PPubMed7 Jul 2016Cited 1×

Analysis of backward differentiation formula for nonlinear differential-algebraic equations with 2 delays

Leping Sun

Abstract

This paper is concerned with the backward differential formula or BDF methods for a class of nonlinear 2-delay differential algebraic equations. We obtain two sufficient conditions under which the methods are stable and asymptotically stable. At last, examples show that our methods are true.

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While investigating nonlinear 2-delayed differential-algebraic equations, we get two sufficient conditions for the stability and asymptotic stability of 2-step BDF methods and think about how to check the conditions with some example.

What’s new

i.Derived two sufficient conditions under which two-step BDF methods are stable for nonlinear 2-delay DDAEs.
ii.Derived two sufficient conditions under which two-step BDF methods are asymptotically stable for nonlinear 2-delay DDAEs.
iii.Provided numerical examples demonstrating that the BDF method converges faster than the implicit Euler method.

How they did it

We investigate a class of nonlinear delay‑differential‑algebraic equations and derive stability and asymptotic‑stability conditions for two‑step BDF methods. Numerical experiments with various step sizes compare the BDF method to the implicit Euler method, showing faster convergence.

What they found

Table 1: With step size h=0.1, max |x_n‑xt_n| = 4.2 × 10⁻³ and max |y_n‑yt_n| = 1.75 × 10⁻².
Table 1: With step size h=0.01, max |x_n‑xt_n| = 4.83 × 10⁻⁵ and max |y_n‑yt_n| = 2.03 × 10⁻⁴.
The BDF method converges much faster than the implicit Euler method, as shown by the error reductions in Tables 1–3.

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