Abstract
This paper presents a two-step hybrid numerical scheme with one off-grid point for the numerical solution of general second-order initial value problems without reducing to two systems of the first order. The scheme is developed using the collocation and interpolation technique invoked on Bernstein polynomial. The proposed scheme is consistent, zero stable, and is of order four($4$). The developed scheme can estimate the approximate solutions at both steps and off-step points simultaneously using variable step size. Numerical results obtained in this paper show the efficiency of the proposed scheme over some existing methods of the same and higher orders.
Highlights
The numerical solution of equation (1.1) coupled with equation (1.2) is still receiving a lot of attention due to the fact that many physical sciences and engineering problems formulated into mathematical equation result to equation of such type
Direct method of solving equation (1.1) has been shown to be more efficient and saves computational time rather than method of reduction to system of first order ordinary differential equation (Brown [4]) and this has led to many scholars to attempt to solve equation (1.1) directly without reduction to system of first order equation
Adeniran and Ogundare [5] propose a one step hybrid numerical scheme with two off grid points for solving directly second over order initial value problems, the scheme can estimate the approximate solution at both step and off step points simultaneously by using variable step size
Summary
Differential equations are important tools in solving real world problems and many physical phenomena are model into second order differential equations, such models may or may not have exact solutions, a need for a numerical solution. Adeniran and Ogundare [5] propose a one step hybrid numerical scheme with two off grid points for solving directly second over order initial value problems, the scheme can estimate the approximate solution at both step and off step points simultaneously by using variable step size. Odejide and Ogundare [6] developed a one step hybrid numerical scheme for the direct solution of general second order ordinary differential equations, the scheme was developed using the collocation and interpolation techniques on the power series approximate solution and augmented by the introduction of one offstep point, in order to circumvent Dahlquist zero stability barrier and upgrade the order of consistency of the method. Okoro [8] use a Bernstein polynomial to develop one step hybrid scheme with one offgrid point via collocation and interpolation techniques for the direct solution of general second order ordinary differential equations. The paper extend the work of Ojo and Okoro [8] by developing a two step hybrid method for solution of equation 1.1&1.2
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