International Journal of Research and Innovation in Applied Science (IJRIAS) |Volume VII, Issue IX, September 2022|ISSN 2454-6194
Umaru, A. H., Donald, J. Z.*, and Skwame, Y.
Department Of Mathematics, Adamawa State University Mubi Nigeria
*Corresponding Author
Abstract: This paper is concerned with the construction of continuous Seven-Step implicit hybrid block Simpson’s Second derivative method for solving initial value problems of second order ordinary differential equations were derived through interpolation and collocation method using maple software. Power series approximation method was used to generate the unknown parameters in the corrector. These Continuous formulations were evaluated at some desired points to give the discrete schemes which constitute the hybrid block method. The constructed block method is consistent, zero-stable and A(α)-Stable. Numerical results obtained using the new block method show that it superior on some system of initial value problems. The study revealed that our new method performed better.
I. INTRODUCTION
A Considerable Literature exists for the hybrid linear multi-step method for the solution of ordinary differential equation of the form (1.1)
where satisfies a given set of initial conditions, Ibijola et al (2011), We assume that the function also satisfies the Lipschitz condition which guarantees existence, uniqueness and continuous differentiable solution, (Jain et al 2014).
For the discrete solution of (1.1) linear multi-step methods has been studied by Lambert (1973). One important advantage of the continuous over the discrete approach is the ability to provide discrete schemes for simultaneous integration. These discrete schemes can as well be reformulated as general linear methods by Butcher (1993). Block method for solving ODEs were first proposed by Milne (1953). The block method are self-starting and can directly be applied to both initial and boundary value problems, Skwame et al (2018) and Donald et al (2009). The block methods show that the proposed block hybrid methods are zero-stable, consistent and A-stable.
In this paper we present Implicit hybrid block Simpson’s Second derivatives method with one off-grid point using