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Item Effects of Fluid and Reservoir Characteristics on Dimensionless Pressure and Derivative of a Horizontal Well in a Bounded Oil Reservoir with Simultaneous Single Edge and Bottom Water Drive(American Journal of Engineering Research (AJER), 2021) Mutili Peter Mutisya, Adewole Stephen Ezizanami, Awuor Kennedy Otieno and Oyoo Daniel Okang’aThis study investigates the effects of fluid, wellbore and reservoir characteristics on dimensionless pressure and dimensionless pressure derivatives at late time flow of a horizontal well in a bounded oil reservoir subjected to a single edge and bottom water drive mechanisms. The properties considered in this paper include the dimensionless well length, dimensionless reservoir width and dimensionless pay thickness. The main objective is achieved by using the source and Green’s functions together with Newman product method. Spline functions for interpolation in curve fitting was used to plot the graphs aided by MATLAB program. Results show that the dimensionless pressure increases with decrease in dimensionless reservoir width and pay thickness. The dimensionless pressure derivative potentially collapses to zero when the dimensionless pressure becomes constant. Higher oil production is indicated by larger magnitudes of dimensionless pressure derivatives. Information in this study will assist in designing and completion of horizontal wells in a bounded reservoir for prolonged enhanced oil production.Item Derivation of Black-Scholes-Merton Logistic Brownian Motion Di erential Equation with Jump Di usion Process(JS Publication., 2019) Andanje Mulambula, D. B. Oduor and B. KwachBlack- Scholes formed the foundation of option pricing. However, some of the assumptions like constant volatility and interest among others are practically impossible to implement hence other option pricing models have been explored to help come up with a much reliable way of predicting the price trends of options. Black-scholes assumed that the daily logarithmic returns of individual stocks are normally distributed. This is not true in practical sense especially in short term intervals because stock prices are able to reproduce the leptokurtic feature and to some extent the volatility smile . To address the above problem the Jump-Di usion Model and the Kou Double-Exponential Jump-Di usion Model were presented. But still they have not fully addressed the issue of reliable prediction because the observed implied volatility surface is skewed and tends to atten out for longer maturities; The two models abilities to produce accurate results are reduced. This paper ventures into a research that will involve Black-Scholes-Merton logistic-type option pricing with jump di usion. The knowledge of logistic Brownian motion will be used to develop a logistic Brownian motion with jump di usion model for price process. MSC: 91GXX, 91G50, 62P05, 97M30.Item Volatility Estimation Using European-Logistic Brownian Motion with Jump Di usion Process(© JSPublication., 2020) Andanje Mulambula, D. B. Oduor, and B. O. KwachVolatility is the measure of how we are uncertain about the future of stock or asset prices. Black-Scholes model formed the foundation of stock or asset pricing. However, some of its assumptions like constant volatility and interest among others are practically impossible to implement hence other option pricing models have been explored to help come up with a much reliable way of predicting the price trends of options. The measure of volatility and good forecasts of future volatility are crucial for implementation, evaluation of asset and derivative pricing of asset. In particular, volatility has been used in nancial markets in assessment of risk associated with short-term uctuations in nancial time-series. Constant volatility is not true in practical sense especially in short term intervals because stock prices are able to reproduce the leptokurtic feature and to some extent the volatility smile . To address the above problem the Jump-Di usion Model and the Kou Double-Exponential Jump-Di usion Model were presented. But still they have not fully addressed the issue of reliable prediction because the observed implied volatility surface is skewed and tends to atten out for longer maturities; the two models abilities to produce accurate results are reduced. This study ventures into a research that will involve volatility estimation using European logistic-type option pricing with jump di usion. The knowledge of logistic Brownian motion will be used to develop a logistic Brownian motion with jump di usion model for price process. MSC: 91GXX, 91G50, 62P05, 97M30.Item A Mathematical Model for Pressure Distribution in a Bounded Oil Reservoir Subject to Single-Edged and Bottom Constant Pressure(IOSR Journal of Mathematics (IOSR-JM), 2020-07) Mutili Peter Mutisya, Adewole Stephen Ezizanami, Awuor Kennedy Otieno and Oyoo Daniel Okang’aWell test analysis of a horizontal well is complex and difficult to interpret. Most horizontal well mathematical models assume that horizontal wells are perfectly horizontal and are parallel to the top and bottom boundaries of the reservoir. As part of effort towards correct horizontal well test analysis, the purpose of this study is to develop a mathematical model using source and Green’s functions for a horizontal well completed in an oil reservoir at late time flow period, where the reservoir is bounded by an edge and bottom constant pressure boundaries. The purpose of the derivation is to understand the effects of well completion, well design and reservoir parameters on pressure and pressure derivative behavior of the well at late flow time, when all these external boundaries are presumed to have been felt. If the model is applied for well test analysis therefore information like reservoir natural permeability distribution, actual external boundary types and even the well completion performance will be decidable easily. Dimensionless variables were used to derive throughout the derivations. Results of the derivation show that the dimensionless pressure and dimensionless pressure derivatives increase with increase in dimensionless well length. This means that higher well productivity is achievable with extended well length when the reservoir is surrounded partially by constant pressure boundaries. Furthermore, the models show that higher directional permeabilities would also encourage higher well productivity at late flow time. The dimensionless pressure derivative will, as a result of a constant dimensionless pressure, potentially collapse gradually to zero at the moment the dimensionless pressure begins to exhibit a constant trend. Finally, the dimensionless pressure and dimensionless pressure derivatives vary inversely with the reservoir dimensionless width at late flow time.Item Characterization of Inner Derivations induced by Norm-attainable Operators(International Journal of Modern Science and Technology,, 2018) M. O. Oyake, N. B. Okelo, O. OngatiIn the present paper, results on characterization of inner derivations in Banach algebras are discussed. Some techniques are employed for derivations due to Mecheri, Hacene, Bounkhel and Anderson. Let H be an infinite dimensional complex Hilbert space and B(H) the algebra of all bounded linear operators on H. A generalized derivation δ: B(H) → B(H) is defined by δA,B(X) = AX −XB, for all X ∈ B(H) and A,B fixed in B(H). An inner derivation is defined by δA(X) = AX −XA, for all X ∈ B(H) and A fixed in B(H). Norms of inner derivations have been investigated by several mathematicians. However, it is noted that norms of inner derivations implemented by norm-attainable operators have not been considered to a great extent. In this study, we investigate properties of inner derivations which are strictly implemented by norm-attainable and we determine their norms. The derivations in this work are all implemented by norm-attainable operators. The results show that these derivations admit tensor norms of operators.