an integral operators gives birth to the modern functional analysi . And the mo t common problem in appli ed mathematic are diffe rential operators which are fruitful source of integral equat ions. Due to various advantages of having integral equation rather than di ffere nti al equation usually we would like to convert and formulate differential equation in to integral equ ation. Many of integral operators encountered in application are bounded operators and many of them are, in fac t, in special classes of bounded operator called compact operators and again the most important classes of compact are the HilbertSchmidt operators.Item Conformal Mappings and the Riemann Mapping Theorem(Addis Ababa University, 2010-06-06) Welay, Kewani; Mohammed, Seid (PhD)It is easy to point out that any function of a complex-variable can be considered as a mapping from one complex plane into another. A great deal of attention is devoted to the study of holomorphic functions. The reason for this is that many problems from the theory of holomorphic functions can be solved according to the following procedures; 1. Solve the problem for the simplest possible type of domain; 2. Express the desired solution in terms of the one already found with the aid of a mapping. A non-constant holomorphic function maps a domain of the z −planeonto another domain f (z) of the w−plane .At points where ' f z 0 such a map has the remarkable property that it is conformal. This means that any two smooth curves intersecting in map into curves which intersect at the same angle in f .By means of conformal mapping, problems of fluid flow, electrostatics and other fields can be mapped into simpler problems of the same general sort in f .Solution of the problem in f then solves the original problem in .Conformal mapping also gives geometrical insight into analytic(holomorphic) questions. This seminar paper includes definitions and basic properties of holomorphic functions ,conformal mappings(the bilinear transformations, The Schwarz-Christoffel transformation),Normal Families, the Riemann Mapping Theorem, which characterizes those domains that can be mapped conformally onto the unit open disk and concludes with one of the consequences of the Riemann mapping theorem.Item Graduate Seminar Report on Convergence of Fourier Series(Addis Ababa,, 2010-06) W/Meskel, Addisu; Mohammed, Seid (PhD)Item Graduate Seminar Report on Harmonic Functions(Addis Ababa University, 2005-06) Geysa, Sidise; Mohammed, Seid (PhD)Item Graduate Seminar Report on Some Basic Properties of Theory of Hardy Space(Addis Ababa University, 2003-06) Dessie, Hunegnaw; Mohammed, Seid (PhD)Item Graduate Seminar Report on the Convergence of Fourier Series of Functions in a Homogeneous Banach Space(Addis Ababa University, 2003-06) Goa, Menegistu; Mohammed, Seid (PhD)Item Harmonic Functions on a Disc and Upper Half Plane(Addis Ababa University, 2010-06-06) Wolde, Moybon; Mohammed, Seid (PhD)This seminar report is devoted to the study of harmonic functions on disc and upper half plane. These functions are closely connected to analytic functions since the real and imaginary parts of analytic function are harmonic functions. The study of harmonic functions is important in physics and engineering, and there are many results in the theory of harmonic functions that are not connected directly with complex analysis. However, in this seminar we consider that part of the theory of harmonic functions that grows out of the Cauchy Theory and Mo ̈bius transformation. These functions have many interesting properties like mean value property (MVP) and the maximum principle. One of the most important aspects of harmonic functions is that they arise as functions that solve a boundary value problem for analytic functions, known as the Dirichlet problem. An example is the problem of finding a function continuous in a closed disk D (upper half plane H)that assumes certain known values on the boundary of the disk D (H) and is harmonic in the interior of the disk D (H). An important tool in the solution is the Poisson Integral formula. This paper is organized in to three chapters. In the first chapter we consider basic properties of real and complex function theory with some topological concept. It includes basic definition and notation, class of analytic function and continuous function and Cauchy theory. The second chapter deals with harmonic function locally as real part of analytic function and the interesting property of harmonic functions such that mean value property (MVP) and the maximum principle and its integral representation, that develops from the theory of conformal mapping. The third chapter deals with the solution of Dirichlet problem on disc and upper half plane and positive harmonic functions.Item Iteration of Rational Function, Properties of Faout Sets(Addis Ababa University, 2012-01-01) Mengistu, Muluken; Mohammed, Seid (PhD)The Central idea of iteration theory is the division of the complex plane in to Fatou set and Julia sets. Especially the paper focuses on properties of Fatou sets of iteration of rational function. We devoted much time on the Fatou and Julia sets due to their particular importance to express properties of Fatou sets. The main objective of this paper is to investigate properties of Fatou sets of iteration of rational function. The paper consists of four main sections: The first section is introductory part, deals with iteration(repeated application) of rational function and some facts on iteration of rational function. In second section, the rational map, conjugacy, valancey, fixed point, Critical point will be defined and discussed; The Extended complex plane, Lipthyz continuous will be defined and elaborated through examples. In the third section, we will come across with equicontinuity and completely invariant sets; not only this formal definition of fatou and Julian sets presented in this section. In last section but the core part of the paper Exceptional points and the back ward orbit will be defined. And then some basic properties of Fatou sets are stated and clarified where the proof of most of these properties are also included. It should be pointed out that these topics are far more in depth than what will be covering here. In fact we can do a whole course on these topics.Item Representation of Positive Linear Functional(Addis Ababa University, 2011-01) Jifara, Worku; Mohammed, Seid (PhD)In analysis, if we consider thespace, the Riesz representation theorem states that, if is a bounded linear functional on spaces, with 1≤<∞ and μ be a -finite measure then there is unique element where 1=1 , such that The main goal of this project is to extend this concept to a positive linear functional on the space of all continuous real valued function which have compact suppor)). That’s if is a positive linear functional on where is locally compact Hausdorff space then there is a unique measure μ such that for all.Item Seminar Report on the Fundamental Group(Addis Ababa Ethiopia, 2010-06) Teshome, Asmelash; Mohammed, Seid (PhD)Item A Singular Boundary Value Problem for a Degenerate Elliptic Partial Differential Equation(Addis Ababa University, 2015-06) Biset, Tesfa; Mohammed, Seid (PhD)A singular boundary value problem associated with the 8-Laplace equation having, in general a non-separable inhomogeneous term is considered. The existence of ground state solution in viscosity sense to such a problem in bounded domains as well as in the whole Euclidean space RN, for N ¥ 2 is established. The investigation of the problem is mainly based on sub-solution and super-solution methods, and existence of principal eigenfunctions to the eigenvalue of Dirichlet type problem 8ufpx; uq; u ¡ 0 in an open set and upxq 0 on B , corresponding to the 8-Laplace equation. Keywords: 8-Laplacian, Viscosity solution, Principal eigenvalue, Principal eigenfunction