Abstract: This paper presents a direct approach for numerical solution of special second order delay differential equations (DDEs) directly without reduction to systems of low orders. The methods were generated using collocation approach via a combination of power series and exponential function. The approximate basis functions are interpolated at the first two grid points and collocated at both grid and off-grid points. The developed schemes and its derivatives were combined to form block methods to simultaneously solve second order Delay Differential Equations (DDEs) directly without the rigor of developing separate predictors. The required methods were obtained for step lengths of five with generalized number of hybrid points (3k). The basic properties of the methods were examined, the methods were found to have high order of accuracy of 21, low error constant, gives large interval of absolute stability, zero stable, consistence and convergent. The developed methods were applied to solve some special second order Delay Differential Equations. The methods also solve an engineering problem namely Matheiu’s equation in order to test for the efficiency and accuracy of the new methods. The results obtained were compared with existing methods in the literature. The results obtained showed better performance than some existing methods. The stability domain of the method is showed in figure 1 whereas the efficiency curve of the application problem for linear and nonlinear is presented in figure 2.Abstract: This paper presents a direct approach for numerical solution of special second order delay differential equations (DDEs) directly without reduction to systems of low orders. The methods were generated using collocation approach via a combination of power series and exponential function. The approximate basis functions are interpolated at the first ...Show More
Abstract: For world's energy demand is to be met in the future, engineers and scientists must work on developing methods and materials for storing and producing power. Since the very discovery of this novel material (Graphene) it has piqued the interest of researchers due to its low cost, reduced weight, unique nano-surface patterns, electrical capabilities, magnetic, spintronics and wide variety of industrial applications. Density functional theory method was used to calculate the electronic and structural properties of graphene sheet nano material using the Quantum Espresso Codes and the Xcrysden was used to visualize the structure and was the optimized. The Energy band gap were found to be zero and 0.25 eV respectively for both pure and doped boron graphene sheet. While the formation energy is 0.84eV and 1.5eV for pure and doped graphene. Also for both the total density of state and projected density of state are estimated to be of 0.29 eV and 0.31eV respectively due to effect of doping. Therefore, doping graphene with Boron is an effective approach to open a band gap for carbon-based next generation devices.Abstract: For world's energy demand is to be met in the future, engineers and scientists must work on developing methods and materials for storing and producing power. Since the very discovery of this novel material (Graphene) it has piqued the interest of researchers due to its low cost, reduced weight, unique nano-surface patterns, electrical capabilities,...Show More
