L-stable Implicit Two-Step Modified Extended Hybrid Block Backward Differentiation Formulae for Solving Nonlinear Second-Order Differential-Algebraic Equations of Higher Indexes

Abstract

Differential-algebraic equations (DAEs) serve as essential extensions of traditional differential equations, incorporating algebraic constraints that are critical for modelling complex mechanical systems and electrical circuits. While numerical solutions for low-index DAEs are well-established, solving higher-order nonlinear systems remains a challenge. This paper presents a two-step L-stable Implicit Modified Extended Hybrid Block Backward Differentiation Formula (IMEHBBDF) specifically designed for Second-Order Nonlinear Differential Algebraic Equations (SONDAEs) of indexes one, two and three. Differential-algebraic equations have been transformed to ordinary differential equations through differentiation-index method. The algorithm was derived using a collocation and interpolation technique to construct an implicit second-derivative formula. This approach generates a unified block of methods that provides simultaneous approximate solutions for the system. Theoretical analysis demonstrates that the IMEHBBDF was consistent, convergent, and achieves an exceptionally high order of fourteen. The efficiency, accuracy and adaptability of the proposed method are validated through the numerical solution of four diverse examples, demonstrating its superiority in handling complex nonlinear DAEs.