Master Dissertations

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http://192.168.18.118:4000/handle/20.500.12661/398

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A review of simple model problems in continuum mechanics with internal body forces
(The University of Dodoma, 2013) Hilal, Awena M.
This study was put in place in order to review simple model problems in continuum mechanics with internal body forces. The main objective of this study is to analyze simple model problems in continuum mechanics with internal body forces, which include the steady flow between parallel plates with internal body forces, which include review of internal body forces with its applications, internal body forces and moments, string with internal body forces, membrane with internal body forces, elasticity with internal body forces, the steady flow between parallel plates with internal body forces, steady flow in a pipe with internal body forces, steady flow between concentric cylinders with internal body forces and Bernoulli equation with internal body forces. In its execution, the study utilized desktop research to enable the collection of information needed for the analysis of simple model problems in continuum mechanics with internal body forces.
A study of existence results on two point boundary value problem by fixed point method, monotone iterative technique and solution matching techniques
(The University of Dodoma, 2015) Afyusisye, Omary
The study of existence results on two point boundary value problems by fixed point methods, monotone iterative techniques, matching solution techniques and integral boundary conditions has been put in place in order to provide a broad understanding to the existence of two point boundary value problems by fixed point methods, monotone iterative techniques and solution matching techniques. The two point boundary value problem for ordinary differential equation has an important role in solving boundary value problems of Ordinary second order differential equations. Also play a great role in solving second-order boundary value problem with integral boundary conditions. The solutions of some linear ODE and boundary integral conditions can be denoted by Green's Function. Some BVP of nonlinear Differential Equation can be transformed into nonlinear integral equations by the kernel of which are Green's Functions of corresponding linear differential equations. The integral can be solved to investigate the property of the positive solutions. The main objective of the study of two point boundary value problems by fixed point methods, monotone iterative techniques, solution matching techniques and integral boundary conditions is to apply in solving positive solutions integral equations and second order differential equation. In this paper we consider the two point boundary value problem for second order linear differential equation (ODE) and integral boundary conditions with some positive solutions. As application, we study the fixed point methods and integral boundary conditions to find the existence of positive solutions.
Carleman estimates to solutions of direct and inverse problems for hyperbolic equations
(The University of Dodoma, 2015) Ngomaitala, Hussein Rajabu
A number of phenomena in modern science can be conveniently described in terms of problem for hyperbolic equation with Carleman estimates to the solution of inverse problem. The purpose of this study is to give a survey of the solution of the inverse problems for hyperbolic equation by Carleman estimates. We extend the results and prove the Carleman estimate focusing on an inverse problem for a simple hyperbolic equation. Also we derive the Lipschitz's stability by energy estimate; we obtain tomographic images by sent x-ray in different directions and measured at different places.
Numerical approximations to solutions of inverse problems for parabolic differential equations
(The University of Dodoma, 2015) Hamad, Hamad Makame
Present work is concerned with solved a coefficient inverse problem of one-dimensional parabolic equation by a higher-order compact finite difference method and we used this a fourth order efficient numerical method to calculate the function u (x; t) and the unknown coefficient a (t) in a parabolic partial differential equation. Also discussed the accuracy and efficiency of the fourth order finite difference formula compare with other finite difference methods such as FTCS explicit scheme, Crank-Necolson algorithm and Back ward time central space scheme. Results show that an excellent estimation on the unknown functions of the inverse problem can be obtained and the fourth order method developed in this work is well-balanced in stability, efficiency and accuracy.
Weights and ranks of causes of road accidents using Analytic Hierarchy Process (AHP) in Dodoma region
(The University of Dodoma, 2015) Omar, Said Hamad
The study of weights and ranks of causes of road accidents using Analytic Hierarchy Process (AHP) in Dodoma region has been conducted in order to provide a broad understanding to the seriousness of problem of Road Accidents in Tanzania. Since the problem of road accidents in Tanzania is highly increasing year after year, various preventive measures have been taken to reduce the rate of road accidents. Thus it is important to understand the actual causes which have high contribution to the road accidents. The main objective of this study is to propose weights and ranks of six selected causes with respect to five categories of motor vehicles associated with road traffic accidents in Tanzania. The five categories of motor vehicles are Lorries, Short Distance Passenger vehicles, Long Distance Passenger vehicles, Private Cars and Motorcycles while the selected causes are over speeding, reckless driving, Proper Use of Light during darkness, Adverse Road Conditions, Drivers‟ Behaviour and Drivers‟ Knowledge. Analytic Hierarchal procedure (AHP) is used as an analyzing tool. Through AHP questionnaire, data were collected from five experienced persons (experts) in road accident analysis. The experts rate the pair-wise comparison in measurement scale from one to five. The comparison scales were analyzed to compute the weights of road accident causes. The results of the study revealed that over speeding which means driving faster than limited speed is ranked as the highest cause in road accidents contribution with the weights 0.2905 followed by reckless driving (with weights 0.2812) while Adverse Road Condition is ranked as the lowest among the six selected road accident causes in this study.
On a global solution of the navier-stokes equations with internal body forces
(The University of Dodoma, 2015) Hamad, Salim Suleiman
The main objective of this study is on a global solution of the Navier-Stokes equations with internal body forces. The internal body forces play very important in both theory and application with Navier-Stokes equations. We discussed about the general solution for ordinary differential equations (ODES) and partial differential equations (PDES). Also they used the MAC model for incompressible flow. The study has been conducted in order to provide abroad understanding the existence of internal body forces with Navier-Stokes equations. The consideration of simple global solution the Navier-Stokes, equations with internal body forces is often possible.
A study on predator- prey systems with immigrant prey with and without harvesting
(The University of Dodoma, 2015) Kinyaga, Justine C.
The study of predator prey model with immigrant prey with and without harvesting has received great attention from both theoretical and mathematical biologists and has been studied intensively and extensively. Different literatures on interaction between species have been surveyed. In this document we establish sufficient stability criteria, criteria for the existence of periodic solution and Hopf bifurcations of a predator prey systems with immigrant prey without and with harvesting of predator. The ecological system in nature can be balanced by introducing an immigrant prey to ensure existence of both preys and predators. Harvesting of predator also can be used as a stabilizing factor as far as they depend on prey for their survival. The approach involves the use of software program (mat lab) in analyzing Equilibrium points, periodic solutions, Hopf bifurcations and its related theorems.
Overview numerical solutions for nonlinear partial differential equations with applications
(The University of Dodoma, 2015) Ndushi, Lwenge
This study is concerned with overview numerical solution for nonlinear partial differential equations. Since it is not easily to iterate the numerical scheme manually, C++ is used to encode the numerical scheme in order to find the numerical solution and Matlab is employed in drawing the figures. Chapter one consists of Introduction of Nonlinear Partial Differential equations, Literature Review, Classification of Partial Differential Equations, Examples of PDEs, Boundary Conditions, Taylor Expansion, Solution of a Partial Differential Equations, Truncation Error, Research Objective, Signification of the Study and Study Limitation. Chapter two contains Formulas of Finite Difference, Consistency, Stability and Convergence, Numerical scheme of Burgers' Equation, Sine Gordon Equation, Kortoweg-de Vries Equation, Gardner Equation and Fisher's Equation. Chapter three contains numerical and graphical results obtained by explicit scheme. In this chapter various examples of Burgers' equations, Sine Gordon Equation, Kortoweg-de Vries Equation, Gardner Equation and Fisher's Equation are discussed. Chapter four deals with Discussion, Conclusion and Recommendations.
Review of applications of partial differential equations in mathematical physics
(The University of Dodoma, 2015) Christopher, Valentine
This dissertation consists of four chapters. The first chapter is about general review of partial differential equations. The second chapter is devoted to the numerical solutions of partial differential equations. The third chapter is about applications of partial differential equations in mathematical physics and the fourth chapter is about mathematical modeling with partial differential equations.
Survey of square root algorithms over finite fields
(The University of Dodoma, 2015) Fumakule, Chokala Malunde
Computing square roots over finite fields is a problem of interest, especially to understanding which algorithm is efficient, and how it works well. There are several known algorithms that computes square roots over finite fields, of all of them the shank’s algorithm is known to be the most efficient. The objective of this dissertation is to survey the square root computing algorithms over finite fields, particularly we consider the the Shank’s algorithm for computing square roots over finite fields. We will write the conceptual explanation and general explanations of the whole algorithm (Shank’s) and finaly show how or why the algorithm works efficiently well.
Overview solutions of nonlinear ordinary differential equations and its applications
(The University of Dodoma, 2015) Mwema, Mohamed
This study is about overview solutions of nonlinear differential equations and its applications. It consists of four chapters where by the first chapter is devoted on the general overview solutions of differential equations. The second chapter deals with numerical solutions of differential equations. It provides the numerical values of the differential equations with initial conditions or boundary conditions within the given range or interval. The third chapter is about the fixed points, their stability and bifurcations. Here we point out the basic theoretical understanding of the nature of solutions, fixed points and stability properties. The fourth chapter consists of some applications of nonlinear differential equations and nonlinear models such as prey-predator models and compartmental models, nonlinear equations in celestial mechanics, Lagrangian mechanics and circuit theories.
Inverse problems for elliptic partial differential equations
(The University of Dodoma, 2015) Ndidi, Dominick Michael
The study of an inverse problems for elliptic partial differential equations has been put in place in order to provide a broad understanding to the existence of inverse problems for elliptic partial differential equations particularly Poisson's equation. The main objective is to describe diferent approaches to inverse problems for elliptic partial differential equations and in particular, the factorization method. Another aim of this work is to study a certain inverse problem for an elliptic partial differential equation and demonstrate the uniqueness and local existence of its solution using the derivation of the special integral equation. Inverse problem for elliptic partial differential equations play very important role in both theoretical part and applications. After the inversion of powerfully computers, more have been achieved through inverse problems for elliptic partial differential equations.
Fourier transform and the fundamental solutions of mechanical models with internal body forces
(The University of Dodoma, 2015) Carl, Collin Henry
In this dissertation, it is considered, the investigation of the Fourier Transform and the fundamental solutions of mechanical models with internal body forces. Although, there are many models, only two models are considered on the review, namely heat conduction and linear isotropic elasticity. For the investigation, the statement of the problem bases only on the linear isotropic elasticity problem. In working with calculations, the Fourier Transform, Inverse Fourier Transform, Residue Theorem, Jordan‟s Lemma, are applied in particular as well as the complex analysis and integration are applied in general. The obtained results of the statement of the problems having two cases, namely first and second cases, satisfy both their relevant equations and corresponding conditions accordingly. Therefore, they are identified as the fundamental solutions of their corresponding cases according to the problem. In the conclusion, the reviews also show that the Fourier Transform is widely applicable in variety of fields such as electromagnetic fields, magnetic resonance imaging (MRI) and quantum physics. Generally, it is also important in mathematics, engineering, and physical sciences. Finally, the investigator of this writing advises more application of the Fourier Transform since there is a lot of problems which are difficult/nearly impossible to solve directly, become easy after using a Fourier Transform. Also, it is reported that the mathematical operations on functions, like derivatives or convolutions, become much more manageable on the far side of a Fourier Transform.
A study on the role of green’s function in solving boundary value problems
(The University of Dodoma, 2015) Kapula, Alinanine
Study of Green’s Functions and applications of BVPs solving different integration has been put in place in order to provide a broad understanding to the existence of Green’s Functions on solving Ordinary Differential Equations with nth order. The Green’s Functions plays an important role in solving boundary value problems of Ordinary Differential Equations. The solutions of some BVPs for linear ODEs can be denoted by its Green’s Function. Some BVPs for nonlinear Differential Equations can be transformed to nonlinear integral equations, the kernel of which are the Green’s Functions of Corresponding linear differential equations. The integral can be solved to investigate the property of the Green’s Functions. The main objective is to analyze the Green’s Function and to show how applied in solving integral equations and ODEs with the application of boundary value problems. In this paper we consider the Green’s Functions for second order linear ODEs with some Three-point boundary conditions. As application, we study the iterative solutions for nonlinear singular second order Three-boundary value conditions, how to find Green’s Functions using a second order three-BVPs.
Comparing the logarithmic least square and eigen value methods in analytic hierarch process by using the best job example.
(The University of Dodoma, 2016) Revocatus, Ponsian
This dissertation centers on comparing two methods based on consistence and ranking preservation of alternatives in analytic hierarchy process (AHP) by using the best job example. These two methods logarithmic least square method (LLSM) and eigenvalue method (EM) are used to develop approximations of ratio scales from a positive reciprocal matrix. The measurement of consistency and rank preservation are the main criteria for comparison of these two methods. The priorities obtained for each method from the combination of the comparison matrices for all criteria with respect to all alternatives, the ranking of the alternatives were given against the four covering criteria, the results show that, the two methods namely LLSM and EM give different results on ranking the alternatives as shown by Saaty and Vargas(1984).
Numerical solutions to parabolic differential equations and application.
(The University of Dodoma, 2016) Mukome, Maswe
Partial Differential Equations (PDEs) of parabolic type arise naturally in the modeling of many phenomenon of in various fields of physics, engineering and economics. The main aim of this research is to study finite difference methods with numerical solutions of this class of equations. Both one and two dimensions have discussed in this research. I have investigated stability and convergence analysis of different schemes and obtained convergence error estimates. It is important to note that convergence results hold also on Finite- element methods but in this study we didn’t worked on in. The discretization in time using Explicit, Implicit, Crank-Nicolson, Richardson and 𝜃 − method are addressed and stability, consistency and convergence estimates analyzed well. The mathematical modeling and Numerical method of non-reactive solute in porous media is also in the scope of this study. Among the applications like time estimates and solutions of dynamic model of reactor have discussed in this research. This fluid dynamics plays a major rule in hydrology, medical science and petroleum industry. Also few problems have discussed in this work to show how these methods like Crank-Nicolson, Explicit and implicit method works in parabolic equations (heat equations). According to Farlow 1982 , Most physical phenomenon, whether in domain of fluid dynamics, electricity, magnetism, mechanics, optics, or heat flow can be described in general by partial differential equations (PDEs); In fact, most of mathematical physics are PDEs. It’s true that the simplification can be made that reduce the equations in equation to ordinary differential equations, but, nevertheless, the complete description of these systems resides in the general area of PDEs. In this research we are aiming to show what parabolic partial differential equations are and how they are solved by Numerical solutions.
Numerical solutions to elliptic differential equations - finite difference approach.
(The University of Dodoma, 2016) Mchette, Jacob Clavery
This dissertation contains materials on numerical solutions based on elliptic differential equations only appropriate for senior level undergraduate or beginning level graduate students. The reader based on this dissertation should have had introductory courses in Calculus, linear algebra and general numerical analysis. A formal course in ordinary or partial differential equations would be useful. In our study, it should be understood that, there are many procedures that come under the name numerical methods. We shall see how the very popular finite difference methods can be used to solve elliptic equations. To begin, we introduce the idea of finite differences. We then show how to use these finite differences to solve a Dirichlet Problem inside a square. However, the numerical algorithm for solving the Dirichlet Problem (Liebmann’s method ) has been included. Moreover, systems of algebraic equations have been solved numerically by an iterative process in order to obtain an approximate solution to the partial differential equation. The iterative methods such as Gauss Seidel iteration method, Gauss Jacobi iteration method and Liebmann iteration method have been discussed. It is also pointed out that the reader will find how numerical solutions to elliptic differential equations are applicable in daily life experience.
Blow-up of solutions to problems for nonlinear hyperbolic equation.
(The University of Dodoma, 2016) Chillingo, Josiah Kidney
Different physical phenomena can be represented in terms of nonlinear problems for partial differential equations; however such problems are often subjected to singularities. Thus it gives rise to a permanent research interest to such problems. In the present study we provide reviews of essential approach applied to Cauchy problems and initial-boundary problems for hyperbolic equations based on latest results in this ﬁeld.
Finite element method with damping control multi-step methods approach to one boundary value problem for the wave equation.
(The University of Dodoma, 2016) Leandry, Leonce
Over the previous years ﬁnite element method (FEM) has become a powerfully tool to approximate solution of differential equations and prove their existence. The purpose of this research is to introduce and describe a number of the ﬁnite element method (FEM) technique applied to problems for partial differential equations (PDEs) with special attentions to the hyperbolic problems in case of wave and damped wave equations. Another aim is to study the one boundary value problem (BVP) for the wave equation and apply damping control multi-step methods integrated into the FEM such as the Newmark method, Backward difference method (BDF) and Hilber-Hughes-Taylor Method (HHT). The ordinary differential equation (ODE) system obtained after applying FEM are then solved by these multi-step methods, where by the BDF-Method and the HHT-Method are second order precision, unconditionally stable and able to dissipate high-modes for some values of the parameters.
Modeling an isotropic charged relativistic matter with linear equation of state.
(The University of Dodoma, 2016) Danford, Petro
We ﬁnd new exact solutions to Einstein-Maxwell ﬁeld equation for charged anisotropy stellar bodies. We are considering the stellar object that is anisotropic and charged with linear equation of state consistent with quark stars. We have new choice of measure of anisotropy and adopted Sunzu’s metric function. The solutions are obtained after considering the transformed Einstein-Maxwell ﬁeld equations for charged anisotropic matter. In our models we regain previous anisotropic and isotropic results as a special case. Exact solutions regained in our models are those by Sunzu, Maharaj and Ray and those by Komathiraj and Maharaj. We have considered the space time geometry to be static spherically symmetry. The exact solutions to the Einstein-Maxwell ﬁeld equations corresponding to our models are found explicitly in terms of elementary functions namely simple algebraic functions. The obtained graphical plots and physical analyses for the gravitational potentials, the matter variables and the electric ﬁeld are well behaved.

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