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Convergence and stability of extended BBVMs for nonlinear delay-differential-algebraic equations with piecewise continuous arguments Chengjian Zhang1,2 · Xiaoqiang Yan1,2 Received: 23 January 2020 / Accepted: 29 July 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020

Abstract Delay-differential-algebraic equations have been widely used to model some important phenomena in science and engineering. Since, in general, such equations do not admit a closed-form solution, it is necessary to solve them numerically by introducing suitable integrators. The present paper extends the class of block boundary value methods (BBVMs) to approximate the solutions of nonlinear delay-differential equations with algebraic constraint and piecewise continuous arguments. Under the classical Lipschitz conditions, convergence and stability criteria of the extended BBVMs are derived. Moreover, a couple of numerical examples are provided to illustrate computational effectiveness and accuracy of the methods. Keywords Delay-differential-algebraic equations · Piecewise continuous arguments · Block boundary value methods · Error analysis · Global stability

1 Introduction Consider the following problem of nonlinear delay-differential-algebraic equations with piecewise continuous arguments (PCAs): ⎧  ⎨ y (t) = f (t, y(t), y(t), z(t)), t ∈ [0, T ], z(t) = g(t, y(t), y(t), z(t)), t ∈ [0, T ], (1.1) ⎩ y(0) = y0 , z(0) = z0 ,  Chengjian Zhang

[email protected] Xiaoqiang Yan [email protected] 1

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

2

Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China

Numerical Algorithms

where · denotes the greatest integer function, y0 and z0 are the given initial values satisfying the regularity condition: z0 = g(0, y0 , y0 , z0 ), and functions f : [0, T ] × Rd1 × Rd1 × Rd2 → Rd1 and g : [0, T ] × Rd1 × Rd1 × Rd2 → Rd2 are assumed to be smooth enough and satisfy the following Lipschitz conditions with constants Li > 0 (1 ≤ i ≤ 6) for all t ∈ [0, T ], y, y, ˆ u, uˆ ∈ Rd1 and v, vˆ ∈ Rd2 : f (t, y, u, v)−f (t, y, ˆ u, ˆ v) ˆ ∞ ≤ L1 y − y ˆ ∞ +L2 u− u ˆ ∞ +L3 v− v ˆ ∞ , (1.2) ˆ ∞ +L5 u− u ˆ ∞ +L6 v − v ˆ ∞ . (1.3) g(t, y, u, v)−g(t, y, ˆ u, ˆ v) ˆ ∞ ≤ L4 y − y Problem (1.1) has been widely used to describe some important phenomena in mechanics, control science, biology, and the other scientific fields (see, e.g., [1, 2]). In particular, after setting z(t) = y  (t), the following problem of neutral differential equations with PCAs: y  (t) = f (t, y(t), y(t), y  (t)), t ∈ [0, T ], y(0) = y0 , can be written as a special form of (1.1): ⎧  ⎨ y (t) = f (t, y(t), y(t), z(t)), t ∈ [0, T ], z(t) = f (t, y(t), y(t), z(t)), t ∈ [0, T ], ⎩ y(0) = y0 , z(0) = z0 .

(1.4)

(1.5)

Although the analytical solution of problem (1.1) has been the subject of investigation, the research on numerical solution of problem