P. Onumanyi’s research while affiliated with University of Ilorin and other places

In this paper we derive finite difference methods by a power series form of multistep collocation for the solution of the initial value problems for ordinary differential equations. By selection of points for both interpolation and collocation, many important classes of finite difference methods are produced including new ones which are generally more accurate (smaller error constants) than the Adams–Moulton Methods with adequate absolute stability intervals for nonstiff problem.

The authors report a hybrid formula of order four for starting the Numerov method applied to the initial-value problem for y '' =f(x,y), over the recently obtained result of order three by P. Onumanyi, U. W. Sirisena and S. O. Adee [Some theoretical considerations of continuous of continuous linear multistep methods for u (v) =f(x,u), v=1,2, Bagale, J. Pure Appl. Sci. 2, No. 2, 1–5 (2002) and by Y. A. Yusuph and P. Onumanyi [New multiple FDMs through multistep collocation for y ' =f(x,y), Abacus 29, No. 2, 92–100 (2002)], based on two different approaches. We illustrate the accuracy of the methods by two numerical examples.

An accurate and efficient reduction method for solutions of partial differential equations is the Lanczos–Chebyshev reduction method. A posteriori error estimation technique based on a modification of the error of the Lanczos economization is proposed. An upper bound to the error in the solution of the differential equation is also presented.

We discuss some theory, which are of immense importance to the study and application of the continuous linear multistep method (CLMM) developed in earlier works for the first and special second order systems of ordinary differential equations (ODEs). The main results are as follows: (i) The associated matrix M is shown to be non-singular and a new proof is given for C = M-1 for the matrix C of coefficients. (ii) An estimation for the global error involved in the implementation of the CLMM approximation formula over sub-intervals that do not overlap is given and shows the convergence of the continuous solution (CLMM). (iii) Two implementations are shown for higher order ODEs and a conjecture for partial differential equations (PDEs) through non-coupled method of lines by collocation.

In recent papers continuous finite difference (FD) approximations have been developed for the solution of the initial value problem (ivp) for first order ordinary differential equations (odes). They provide dense output of accurate solutions and global error estimates for the ivp economically. In this paper we show that some continuous FD formulae can be used to provide a uniform treatment of both the ivp and the boundary value problem (bvp) without using the shooting method for the latter. Higher order accurate solutions can be obtained on the same meshes with constant spacing used by one-step method without using the iterated deferred correction technique. No additional conditions are required to ensure low order continuity and this leads to fewer necessary equations than those required by most of the popular methods for bvps. No quadratures are involved in this non-overlapping piecewise continuous polynomial technique. Some computed results are given to show the effectiveness of the proposed method and global error estimates.

In this paper, a method is described for obtaining an estimate of the error of the Tau Method for ordinary differential equations; it is based on a modification of the error of Lanczos economization process. Perturbing the integrated error equation does not appear to improve the accuracy of the estimate significantly, while perturbing the homogeneous boundary conditions can lead to an increase in the accuracy of the estimate. In addition, the estimate is represented in terms of Canonical polynomials without any appreciable loss of accuracy. Several examples are given to illustrate the effectiveness of the method, including the incomplete gamma, exponential and Bessel functions.

This paper concerns the numerical solutions of two-point singularly perturbed boundary value problems for second order ordinary differential equations. For the linear problem, Canonical polynomials are constructed as new basis for collocation solution in the smooth region which is superposed with an exponential function in the boundary layer region. Numerical examples are given which show that the exponential fitting leads to a numerical asymptotic procedure. Extension to nonlinear problems is demonstrated by an example.

Cubic basis functions in one dimension for the solution of two-point boundary value problems are constructed based on the zeros of Chebyshev polynomials of the first kind. A general formula is derived for the construction of polynomial basis functions of degree r, where 1 less than equivalent to r greater than infinity . A Galerkin finite element method using the constructed basis functions for the cases r equals 1, 2 and 3 is successfully applied to three different types of problem including a singular perturbation problem.