L-stable Extended One-Step Hybrid Block Linear Multistep Methods for Solving Nonlinear Semi-Explicit Hessenberg DAEs Indexes with a Wider Range

Abstract

Differential algebraic equations (DAEs) are natural extension of differential equations that also involve algebraic equations. Several numerical methods have been developed to solve these DAEs, mostly for low-index systems. The methods are RKMs and others. SEHDAEs are important type of DAEs that are commonly used in the modeling of mechanical systems and electrical circuits. These methods either lack a rigorous theoretical analysis for higher-index systems. The LMM preserves the self-starting characteristics over existing methods and also offers the advantage of approximating the solution at multiple points in which the existing numerical and analytical methods cannot perform. EHBBDF of order nine was presented in one step for the numerical solution of SEHDAEs of lower and higher index. The main objectives of this work are to develop a new EHBBDF, investigate numerical properties, and compare the numerical results with existing methods for the general solution of SEHDAEs and it was carried out with MAPLE 2016 software. Collocation and interpolation techniques are used to construct the approach which yields the main and supplementary techniques. The formula is then used to create a single block of methods that simultaneously provide the approximate solutions for the SEHDAEs of lower and higher indexes. The new method (EHBBDF) was revealed consistent, convergent, and L-stable. The effectiveness and precision of the novel approach are assessed using four numerical examples. In specific, all differential variables of nonlinear SEHDAEs suggests the potential for extension of index 4 or higher.