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We present a study on numerical solutions of nonlinear ordinary differential equations by applying Runge-Kutta-Fehlberg (RKF) method, a well-known adaptive Runge-kutta method. The adaptive Runge-kutta methods use embedded integration formulas which appear in pairs. Typically adaptive methods monitor the truncation error at each integration step and automatically adjust the step size to keep the error within prescribed limit. Numerical solutions to different nonlinear initial value problems (IVPs) attained by RKF method are compared with corresponding classical Runge-Kutta (RK4) approximations in order to investigate the computational superiority of the former. The resulting gain in efficiency is compatible with the theoretical prediction. Moreover, with the aid of a suitable time-stepping scheme, we show that the RKF method invariably requires less number of steps to arrive at the right endpoint of the finite interval where the IVP is being considered. 

Urban development is occurring in many Sub-Saharan Africa cities and rapid urbanization is underway in the East African city of Addis Ababa, Ethiopia. In an effort to address urban poverty and increase homeownership opportunities for low and middle-income residents, the City Administration of Addis Ababa initiated a large-scale housing development project in 2005. The project has resulted in the completion of 175,000 units within the city with 132,000 more under construction. To understand the impacts of both rapid growth and the housing program’s impact on the city’s urban form, we compared the type and distribution of land uses in Addis Ababa, Ethiopia, between 2006 with 2016 using hand-digitized, ortho-rectified satellite images in Geographic Information Systems (GISs). While residential density has increased, overall density has decreased from 109 people/ha to 98 people/ha. We found that between 2006 and 2016, land occupied by residential housing increased from 33% to 39% and the proportion of informal housing decreased from 57% to 38%. Reflecting the country’s economic prosperity, there was a dramatic increase in the presence of single family housing, particularly on the city’s western side. In 2006, only 1% of residential areas were occupied by high-rise condominiums (4 floors or greater) and this increased to 11% by 2016. The majority of the new, higher density residential developments are located near the eastern edges of the city and this outlying location has significant implications for residents, infrastructure construction, and future development.