Recommended by Michael EvansThe purpose of this paper is to produce an efficient zero-stable numerical method with the same order of accuracy as that of the main starting values predictors for direct solution of fourth-order differential equations without reducing it to a system of first-order equations. The method of collocation of the differential system arising from the approximate solution to the problem is adopted using the power series as a basis function. The method is consistent, symmetric, and of optimal order p 6. The main predictor for the method is also consistent, symmetric, zero-stable, and of optimal order p 6.

Problem statement: The conventional methods of solving higher order differential equations have been by reducing them to systems of first order equations. This approach is cumbersome and increases computational time. Approach: To address this problem, a numerical algorithm for direct solution of 5th order initial value problems in ordinary differential equations (odes), using power series as basis function, is proposed in this research. Collocation of the differential system is taken at selected grid points to reduce the number of functions to be evaluated per iteration. A number of predictors and their derivatives having the same order of accuracy with the main method are proposed. Results: The approach yields a multiderivative method of order six. Numerical examples solved show increased efficiency of the method with increased number of iterations, converging to the theoretical solutions. Conclusion/Recommendations: The new mutiderivative method is efficient to solve linear and nonlinear fifth order odes without reduction to system of lower order equations

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In this paper we derived a continuous linear multistep method (LMM) with step number k = 5 through collocation and interpolation techniques using power series as basis function for approximate solution. An order nine p-stable scheme is developed which was used to solve the third order initial value problems in ordinary differential equation without first reducing to a system of first order equations. Taylor's series algorithm of the same order was developed to implement our method. The result obtained compared favourably with existing methods.

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This research focuses on the delineation of subsurface basement granitic structures suitable for engineering construction materials for the sitting of quarry industry in the area. The key objective of the study was to locate and delineate the depths of burial to the subsurface granite rock bodies and the regolith thickness overlain the bedrock unit. 14 resistivity profile lines with a surveyed length of 200 m and electrode spacing of 5 m, were carried out with the application of electrical resistivity tomography software, to image the subsurface structural units around this area, utilizing pole-dipole electrode configurations method towards assisting the Engineers in obtaining information on the subsurface geological features in this part of the Peninsula Malaysia. The focus is on characterizing engineering construction materials suitable for sitting the quarry industry, determination of the longitudinal conductance and coefficient of anisotropy of subsurface lithological units that determines the competency of the bedrock underneath the area from the geoelectric parameters obtained through the interpretations of the RES2DINV ERT images. The depth of bedrock unit as delineated from the results ranged from about 5 m to 100 m while the resistivity values recorded was greater than 6000 Ω-m in most of the profiles. Groundwater bearing channels that would serve the factory needs was delineated alongside the granitic rock unit. These results make the subsurface granitic bedrock unit to be adjudged competent and suitable enough as quarry construction materials for sitting the factory in the area.

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