Mathematics

Permanent URI for this collection

http://etd.aau.edu.et/handle/123456789/31

Browse

Now showing 1 - 20 of 44

Asymptotic Solutions of Ordinary Differential Equations Having an Irregular Singular Point
(Addis Ababa University, 2010-06-06) Yesuf Jemal; Abdi Tadesse (PhD)
The problem of finding a basis solution of linear homogeneous ordinary differential equations with variable coefficients is, in general difficult. Indeed a considerable effort was directed at constructing solutions of specific equations with non constant coefficients. The solutions of differential equations with analytic coefficients were obtained from a theory called the method of power series. Most importantly is the case in which the equation has a singular point. In the special case of a so called regular (weak) singular point we can apply the method of Frobenius to construct convergent series solutions in a neighborhood of this point. In general this cannot be done for an irregular (strong) singular point. Treatment of this case is the main aim of this seminar. The first chapter gives the background information about the problem under investigated. The second chapter constructs a method of approximating the solutions of differential equations in a neighborhood of an irregular singular point. The final chapter extends these approximations to asymptotic expansions.
Asymptotics for Bessel Functions
(Addis Ababa University, 2014-08) Esubalew Tangut; Abdi Tadesse (PhD)
The main objective of the present paper is to address how to solve Bessel's equation and describe its solution, Bessel functions. Additionally we discuss the approximate solution of Bessel equations as x tends to infinity(for larger interval of x).
Bessel’s Differential Equation and the Aging Spring Problem
(Addis Ababa University, 2014-08) Belayneh Tilahun; Abdi Tadesse (PhD)
In this paper we study the aging spring equation without damping X= (t) + e−at+b x(t)=0 , or the most general one with damping (t) + dx= (t) + e−at x(t)=0 which is a special cases of homogeneous linear second order ODE with variable coefficients. We will show that, with a suitable change of variables, this equation of aging spring problem can be transformed into Bessel differential equation and hence the solution can be expressed in terms of Bessel functions
The Cauchy- Problem With the Heat Equation
(Addis Ababa University, 2014-06-06) Eshetu Abebe; Abdi Tadesse (PhD)
This report attempts to study solution of explicit linear PDE of second order, the heat equation u k u f t With certain symmetry condition on the solution u .In this regard, some sort of scaling of variables is introduced and pertinent scaling transformation T that leaves the ratio t x 2 unchanged is shown to usher to a solution of the form: t x u x t v 2 , Which is radial and hence a symmetric function.
Center Manifold Analysis for Static Bifurcation and Normal Forms
(Addis Ababa University, 2012-02) Admasu Muluken; Abdi Tadesse (PhD)
This paper addresses the dependency of solution of differential equation on initial data and control parameter. To reduce the dimension of the system, we use center manifold reduction method. Mainly, the paper focus on the cases of static bifurcations occur and non bifurcation. We transformed the fixed points and control parameter to zero. This greatly helps us to use lower order approximation of Taylor series. Finally, by using center manifold reduction and Taylor series expansion we construct the normal form of such types of bifurcations
Center Manifold Analysis of Hopf Bifurcation for Delayed Lienard Equation
(Addis Ababa University, 2011-01-01) Amsalu Hafte; Abdi Tadesse (PhD)
Lie nard equation serves as the elegant models for oscillating circuits. This paper addresses the stability property of a class of delayed lie nard equations. This project uses operator differential equation formulation to reduce the infinite dimensional delayed lie nard equation onto a two dimensional manifold on the critical bifurcation. Based on the reduced two dimensional systems, the so called Poincare-Lyapunov constant is analytically determined, which determines the criticality of the Hopf-bifurcation.
Discrete Dynamical Systems
(Addis Ababa University, 2011-01-25) Cherkos Meseret; Abdi Tadesse (PhD)
This project is based on equations that allow us to compute the value of a function recursively from a given set of values. In the main, Hartman-Grobman theorem is proved by using different lemmas and other theorems. Population modeling with difference equations by converting to discrete version is briefly discussed. The notions of fixed and periodic points of dynamical systems are considered.
Distribu- tional solution of Con uent Hypergeometric Diferential equation
(Addis Ababa University, 2015-10-01) Alemu Seyome; Abdi Tadesse (PhD)
This project deals with distributional solutions y(x) =∑ an (n) (x) n=0 of the conuent hypergeometric di_erential equation.Such di_erential equations also have solutions of the form 1F1(p;m; x) = 1 +∑ (p)nxn (m)nn where (p)n is the pochhammer's symbol
Duhamel’s Principle and the Method of Descent for the Wave Equation
(Addis Ababa University, 2014-06) Hailu Tafesse; Abdi Tadesse (PhD)
In this project we present investigation of the linear wave equation with the unknown function u, subject to prescribed initial and/or boundary data, where Δ is n-dimensional Laplacian. In 1d, the solution of IVP is rendered by first reducing it into lower order PDE and then appealing to the method of characteristics, while, for BVP the method of reflection is employed to yield the pertinent solution. In higher dimension, explicit solution of IVP is derived as based on the method of spherical mean and the method of descent. In the sequel, Duhamel’s principle is used to get the solution of non-homogeneous wave equation from the associated homogeneous wave equation
Eigenpairs of Fredholm Integral Operators With Separable Kernel
(Addis Ababa University, 2012-02-02) Atnaw Malede; Abdi Tadesse (PhD)
An integral equation is an equation in which the unknown function appears in the integrand. There are different types of integral equations, among these Fredholm integral equation is an integral equation with constant limit of integration. Depending on the appearance of the unknown function we have three kinds of Fredholm integral equations, first kind, second kind and third kind. In this project we discuss the techniques of solving Fredholm integral equations. Even though, it is difficult to find the solution of each kind, we consider the solution of the first kind in terms of a linear combination of eigenpairs of separable kernel. The solution of the second kind is also determined by using different decomposition methods.
Existence and Approximation of Solution of Explicit Second Order Nonlinear Neumann Problems
(Addis Ababa University, 2015-01) Sinshaw Simachew; Abdi Tadesse (PhD)
In this paper we shall present the basic theory of Neumann boundary value problem (BVP) together with a discussion of some of the powerful methods that are used to solve the Neumann boundary value problem (NBVP). Especialy the paper focuses on the upper and lower solutions of NBVP. .The main objective of this paper is to investigate the existence and approximation of solutions of second order nonlinear Neuman problems. 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 quasilinearization technique, and generalization quasilinearization technique will be deffned and discussed;upper and lower solutions of Neumann BVP and elaborate through examples. And then some basic properties of NBVP (theorems on NBVP) are stated and clari- fied where the proofs of most of these properties are also include. Having familiarized with NBVP in the second section, in the third section,(the core part of the paper), we will come across with Generalized quasilinearization technique
Existence of Periodic Solutions for a Class of Second Order Odes With Periodic Data
(Addis Ababa University, 2014-03) Weldemichael Sahle; Abdi Tadesse (PhD)
In this project we will see the existence of periodic solution(s) to the second order ODE of the form: x00(t) + a(t)x0(t) = g(t; x) =f(t; x(t); x0(t)) by means of Schauders Fixed Point Theorem where a is a continuous !- periodic function , g(t; u), f(t; u; v) are !-periodic functions in t for u = x(t), v = x0(t) and ! > 0. The method of proof is composed of two steps, the _rst step is to transform the original equation into integrodi _erential equation through a linear integral operator and the second step is an application of the Schauder's Fixed Point Theorem. Keywords: Periodic solution; Schauder's _xed point theorem; Fundamental matrix
Explicit Finite Difference Scheme for the Finite String
(Addis Ababa University, 2014-08) Amsalu Solomon; Abdi Tadesse (PhD)
This project provides a practical overview of numerical solutions to the wave equation of finite string using the finite difference method. The second or- der centered differences for time and space is applied to a simple problem involving the one-dimensional wave equation of finite string which lead to an explicit numerical scheme. It also allow the reader to experiment with the consistency, stability and convergency of explicit finite difference scheme for finite string and an example with working Matlab code for the scheme is presented.
Explicit finite Difference Schemes using Ghost points for the heat equation with insulated ends Explicit finite Difierence Schemes using Ghost points for the heat equation with insulated ends
(Addis Ababa University, 2015-06) Tezera Tesfu; Abdi Tadesse (PhD)
In this project report, explicit ffnite di_erence scheme for the linear heat equation in an insulated bar in one space dimension is considered. This method is used to solve the partial derivatives in the partial di_erential equations at each grid point that are derived from neighboring values by using Taylors theorem . The forward - time centered - space(FTCS) and explicit schemes are developed. To find the boundary equations of the schemes ghost points are introduced in central di_erence approximation . The MATLAB implementation allow the readers to experiment with the stability limit of the forward time , centered space(FTCS)schemes key words- finite diference, explicit scheme, heat equation in an insulated bar , Ghost point
Exponential Estimates in Retarded Functional Differential Equations
(Addis Ababa University, 2012-02-02) Tadesse Dereje; Abdi Tadesse (PhD)
So far, many things were said about differential equations without time delay, the so called ordinary differential equations or partial differential equations and their solutions. Different techniques have been used to show the existence and uniqueness of the solution of initial value problem of these equations. Since time delay occurs naturally in just about every interaction of the real world, here in this project report we treat some differential equations with time delay, the so called linear functional differential equations or linear differential difference equations in general and for these equations, a complete theory can be developed using very basic tools. The project serves as an introduction to the more general types of differential difference equations. It also is intended to bring out the roles of the characteristic equation and Laplace transform to determine the solution of linear differential difference equations and to emphasize some of the differences between retarded and neutral equations.
Fiinete,Simple and Undirected Graph Encoding
(Addis Ababa University, 2014-03) Demissie Tsedeke; Abdi Tadesse (PhD)
In this project we will see the existence of periodic solution(s) to the second order ODE of the form: x00(t) + a(t)x0(t) = g(t; x) f(t; x(t); x0(t)) by means of Schauders Fixed Point Theorem where a is a continuous !- periodic function , g(t; u), f(t; u; v) are !-periodic functions in t for u = x(t), v = x0(t) and ! > 0. The method of proof is composed of two steps, the _rst step is to transform the original equation into integrodi erential equation through a linear integral operator and the second step is an application of the Schauder's Fixed Point Theorem. Keywords: Periodic solution; Schauder's _xed point theorem; Fundamental matrix
Generalized Solution of Inhomogeneous Differential Equations
(Addis Ababa University, 2010-06) Tesfaye Yohannes; Abdi Tadesse (PhD)
We study the method to find a generalized solution to the linear inhomogeneous differential equations of different orders. Here first, second and fourth order linear inhomogenous differential equations are considered. We use generalized function for finding the generalized solutions of linear inhomogenous differential equations. Green’s function plays a great role for solving such differential equations
Generalized Solutions of Boundary Value Problems With Jump Discontinuity (Submitted in Partial Fulfillment of the M. Sc. Degree in Mathematics)
(Addis Ababa University, 2012-02) Shiferraw Abdissa; Abdi Tadesse (PhD)
We study on how to find a generalized solution of boundary value problems of different orders with jump discontinuity. Here first, second order linear inhomogeneous differential equations are considered and then we extend it to the nth order. Green’s function plays a great role for solving such differential equations. Classical theory is based on solving first- or higher-order derivatives with jumps on both sides of the boundaries and then attempting to satisfy the boundary conditions. Our aim is to develop the vector analysis of functions with jump discontinuities across surfaces and boundaries that cannot be solved by classical techniques. With the help of this we can solve many unsolved problems in the potential, scattering and wave propagation theories. Furthermore, problems whose solutions are already known can be solved by this method in a very simple fashion. We can also show that there is a new solution of linear homogeneous systems of differential equations with singular in coefficients, in the space of generalized functions other than the classical solutions
Graduate Seminar Report on the Hartman-Grobman Theorem and Planar Systems
(Addis Ababa University, 2012-02) Balcha Solomon; Abdi Tadesse (PhD)
The purpose of this study was to investigate the overall qualitative behavior of two dimensional linear systems such as classification of the systems from the dynamical point of view and in particular. To obtain insight in the classification of fixed point using trace and determinant of the coefficient matrix of planar systems; To develop classification criteria using trace and determinant of the coefficient matrix of the system, and also the way how to draw the trace determinant plane is discussed. To discuss the more subtle issue of topological equivalence (conjugacy) of these systems. Starting from simple planar linear systems and then we develop insight in investigating the relationship between two system that are topological conjugate and equivalent. Moreover, there is very important theorem mentioned in this project, the Hartman-Grobman theorem, which results in the local behavior theory of differential equations. The theorem shows near the hyperbolic fixed point, the non-linear system has the same qualitative structure as its linearization
Greens Functions and Boundary Value Problems
(Addis Ababa University, 2014-08) Eshetu Assaye; Abdi Tadesse (PhD)
In this paper, we investigate some boundary value problems for two dimensional harmonic functions. That is basic introduce new tools for solving Dirichlet problems, Poisson’s equations and Neumann problems with Green’s function G( x; y; x0 y0) =1/2in( x –y0)2+( x – y0)2) + h (x ,y, x,0 y0) Where h is harmonic on the region and h(x, y, x0 , y0 ) =-1/2in ( x – x0)2 + y - y 0)2)on the boundary  . Roughly speaking Green’s function for a given region Ω and that can be used to solve any Dirichlet problems or Poisson problems on Ω. In the same way that the Poisson’s kernel on the real line can be used to solve Dirichlet problems in the upper half plane

Browse

Browsing Mathematics by Author "Abdi Tadesse (PhD)"

Results Per Page

Sort Options