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

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.

Boundary value problems, interpolation, collocation, differential equations, numerical methods.