School of Mathematics, Statistics and Actuarial Sciencehttps://repository.maseno.ac.ke/handle/123456789/13162024-03-28T09:26:15Z2024-03-28T09:26:15ZFinite element method solution for steady magnetohydrodynamic flow in a straight horizontal pipe of elliptical cross sectionKWEYU, Davidhttps://repository.maseno.ac.ke/handle/123456789/56842023-04-28T07:59:21Z2022-01-01T00:00:00ZFinite element method solution for steady magnetohydrodynamic flow in a straight horizontal pipe of elliptical cross section
KWEYU, David
Velocity profile and temperature distribution for Magnetohydrodynamic (MHD) flow in a straight horizontal pipe of elliptical cross section has been investigated. Many researchers have carried out research on pipes of circular, square, rectangular, annular and elliptical cross sections in magnetohydrodynamics because there are many applications. Their studies concentrated on a given cross section as a different entity with fluid being driven by pumps. In this study, investigation is done on a circular pipe as it changes into an elliptical pipe when fluid is propelled by gravitational force. The main purpose of the study is to find out which pipe between one whichhasacircularcrosssectionandanotherofellipticalcrosssectionismorebeneficial. Effects of velocity profile and temperature distribution on the pipe as it changes cross section from circular to elliptical are investigated. Governing equations, partial differential equations (pdes), are formulated, non dimensionalised, expressed in terms of stream function and transformed into ordinary differential equations (odes) using similarity transformation. The odes are solved by Finite Element Method in conjunction with Mathematica version 12.0. The objectives of the study are: To model Finite Element Method solution for steady Magnetohydrodynamic flow in a straight horizontal pipe of elliptical cross section. To formulate governing equations (pdes) in cylindrical coordinates (r,θ,z) comprising Navier-Stokes equations, Ohm’s law of electromagnetism, equation of continuity, cross section of elliptical pipe and heat energy equation. To solve by Finite Element Method the ordinary differential equations (odes) formed when non dimensionalisation and similarity transformation are carried out on the governing equations. To determine the effects of dimensionless numbers of Hartmann number, Reynolds number, Eckert number and Prandtl number as well as other physical quantities of gravitational force and aspect ratio on fluid velocity and temperature. To find out the repercussions of velocity and temperature on a pipe as it transits from circular to elliptical cross section. Finite Element Method (FEM) is embraced instead of other methods like Finite Difference Method (FDM) because FEM is able to handle complicated geometries and boundaries with relative ease while other methods are restricted to handle rectangular shapes. Also many of the real life medical, engineering, astrophysics, etc problems can be solved in weak form, which FEM encompasses compared to strong form, which other methods employ. Results are displayed as tables and graphs and reveal that: Increase in Hartmann number, 1.0≤Ha≤40.0, increases temperature but retards velocity. Rise in Reynolds number, 0.5≤Re≤8.0 and aspect ratio, 1≤α ≤1.6, leads to rise in both velocity and temperature. An upsurge in gravitational force, 0.00002≤λθ ≤0.00008, results in an upsurge in velocity. Temperature increases when Eckert number, 1≤Ec≤40 , increases but decreases when Prandtl number, 0.5≤Pr≤2.0, is raised. In all scenarios, velocity and temperature are maximum at the centre of pipe but diminish to zero at the periphery. Spike in aspect ratio leads to rise in velocity which results in increase in temperature. A pipe of elliptical cross section will be more convenient where there is limited space in the vertical direction due to existing structures yet there is demand in increase in productivity. This is in comparison to circular shape. A pipe of elliptical cross section has greater capacity for the same depth of flow. It is envisaged that the conducting fluid is flowing as a coolant at a nuclear power plant or as molten metal at a metallurgical process. A pipe of elliptical cross section would therefore be moreproductive inindustrialprocessesthanone whichiscircularaccordingto thefindingsofthis dissertation.
2022-01-01T00:00:00ZArima and vector autoregressive model evaluation in forecasting rainfall: a case of KisumuMawora, Thomas Mwakudisahttps://repository.maseno.ac.ke/handle/123456789/55712022-12-20T12:32:31Z2022-01-01T00:00:00ZArima and vector autoregressive model evaluation in forecasting rainfall: a case of Kisumu
Mawora, Thomas Mwakudisa
Time Series Analysis has been used over the decades in data analysis and forecast
ing. Auto Regressive Integrated Moving Average (ARIMA) models have been fit on
economic data and engineering data. The models have also been used in analysis of
climate data. Previous studies have focussed on temperature data from National Mete
orological Stations where summarized monthly values were used. In this study, we used
daily rainfall data from Kenya Meteorological Services Station in Kisumu. The objec
tives included univariate time series modelling using ARIMA on long term rainfall data
for daily, monthly, seasonal and annual data and forecasting rainfall for the different time
periods. The other objective was to compare forecast from univariate ARIMA to Vector
Autoregression (VAR) when rainfall, minimum and maximum temperature values are
included in model. ARIMA models were fit on the KMS rainfall data, and VAR models
were fit on temperature, minimum and maximum rainfall data from KMS. Finally, farm
ers’ local rainfall data was compared to that of KMS for independence. Results showed
that forecasts under VAR did not give a more precise forecast of future rainfall than
ARIMA. Further, that there was not enough statistically significant evidence to suggest
that rainfall data from KMS and farmers’ locale were independent.
2022-01-01T00:00:00ZLie symmetry solutions of the Generalized burgers equationODUOR, Okoya Edmund Michaelhttps://repository.maseno.ac.ke/handle/123456789/43292021-11-08T09:05:57Z2005-01-01T00:00:00ZLie symmetry solutions of the Generalized burgers equation
ODUOR, Okoya Edmund Michael
Burgers equation: u, + UUx = luxx is a nonlinear partial differential equation which arises
in model studies of turbulence and shock wave theory. In physical application of shock
waves in fluids, coefficient 1 ,has the meaning of viscosity. For light fluids or gases the
solution considers the inviscid limit as 1 tends to zero. The solution of Burgers equation
can be classified into two categories: Numerical solutions using both finite difference and
finite elements approaches; the analytic solutions found by Cole and Hopf In both cases
the solutions have been valid for only 0 ~ 1 ~ 1. In this thesis, we have found a global
solution to the Burgers equation with no restriction on 1 i.e. 1 E (- 00 , 00). In pursuit
of our objective, we have used, the Lie symmetry analysis. The method includes the
development of infinitesimal transformations, generators, prolongations, and the invariant
transformations of the Burgers equation. We have managed to determine all the Lie
groups admitted by the Burgers equation, and used the symmetry transformations to
establish all the solutions corresponding to each Lie group admitted by the equation.
These solutions, which are appearing in literature for the first time are more explicit and
more general than those previously obtained. This is a big contribution to the
mathematical knowledge in the application of Burgers equation.
2005-01-01T00:00:00ZThe almost holomophic functional calculusODHIAMBO, Paul Olechehttps://repository.maseno.ac.ke/handle/123456789/43272021-11-05T09:52:30Z2008-01-01T00:00:00ZThe almost holomophic functional calculus
ODHIAMBO, Paul Oleche
2008-01-01T00:00:00ZUnit groups of certain classes of commutative finite ringsOWINO, Maurice Oduorhttps://repository.maseno.ac.ke/handle/123456789/43202021-11-04T10:07:36Z2009-01-01T00:00:00ZUnit groups of certain classes of commutative finite rings
OWINO, Maurice Oduor
2009-01-01T00:00:00ZSome aspects in the study of invariant subspacesSIMIYU, Achiles Nyongesahttps://repository.maseno.ac.ke/handle/123456789/43182021-11-04T09:46:12Z2010-01-01T00:00:00ZSome aspects in the study of invariant subspaces
SIMIYU, Achiles Nyongesa
2010-01-01T00:00:00ZStudy of non-normal operators in a complex Hilbert Spacejustus, kithekahttps://repository.maseno.ac.ke/handle/123456789/43082021-11-04T07:40:02Z2009-01-01T00:00:00ZStudy of non-normal operators in a complex Hilbert Space
justus, kitheka
2009-01-01T00:00:00ZThe Derivation of a Logistic Nonlinear Black Scholes Merton Partial Differential Equation: European OptionNYAKINDA, Joseph Otulahttps://repository.maseno.ac.ke/handle/123456789/42672021-08-02T08:18:46Z2011-01-01T00:00:00ZThe Derivation of a Logistic Nonlinear Black Scholes Merton Partial Differential Equation: European Option
NYAKINDA, Joseph Otula
\onlinear Black-Scholes equations have been increasingly attracting
interest over the last twenty years. This is because they provide more
accurate values by taking into account more realistic assumptions, such
as transaction costs, illiquid markets, risks from an unprotected portfolio
or large investor's preferences, which ruay have an impact on the stock
price, the volatility, the drift and the option price itself. Recent models
have been developed to take into account the feedback effect of a fund
hedging strategy Or of the transaction costs of large traders tv[ost of these
models cue represented by nonlinear variations of the well known Black-
Scholes Equation.On the other hand, asset security prices may naturally
not shoot up indefinitely (exponentially) leading to the use of Verhlusts
Logistic equation. The objective of this study was to derive a Logistic
Nonlinear Black Scholes f\. lertou Partial Differential equation by considering
transaction costs (which \\ere oVBrlooked in the derivation of the
classical Black Scholes model) and incorporating the Logistic geometric
Brownian motion.The methodology involved, analysis of the geometric
Brownian motion, review of logistic models, Ito's process and lemma,
stochastic volatility models and the derivation of the linear and nonlinear
Black-Scholes-Merton partial differential equation. Illiquid markets have
also been analyzed alongside stochastic differential equations. The result
of this study may enhance reliable decision making based on a rational
prediction of the future asset prices given that in reality the stock market
may depict a non linear pattern.
2011-01-01T00:00:00ZMathematical Models for Malaria Co-Infections With Persistent Pediatric Infections in KenyaLAWI, George Owuorhttps://repository.maseno.ac.ke/handle/123456789/42152021-07-29T07:15:28Z2013-01-01T00:00:00ZMathematical Models for Malaria Co-Infections With Persistent Pediatric Infections in Kenya
LAWI, George Owuor
Despite many years of study and advanced biological, medical and mathematical
understanding of diseases together with commitment to child
survival, malaria and persistent infectious diseases of childhood continue
to inflict the developing nations, especially the Sub-Saharan Africa in
large proportions. In 1990 the Kenyan under-five mortality rate was reported
as 97 deaths per 1000 live births, but in 2006 it had increased to
121 deaths per 1000 live births. Kenya is thus among the countries with
least progress towards Millennium Development Goal Four (MDG 4) of
32 deaths per 1000live births in 2015. In malaria endemic places, malaria
co-infections with persistent infections like meningitis, pneumonia and rotavirus
are common. Furthermore, these diseases have a high symptom
overlap with malaria thus frequently leading to clinical misdiagnosis and
its associated problems.
The objective of the study was to develop and analyse, using the stability
concepts of differential equations, deterministic mathematical models
for the co-infection of malaria with meningitis, pneumonia and rotavirus
among Kenyan children under the age of five years. This is because children
in this age group have not developed sufficient immunity and are
thus more vulnerable to infection.
The symptom overlap between malaria and these persistent infections,
in resource scarce settings typical of the developing world, is a cause
for concern. This is because in such settings diagnosis is often clinically
done. Our analysis indicate that protection against a second infection is
desirable in minimizing the effects of co-infection. Without laboratory diagnosis,
the presence or absence of a co-infection may not be established
2013-01-01T00:00:00ZAnalytic Solution of a Nonlinear Black-Scholes Partial Differential EquationEYANG'AN, Esekon Josephhttps://repository.maseno.ac.ke/handle/123456789/41992021-07-28T07:40:28Z2011-01-01T00:00:00ZAnalytic Solution of a Nonlinear Black-Scholes Partial Differential Equation
EYANG'AN, Esekon Joseph
The assumptions under which the standard Black-Scholes equation
has been derived are restrictive (e.g. liquid and frictionless markets).
When illiquidity and market friction are introduced into the
market, financial models based on these assumptions fail. Nonlinear
equations for modelling illiquid markets have been solved numerically.
Numerical techniques give approximate solutions. Recently,
Lie group symmetry analysis has been used to solve the same. Although
Lie group symmetry analysis is very useful in determining
all the solutions of a given nonlinear equation, it has been established
that any small perturbation of an equation disturbs the group
admitted by it. This in effect reduces the practical use of symmetry
group analysis. Our objective is to find an analytic solution of
a nonlinear Black-Scholes equation for modelling illiquid markets.
The methodology involved transformation of the nonlinear Black-
Scholes equation into a groundwater equation. This yields Ordinary
Differential Equations which have been solved. Using substitutions
and integration led to an analytic solution of the nonlinear Black-
Scholes equation. In a real market situation, this solution may help
in finding how typical prices of derivatives can be described hence
contributing significantly to the field of Financial Mathematics.
2011-01-01T00:00:00Z