Solving the Boundary Value Problems for Second-Order Ordinary Differential Equations using a Class of Obrechkoff-Type Block Methods

Taiwo E. Fayode¹, Bola T. Olabode², Emmanuel A. Areo², Ezekiel O. Omole³ and Oluwasegun M. Ibrahim^{4, 5,}

¹Bamidele Olumilua University of Education, Science and Technology, Ikere-Ekiti, Nigeria

²Federal University of Technology Akure, Nigeria

³Landmark University, Nigeria

⁴University of Texas at Austin, USA

⁵STEAM City Initiative, Nigeria;African Institute for Mathematical Sciences, Rwanda

Cite this paper:
Taiwo E. Fayode, Bola T. Olabode, Emmanuel A. Areo, Ezekiel O. Omole and Oluwasegun M. Ibrahim. Solving the Boundary Value Problems for Second-Order Ordinary Differential Equations using a Class of Obrechkoff-Type Block Methods. American Journal of Applied Mathematics and Statistics. 2025; 13(2):30-37. doi: 10.12691/ajams-13-2-2

Abstract

Employing a combination of collocation and interpolation techniques, this research introduces a novel set of Obrechkoff-type methods designed to address second-order boundary value problems (BVPs) characterized by Neumann and Dirichlet boundary conditions. The methodology involves placing the derivative function equation at all grid points and interpolating the basis function at only two locations, resulting in the development of a series of continuous multistep techniques with variable step numbers. For numerical implementation, our approach employs block mode techniques. We investigate the order, consistency, stability, and convergence of these algorithms to ensure their robustness and reliability. To evaluate the efficacy and precision of the proposed method, comprehensive testing is conducted using Obrechkoff-type problems. Notably, the numerical solutions demonstrate improved performance compared to traditional methods, highlighting the potential of our approach to deliver enhanced accuracy in solving second-order BVPs.

Keywords:
Boundary value problems interpolation collocation differential equations numerical methods.

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