Mathematics

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Bessel’s Differntial Equation with Integer Order
(Addis Ababa University, 2024-09) Gebrye T/Mariam; Kassahun Workalemahu
Dif erential equations play an important role in solving real life problems, and we knowthatsolving dif erential equation is not simple in general. In this thesis we take the Bessel dif erentialequation and discuss the methods to solve the dif erential equation using Frobenius method. Webegin by finding Bessel functions Jn(x) and Yn(x) which are two solutions of Bessel dif erentialequation named as Bessel functions of first and second kind will also be described, whennisaninteger. For our purpose we consider some properties and proofs of recurrence relations, generating functions and special functions. Finally we discuss about the vibration of circularmembrane and its solutions.
Some Applications of Elliptic Functions and Elliptic Integrals
(Addis Ababa University, 2024-09) Ayana Mossie; Bahru Tsegaye
One of the most important applications of elliptic integrals of the form ∫𝑅(𝑥.√𝑝𝑛(x)) dx where R is rational algebraic function and 𝑝𝑛(x) is polynomials of degree 3 or 4 with no repeated roots. Nowadays, incomplete and complete elliptic integrals of the first kind are estimated with high accuracy using advanced calculators. In this paper, several techniques are discussed to show how definite integrals of the form ∫𝑅(𝑥.√𝑝𝑛(x)) dx can be converted to elliptic integrals of the first kind, and hence be estimated for optimal values. Indeed examples are provided in each step to help clarifications. Finally I have used eq.6 and eq.23 which is an important equations to show my analysis of ascending and descending transformation by taking initial values of F(∅0,k0) to provide a 30 decimal place values of numerical evaluations.
Boundary-Domain Integral Equations for Variable-Coefficient Mixed BVP in 2−Dimensional Unbounded Domain
(Addis Ababa University, 2024-09) Eshetu Seid; Tsegaye Ayele
In this thesis, the direct segregated Boundary Domain Integral Equations (BDIEs) for the Mixed Boundary Value Problems (MBVPs) for a scalar second-order elliptic Partial Differential Equation (PDE) with variable coefficients in an unbounded (exterior) 2D domain is considered. In this thesis, we formulate the exterior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second-order divergent elliptic PDE with variable coefficients. The aim of this work is to reduce the MBVPs to some direct segregated BDIEs with the use of an appropriate parametrix (Levi function). We examine the characteristics of corresponding parametrix-based integral volume and layer potentials in some weighted Sobolev spaces, as well as the unique solvability of BDIEs and their equivalence to the original MBVPs. This analysis is based on the corresponding properties of the MBVPs in weighted Sobolev spaces that are proved as well.
Growth Estimate of Composition Operator on Hilbert Space of Dirichlet Series
(Addis Ababa University, 2024-09) Marta Ayele; Hunduma Legesse
Motivated by a theorem of Gordon and Hedenalm (1), the study of composition operators acting on various scales of function spaces of Dirichlet series has arisen intensive interest. In this thesis, we characterize the composition operators in Hilbert space and growth estimate of the composition operator in Hilbert space of Dirichlet series.
On the General Eccentric Connectivity Index of Graphs
(Addis Ababa University, 2024-08) Hana Adugna; Mesfin Masre
The general eccentric connectivity index of graphs is the main topic of this study. For a connected graph G, the general eccentric connectivity index of graph G is defined by ECIa(G) = Σ v∈V(G) eccG(v)daG (v) for a ∈ R, where the degree of v in G is dG(v), the eccentricity of vertex v is eccG(v), and the vertex set of G is V(G). In this thesis, we study the general degree-eccentricity index of graphs. Among all the unicyclic graphs of a particular order and matching number, we identify the unicyclic graphs with the largest and smallest general eccentric connectivity index.
Applications of Green’s Function to Dirichlet Problems
(Addis Ababa University, 2024-09) Chali Bekele; Bahru Tsegaye
The aim of this paper is to present a new definition of the Green function of the Dirichlet problem for the Laplace equation prompted by the theory of ordinary Differential equation and Partial Differential Equations. A Green function, a mathematical function that was introduced by George Green in 1793-1841. Green function use for solving Ordinary and Partial differential equations in different dimensions for both time dependent and time independent problems. Green function is used in many theories such as quantum field theory, Electrodynamics and statistical field theory to refer various types of functions
Weak Idempotent Nil-neat Rings
(Addis Ababa University, 2024-08-31) Biadiglign Asmare; Kolluru Venkateswarlu; Tilahun Abebaw
We introduce the concept of a weak idempotent nil-clean ring which is a generalization of weakly nil-clean ring. We give certain characterizations for a weak idempotent nil-clean ring in terms of the Jacobson radical and nil-radical. In addition to this, we prove that n n upper (lower) triangular matrix over a ring R is weak idempotent nil-clean if and only if so is R. We introduce the concept of a strongly weak idempotent nil-clean ring which is a generalization of a strongly weakly nil clean ring. We characterize strongly weak idempotent nil-clean rings in terms of the set of nil-potent elements, homomorphic images, and Jacobson radicals. Moreover, we give necessary and sufficient conditions of a strongly weak idempotent nil-clean ring in relation to periodic rings, and also we give a characterization between strongly weak idempotent nil-clean rings and strongly -regular rings and strongly clean rings element wise. Furthermore, we prove that a strongly weak idempotent nil-clean ring R with 2 2 J(R) satisfies nil-involution property. We define the concept of a weak idempotent nil-neat ring which is the generalization of a weakly nil-neat ring. We characterize reduced weak idempotent nil-clean rings. Also, we give a characterization of weak idempotent nil-neat rings in terms of semiprime ideals, maximal ideals and Jacobson radicals. Moreover, we prove that every nonzero prime ideal of a strongly weak idempotent nil-clean ring is maximal. Finally, we investigate the condition for which the group ring R[G] becomes a weak idempotent nil-clean ring and a weak idempotent nil-neat ring.
Mathematical Modeling of CRYPTOSPORIDIOSIS Disease
(Addis Ababa University, 2016-08) Endalkachew Tazeze; Manalebish Debalike
The mathematical consequence of the disease-free equilibrium being locally asymptotically stable with the related epidemic fundamental reproduction of cryptosporidiosis is studied in the mathematical model for cryptosporidium infection. The infection model from the sensitivity analysis 0 is more sensitive to δ and crypto parameters when Rcr0>1. For the cryptosporidiosis reproductive number ( Rcr0) >1 is less sensitive. this is explanation of cryptospridiosis. Cryptosporidium is organized with water born transmission implement through dejection –oral path in many recreational water facilities. The homotopy decomposition method (HDM), a recently discovered analytical technique, is used to get the approximation or estimate solution by the numerical result. The etiological agent of cryptosporidiosis, an intestinal disease marked by frequent watery diarrhea, is Cryptosporidium, a protozoan parasite belonging to the species Ape complex. There are more than 30 known species of Cryptosporidium. The species most often linked to human infection include Cryptosporidium parvum and Cryptosporidium hominidi.Contact with diseased animals has been linked to the zoonotic transmission of cryptosporidiosis in humans in several cases, while a cryptosporidium infection by itself is not deadly, cryptosporidiosis which can lead to periodic epidemic disease—remains a significant worldwide cause of morbidity and mortality if an immune system is weakened.
Global Solution Theorems for Odes
(Addis Ababa University, 2016-08) Tuba Negesso; Tadesse Abdi
This thesis explores the globalization of the implicit function theorem (IFT) within the context of global solution theorems for ordinary differential equations (ODEs). Traditionally, the IFT provides powerful local results, but its global applicability has been less thoroughly examined. By extending the IFT to a global setting, this work develops new theoretical frameworks for understanding and solving ODEs on a broader scale. The study introduces generalized formulations of the IFT, derives significant global bifurcation results, and applies topological methods such as the Leray-Schauder degree to substantiate these results. Emphasis is placed on deriving conditions under which global solutions can be effectively analyzed and obtained.
On System of Ordinary Differential Equations and the Trace-Determinant Plane
(Addis Ababa University, 2016-08) Temesgen Bekele; Taddese Abdi
This thesis provides an advanced analysis of systems of ordinary differential equations (ODEs) using the trace-determinant plane as a central framework. The trace-determinant plane, defined by the trace and determinant of the Jacobean matrix of a linear system, serves as a powerful geometric tool for understanding and classifying the qualitative behavior of dynamical systems. The research begins with a detailed examination of the trace-determinant plane in the context of linear systems. By exploring the relationships between the trace (sum of Eigen-values) and determinant (product of Eigen-values) of the Jacobean matrix, the thesis elucidates the various types of equilibrium points and their stability characteristics. The trace-determinant plane is shown to map regions of distinct dynamic behaviors, including nodes, saddles, spirals, and centers, offering a comprehensive classification framework. Building on this foundation, the thesis extends the analysis to nonlinear systems by employing linearization techniques. It demonstrates how the trace-determinant plane can be utilized to approximate the local dynamics around equilibrium points and provides insights into the global behavior of nonlinear systems. The impact of parameter variations on the trace-determinant plane is also investigated, revealing how bifurcations and stability changes can be visualized and interpreted. A significant contribution of this thesis is the integration of computational methods with theoretical analysis. Algorithms for plotting the trace-determinant plane and analyzing system stability are developed and applied to a range of practical examples, including control systems, ecological models, and mechanical systems. These case studies illustrate the practical utility of the trace-determinant plane in both theoretical and applied contexts.
On Application of Network Simplex Method to Solve Minimum Cost Network Flow Problem
(Addis Ababa University, 2024-09) Getachew Beyene; Mesfin Masre
The problem of the minimum cost flow (MCF) is to send a flow from a set of supply nodes to a set of demand nodes via the arc of a network at a minimum total cost, without violating the lower and upper limits of the flow through the arc. The framework work of the MCF is particularly broad and can be used to model a number of specialized network problems, including assignment, transport and transfer problems, the shortest path problem and the maximum flow problem. The network simplex method is described to solve the minimum cost network flow problem, one of the most fundamental and a significant problem of network optimal design, and is applied to the network flow programming problem using simplex algorithms. The Network Simplex method describes the basic solutions for the problem of network flow programming and provides procedures for calculating the basic and double solutions associated with a given basis to find the optimal solution.
A Study on Ideal with Skew Derivations of Prime Rings
(Addis Ababa University, 2024-09) Gizat Alemu; Tilahun Abebaw
In this work, we apply the theory of generalized polynomial identities with automorphism and skew derivations to investigate the commutativity of a ring R satisfying certain properties on some appropriate subset of R.Let R be a prime ring and set [x,y]1=[x , y]=xy-yx for all x,y ∈ R and inductively[x,y]k=[[x,y]k-1,y] for k> 1 . We apply the theory of generalized polynomial identities with automorphism and skew derivations to obtain the following result: Let R be a prime ring and I a nonzero ideal of R. Suppose that .(𝛿 , 𝜑) is a skew derivation of R such that 𝛿([ x,y] =[x,y]n• for all x , y ∈ I , then R is commutative.
Mathematical Model and Analysis of the Impact of Infective Immigrants on the Transmission of Cholera Disease with Control Strategy
(Addis Ababa University, 2024-08) Desta Worku; Manalebesh Debalkie
Cholera is water born infectious disease caused by ingesting food or drinking water contaminated with a bacterium called vibrio cholera, characterized by extreme diarrhea and vomiting. In this thesis, we present mathematical model and analysis of the impact of infected immigrants on the transmission of cholera disease by adding some control strategy. In our work, we considered public health education, medical treatment and water treatment as the control strategy that limits the disease. The positivity and boundedness of the model system, and the existence and stability of the equilibrium points are discussed. Using next generation matrix, the reproduction number is calculated. Numerical simulation done using the MATLAB software code ode45 that shows when health education is effective or apply strongly, the number of infective are decrease faster, implying that health education and awareness are vital in controlling the spread of the cholera disease together with water treatment and medical treatment.
Endemic Malaria Dynamics with Variable Human and Mosquito Populations
(Addis Ababa University, 2024-08) Talelign Tatek; Kassahun Workalemahu
A malaria disease transmission dynamics analysis is done using a deterministic differential equation model for endemic malaria dynamics that incorporates a variable human and mosquito population. Disease free and endemic equilibrium points are calculated and their stability is discussed. For this analysis, a threshold parameter 𝑅 ̃ 𝑜 is calculated. The disease can continue only if 𝑅 ̃ 𝑜 is greater than 1and the disease-free equilibrium point is globally stable when 𝑅 ̃ 𝑜 is less than 1.We have also shown that there exists an endemic equilibrium point for the model. Numerical stimulation is used to demonstrate the population dynamics and its dependence in the threshold parameter.
Fixed Point Theorems for Generalized ( a- w- y- F)-Rational Contraction Type Mappings in a Complete B- Metric Spaces and its Applications in Ordinary Differential Equations
(Addis Ababa University, 2024-09) Deme Sani; Dula Tolera
In this thesis we introduce Fixed point theorems for generalized ( α-w -y -F)- rational contraction type mappings in b-metric spaces and prove the existence and unique fixed point theorems for such mappings and also we give its application in ordinary differential equations. Our result Generalizes many fixed point theorems in the literature.
Mathematical Modelling of Malaria Transmission Dynamics with Optimal Control Strategies
(Addis Ababa University, 2024-08) Dejen Agachew; Manalebish Debalke
Malaria is a tropical disease caused primarily by Plasmodium falciparum, which has been humanity’s major adversary to this day. This research proposes a malaria model that incorporates the use of treated mosquito nets as a disease control approach, which is then turned into proportions to estimate the worldwide impact of ITNs on malaria prevalence. Using a matrixtheoretic approach to construct a Lyapunov function results in a malaria-free equilibrium state that is globally asymptotically stable if the control reproduction number,Rm < 1. This suggests that malaria can be controlled or eradicated beneath a certain threshold amount,Rm. A malaria-persistence equilibrium state occurs and is globally stable for Rm > 1, utilizing the geometric theoretic technique with the Lozoskii measure. Numerical experiments show that the prevalence of infection can be reduced to zero if the fraction of vulnerable persons using treated mosquito nets exceeds a particular threshold number.
Laplace Adomian Decomposition Method to Solve Non Linear Partial Differential Equation
(Addis Ababa University, 2024-10-03) Thomas Kebede; Tesfa Biset
The Laplace-Adomian Decomposition Method (LADM) is an effective technique for solving nonlinear heat equations, which are crucial in various scientific and engineering applications. By combining the Laplace transform with Adomian’s Decomposition Method, LADM simplifies the resolution of nonlinearities and boundary conditions, transforming complex equations into manageable subproblems solved iteratively. This approach enhances computational efficiency and convergence speed without linearization or discretization. LADM is also successfully applied to the Porous Medium Equation (PME) and Fast Diffusion Equation (FDE), which describe physical processes like fluid flow through porous media and diffusion. The method demonstrates high accuracy and practicality, making it a valuable tool for tackling complex nonlinear problems.
Solution of Nonlinear Wave Equations Using Natural Decomposition Method
(Addis Ababa University, 2024-09) Henok Adane; Tesfa Biset
In this thesis, the Natural Decomposition Method, a combination of the Natural Transform Method and the Adomian Decomposition Method is used to obtain solutions to nonlinear wave equations. The method yields an exact solution in a form of a rapidly convergent series with easily computable components. The results we obtained have been compared with the approximate solution of nonlinear wave equations with other solutions by the methods of NDM and ADM. The natural decomposition method works perfectly for nonlinear wave equations.
Numerical Solutions of Nonlinear Heat Equations Using Adomain Decomposition and Natural Decomposition Methods
(Addis Ababa University, 2024-09) Endalew Kebede; Tesfa Biset
The Adomian decomposition method (ADM) is a powerful method which considers approximate solution of a non-linear equation as an infinite series which usually converges to the exact solution in the form of a rapid convergent series with easily computable components.Accrdingly, it is found to study the method to obtain solution of nonlinear heat equations. The method is a combination of Natural Transform Method (NTM) and Adomian Decomposition Method (ADM) which gives an exact solution. The exact solutions we obtained have been compared with the approximate solution of the nonlinear heat equations. ADM is able to solve this type of equations effectively and accurately without any need for discretization, perturbation, transformation and linearization.
Elliptic Problems and the Variational form
(Addis Ababa University, 2024-09) Wubu Getahun; Tadesse Abdi
In this thesis, given a bounded domain Ω⊂𝑅𝑛. we focus on the variational form ∫ Ω{𝛼∇𝑢⋅∇𝜙+𝑎𝑢𝜙}𝑑𝑥=∫ Ω𝑓𝜙𝑑𝑥,𝜙∈𝐻01(Ω) Of an elliptic problem, −𝛼Δ𝑢+𝑎(𝑥)𝑢=𝑓,𝑖𝑛 Ω With various conditions (Dirichlet, Neumann, and Robin) on boundary 𝜕Ω. We provide some general results on uniqueness and also find sub (supper)-solutions with in the frame work of variational setting.

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