A Novel and Efficient Thirteen-Order Implicit Extended Hybrid Block Backward Differentiation Formula for Solving of First- and Second- Order Ordinary Differential Equations

Abstract

This study introduces a novel three-step implicit extended hybrid block backward differentiation formula (IEHBBDF) designed for the resolution of first and second-order stiff ordinary differential equations (ODEs). The method employs power series polynomials as basis functions and was developed using MAPLE 2016 software. The formulation includes discrete and continuous schemes, with evaluations conducted at both grid and off-grid points. The numerical properties of L-stability and zero-stability are analyzed rigorously. The numerical results obtained from testing standard first and second order ordinary differential equations (ODEs) across a broader range of intervals indicate that the method is efficient, highly accurate, and appropriate for ODEs. The scheme demonstrated effectiveness, reliability, and consistency in numerical problems.