Hybrid special class for solving differential-algebraic equations

This paper introduces three classes of hybrid methods to solve systems of differential algebraic equations (DAEs). These classes are based on a free parameter class of linear multistep method (LMM). Two classes contain one step point and one stage point (off‐step point) of the first derivative of the solution. The third one contains two step points and one stage point of the first derivative of the solution. Parameters are selected to improve the absolute stability regions. The proposed solution methodologies have larger stability regions compared to the backward differentiation formulae (BDF), the extended backward differentiation formulae (EBDF), and the Hybrid extended backward differentiation formulae (HEBDF). The constructed first class is A−stable of the orders 2 to 5, and A(α)−stable of the orders 6 to 11. The second class is A−stable of the orders 2 to 5 and A(α)−stable of the orders 6 to 10. The last class is A−stable of the orders 3 to 4 and A(α)−stable of the orders 5 to 9. The A− stable methods of the three classes are L−stable. Numerical tests are conducted to validate the performance of the proposed technique.