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In this paper, block procedure for some k-step linear multi-step methods, using the Legendre polynomials as the basis functions, is proposed. Discrete methods were given which were used in block and implemented for solving the initial value problems, being continuous interpolant derived and collocated at grid points. Some numerical examples of ordinary differential equations were solved using the derived methods to show their validity and the accuracy. The numerical results obtained show that the proposed method can also be efficient in solving such problems.

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Abstract. This paper presents the result of a two-dimensional model for predicting groundwater hydraulic head in Nariya Village, Kaduna, Nigeria. Vertical Electrical Sounding (VES) was employed in investigating the subsurface water level at six different points (500m to 1km apart), and the result obtained was used to fit a two-dimensional polynomial, using Mathcad 14, which satisfactorily predicted the groundwater hydraulic head at other locations on the site. Surface and Piezometric surface contour maps were plotted using the polynomial with Maple 14 software, and the modeled result showed good agreement with the interpreted field data with less than 4% error margin.

In this paper, an implicit k-step linear multi-step methods using the Laguerre polynomials as the basis functions is proposed. The given discrete methods were used in block and implemented for solving the initial value problems, being continuous interpolant derived and collocated at grid points. Numerical examples of initial value problems (IVPs) of ordinary differential equations (ODEs) were solved using the derived methods, and it is observed that the results obtained converged faster and the consistency and the zero stability are validated.