Asian Journal of Mathematical Sciences(AJMS)

On Extension Theorems in Ordinary Differential Systems of Equations

Eziokwu, Chigozie Emmanuel — 2023-02-08

Given the differential system x′ = f (t, x); f: J × M → Rn, we establish the Peano’s theorem on existence of the solution plus Picard Lindelof theorem on uniqueness of the solution. Using the two, we then worked on the extendibility of the solutions whose local existence is ensured by the above in a domain of open connected set producing the following results; If D is a domain of R × Rn and F: D → Rn is continuous and suppose that (t0, x0) is a point D and if the system has a solution x(t) defined on a finite interval (a, b) with t ∈ (a, b) and x(t0) = x0, then whenever f is bounded and D, the limits
xa
xtta+()=+lim()
xb
xttb−()=−lim()
Exists as finite vectors and if the point (a, x(a+)),(b, x(b-)) is in D then x(t) is extendable to the point t = a (t = b). In addition to this are other results as could be seen in major sections of this work.

Simulation of the Movement of Ground Water in a rectangular jumper with a screen

E. N. Bereslavsky — 2023-02-08

Within the frame work of planar steady-state filtration of incompressible fluid according to Darcy’s law, an exact analytical solution of the problem of flow in a rectangular cofferdam with a screen in the presence of evaporation from the free surface of groundwater is given. The limiting cases of the considered motion – filtration in unconfined reservoir to imperfect gallery, as well as the flow in the absence of evaporation – are noted.

On Application of the Newton-Raphson’s Fixed Point Iterative Method in the Solution of Chemical Equilibrum Problems

Eziokwu C. Emmanuel — 2023-02-08

In this work, we discussed the solution of a chemical equilibrium problem aiming to obtain it’s fixed point. To do this, the preliminary and basic ideas introducing the fixed point theory were X-rayed and the Newton-Raphson’s iterative method for solving the system of non-linear equations discussed; then, the problem of the chemical equilibrium involving principal reactions in the production of synthesis gas by partial oxidation of methane with oxygen was stated. Using a computer program, the O reactant ratio that produces an adiabatic equilibrium temperature was obtained by developing a system of seven simultaneous non-linear equations that have the form which we now solve using the Newton-Raphson’s method described in section 2.2 and hence the desired fixed point of the chemical equilibrium problem.

Exploring the Methods of Cointegration Procedures in Dynamics of HIV/AIDS and Related Infections

S. A. Adeyemi Gidado — 2023-02-08

Introduction: Cointegration has become an important property in contemporary time series analysis. Time series often have trends - either deterministic or stochastic. Material and Methods: This research work seeks to determine the inherent long run relationship between the human immunodeficiency virus (HIV) and other diseases and the circumstances when it is reasonable to expect that two or more diseases may be cointegrated. That is, if at least one of the processes is driving the other and if the diseases are being driven by the same underlying process. Result: The data (Prevalence of HIV/Acquired Immunodeficiency Syndrome [AIDS] Incidences) modeled in this study were obtained from the National Agency for the Control of AIDS. The stationarity characteristics of the study variables were investigated using Augmented Dickey-Fuller test, the long-run relationship between HIV and other two diseases was determined using Engle-Granger cointegration, Phillips-Ouliaris, and Johansen testing procedure while the Granger causality test was also performed to determine the short run relationship of the variables. Conclusion: Results showed that the series are integrated of order two; HIV, Hepatitis, and TB are found to be strongly and significantly positively correlated, the data series are considered to be stationary after the second differences. Furthermore, the Granger causality tests show that HIV “Granger causes” Tuberculosis and Hepatitis in Nigeria. However, Phillips-Ouliaris Test for Cointegration Determines the strongest cointegration level among HIV/AIDS and other infections. Hence, it is the most robust test for testing cointegration between HIV and tuberculosis, and HIV and Hepatitis disease.