A One Step Method for the Solution of General Second Order Ordinary Differential Equations

In this paper, an implicit one step method for the numerical solution of second order initial value problems of ordinary differential equations has been developed by collocation and interpolation technique. The introduction of an o step point guaranteed the zero stability and consistency of the method. The implicit method developed was implemented as a block which gave simultaneous solutions, as well as their rst derivatives, at both o step and the step point. A comparison of our method to the predictor-corrector method after solving some sample problems reveals that our method performs better

Two Steps Block Method for the Solution of General Second Order Initial Value Problems of Ordinary Differential Equation

In this paper, an implicit block linear multistep method for the solution of ordinary differential equation was extended to the general form of differential equation. This method is self starting and does not need a predictor to solve for the unknown in the corrector. The method can also be extended to boundary value problems without additional cost. The method was found to be efficient after being tested with numerical problems of second order

Solving General Second Order Ordinary Differential Equations by a One-Step Hybrid Collocation Method

A one-step hybrid method is developed for the numerical approximation of second order initial value problems of ordinary differential equations by interpolation and collocation at nonstop and step points respectively. The method is zero stable and consistent with very small error term. Numerical experiment of the method on sample problem shows that the method is more efficient and accurate than the results obtained from our earlier methods

Modified Block Method for the Direct Solution of Second Order Ordinary Differential Equations

The direct solution of general second order ordinary differential equations is considered in this paper. The method is based on the collocation and interpolation of the power series approximate solution to generate a continuous linear multistep method. We modified the existing block method in order to accommodate the general nth order ordinary differential equation. The method was found to be efficient when tested on second order ordinary differential equation

Four Steps Implicit Method for the Solution of General Second Order Ordinary Differential Equations

Four steps implicit scheme for the solution of second order ordinary differential equation was derived through interpolation and collocation method. Newton polynomial approximation method was used to generate the unknown parameters in the corrector. The method was tested with numerical examples and it was found to be efficient in solving second order ordinary differential equations

In-Group Effects of News Use on African Americans

The topic of this thesis is how frequent news exposure affects the Black community’s perceptions of the world and trust in institutions. The purpose of this thesis is to uncover whether African Americans with more news exposure are more likely to view the world with skepticism and fear when compared to those with less news exposure. My hypotheses predicted that African Americans who have had frequent exposure to news will perceive the world as a meaner and scarier place (H1), will exhibit less trust in police officers (H2), and will exhibit less trust in news media (H3) than those who have had less news exposure. I also took into account how strength of racial identity and prior contact with law enforcement moderated these relationships. I ran a series of regression tests that revealed support for H1 and H2, but no support for H3. I found that higher levels of news exposure for African Americans predicts mean world perceptions and feelings of paranoia around police officers, but does not predict lack trust in the news media. I also found that racial identity and prior contact with law enforcement partially moderated these relationships. The possibility that participants who consume news frequently do so because they have a lot of trust in the news media is discussed, as well as other implications. Limitations and opportunities for future study are also considered

Numerical Simulation of Electric Field and Charge Structure within an Isolated Thunderstorm

The study of electrical structure of thunderstorm is a deterministic factor in characterizing the behavior of lightning. It is impossible to measure the value of electric field at every point within the thunderstorm, even when the in situ measurements are made, they are done along the path of balloon-borne instruments or other methods employed in the measurement. The average data obtained from past measurements were employed in determining the electric field generated by the thunderstorm as observed along the horizontal distance using the cylindrical charge model. The data consisted of a screening charge layer firstly located “between” 1.8-2.0 km and later between 10.0-10.2 km respectively above the ground level (agl) to see the effect of positioning on the resultant computation. The lower positive charge center with radius ranging from 0.5 km to 2.0 km was placed at various heights between 2.0 and 6.0 km. This is to determine the effect of cloud base height on the resulting field profile. The main negative charge is accommodated between 4.0 and 7.5 km and the upper positive charge layer existed between 7.5- 10.0 km agl for a cloud base height of 2.0 km and 6.0- 9.5 km agl for a negative charge layer, 7.5- 11.0 km agl for upper positive charge with a cloud base of 4.0 km high. The resulting potential gradient (PG) profiles were obtained for spatial distribution with screening charge layer playing no active role in the thunderstorm electrification. The effects of wind shear due to increasing wind speed with height were also investigated to have a clear picture of the cloud charge dynamics; hence the vertical PG patterns produced by a moving thunderstorm for various cloud cell life-times were modeled. The results showed that a typical tropical thunderstorm can exhibit charge layer more than the normal tripolar structure, and the pocket of lower positive charge can be more than one or larger than the usual size. Wind shear was found to enhance lightning discharges at the surface of charge discontinuities. Keywords: Potential gradient, Wind shear, Cloud charge, Lower positive charge center, Noninductive charging.

Continuous Implicit method for the Solution of General Second Order Ordinary Differential Equations

Abstract Methods of collocation and interpolation were adopted to generate a continuous implicit scheme for the solution of second order ordinary differential equation. Newton polynomial approximation method was used to generate the unknown parameter in the corrector. This enables us to solve both initial and boundary value problems

Reduction of Uranium by Bacterial Products

The Old Rifle Mill Processing site at Rifle, CO, contains uranium contaminated groundwater. The presence of uranium is one of the major problems at Department of Energy legacy sites. There is an initiative for attenuation of uranium by the Department of Energy. Uranium undergoes oxidation/reduction reactions with the substances at the site. Uranium’s oxidation state determines its solubility and mobility in the aquifer. The oxidation reduction pathways at this site have mineralogical, microbial and geochemical components. Understanding the oxidation/reduction pathways of these components will allow for better predictions of the changes and movement of uranium. Sulfide [S2- ] and Ferrous [Fe2] ions are products of microbial activity. These ions can reduce uranium [U(VI)], but bicarbonate ions [HCO3-] in the aquifers slows down the reaction. However, we believe that organic matter in the environment enhances U(VI) reduction by Fe2+and S2- in the absence of microbes. To address this, U(VI) was mixed with Fe2+ or S2- in autoclaved biomass from Rifle and artificial groundwater for seven to eleven days. Aqueous samples from the vials will be analyzed for Uranium presence using ICP-MS [Inductively Coupled Plasma- Mass Spectrometry]. The biomass from each of the samples will be analyzed using XAS [X-Ray Absorption Spectroscopy] to determine the ionization state of Uranium. Based on the data, we can conclude that there is a significant decrease of the concentration of uranium from the reaction when the biomass was not sterilized. There were no differences within in the vials that had Fe2+ or S2-. The XAS data shows a mixture of U(IV) and U(VI) in the biomasss and more U(IV) in the biomass that was not sterilized. This suggests that the bacterial products alone is not completely responsible for the reduction of uranium

Three Steps Hybrid Block Method for the Solution of General Second Order Ordinary Differential Equations

Block method is adopted in this paper for the direct solution of second order ordinary differential equations. The method is derived by collocation and interpolation of power series approximate solution to give a continuous hybrid linear multistep method which is implemented in block method to derive the independent solution at selected grid points. The properties of the derived scheme were investigated and found to be zero-stable, consistent and convergent. The efficiency of the derived method was tested and found to compare favourably with the existing methods