Open Access Original Research Article

Achiles Nyongesa Simiyu, Philis Alosa, Fanuel Olege

Analytic dependence on a complex parameter appears at many places in the study of differential and integral equations. The display of analyticity in the solution of the Fredholm equation of the second kind is an early signal of the important role which analyticity was destined to play in spectral theory. The definition of the resolvent set is very explicit, this makes it seem plausible that the resolvent is a well behaved function. Let T be a closed linear operator in a complex Banach space X. In this paper we show that the resolvent set of T is an open subset of the complex plane and the resolvent function of T is analytic. Moreover, we show that if T is a bounded linear operator, the resolvent function of T is analytic at infinity, its value at infinity being 0 (where 0 is the bounded linear operator 0 in X). Consequently, we also show that if T is bounded in X then the spectrum of T is non-void.

Open Access Original Research Article

Paul Kiplimo Tarus, Wesley Cheruiyot Koech

Mathematical models and there parameters are essential for designers to predict the close loop behaviors of the plant so that systems are stable. A block model is develop in the MATLAB/simulink for the DC Motor-Gear-AC-Generator mathematical model in this paper, the block built is used to estimate the parameters in the estimation node using the gradient descent, simplex search and nonlinear least square algorithm. Gradient descent curve match that of the experimental data and its values are used in the DC Motor-Gear-AC Generator mathematical model.

**Objective:**

- To built block simulink
- Estimate the parameters of the DC Motor-Gear-Generator mathematical model.

Open Access Original Research Article

Purity Mberia, Stephen Karanja, Mark Kimathi

Numerical analysis of fluid flow is anchored on the laws of conservation. A challenge in solving the momentum equation arises due to the unavailability of an explicit pressure equation. To avoid solving the pressure term most researchers have eliminated it by cross differentiating the x and the y two dimensional momentum equations and subtracting them. This method introduces more variables to be solved in comparison to the primitive variables and is restricted to two-dimensional flows as streamlines do not exist in three-dimension. This method thus presents a serious limitation in analysis of fluid flow. In this study an equation for computing pressure has been developed using pressure - velocity coupling and used in solving the governing equations. The performance of three pressure velocity schemes namely; the Semi Implicit Method for Pressure linked Equation (SIMPLE), SIMPLE Revised (SIMPLER) and SIMPLE Consistent (SIMPLEC) for laminar buoyancy driven flow has been tested in order to establish the scheme that gives results consistent with bench mark data. The equations governing the flow are solved iteratively using finite volume method together with the central difference interpolating scheme. The solutions are presented for Rayleigh numbers of 10^{3}, 10^{4}, and 10^{5}. This resulted in the velocity profiles for the SIMPLE, SIMPLER, and SIMPLEC algorithm for a Rayleigh number of 10^{4} and 10^{5} converging to the same path. At a Rayleigh number of 10^{3 }however, SIMPLER algorithm undergoes a degradation in convergence with grid refinement at the baffle region. Results predicted by using the SIMPLEC algorithm are thus able to effectively compute the velocity of fluid flow in a differentially heated square enclosure with baffles for both low and higher Rayleigh numbers irrespective of the grid size.

Open Access Original Research Article

Kunio Takezawa

When data are found to be realizations of a specific distribution, constructing the probability density function based on this distribution may not lead to the best prediction result. In this study, numerical simulations are conducted using data that follow a normal distribution, and we examine whether probability density functions that have shapes different from that of the normal distribution can yield larger log-likelihoods than the normal distribution in the light of future data. The results indicate that fitting realizations of the normal distribution to a different probability density function produces better results from the perspective of predictive ability. Similarly, a set of simulations using the exponential distribution shows that better predictions are obtained when the corresponding realizations are fitted to a probability density function that is slightly different from the exponential distribution. These observations demonstrate that when the form of the probability density function that generates the data is known, the use of another form of the probability density function may achieve more desirable results from the standpoint of prediction.

Open Access Original Research Article

Jikang Bai

When the eigenvalue \(\lambda\) of a higher order homogeneous linear differential equation with constant coecients is the repeated root of multiplicity \(k\), the differential equation has exactly \(k\) linearly independent solutions. Different textbooks often use different ways to deal with this part of the content. "Advanced Mathematics" by Tongji University and "Ordinary Differential Equations" by Sun Yat-sen University are two commonly used textbooks for science and engineering majors and mathematics majors of universities in China. The former directly gives the conclusion, while the reasoning skills of the latter are not easily understood and mastered by many students, which leading to the degeneration of the mastery of this part into "knowing the conclusion" and "being able to use the conclusion". This is contrary to the principle that knowledge is only a carrier and teaching must focuses on ability cultivation. We discuss two new methods based on operator decomposition and the solving method for first order linear differential equation. This method is easier to understand and grasp, and can be processed in the same way for real and complex eigenvalues.

Open Access Original Research Article

B. J. Omowo, I. O. Longe, C. E. Abhulimen, H. K. Oduwole

In this paper, we verify the convergence and stability of implicit (modified) finite difference scheme. Knowing fully that consistency and stability are very important criteria for convergence, we have prove the stability of the modied implicit scheme using the von Newmann method and also verify the convergence by comparing the numerical solution with the exact solution. The results shows that the schemes converges even as the step size is rened.