On the convergence of multistep collocation methods for integral-algebraic equations of index 1
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On the convergence of multistep collocation methods for integral-algebraic equations of index 1 Tingting Zhang1 · Hui Liang2 · Shijie Zhang1 Received: 15 May 2020 / Revised: 25 August 2020 / Accepted: 21 September 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract The multistep collocation method is introduced to solve integral-algebraic equations of index 1. The existence and uniqueness of the multistep collocation solution are proved. The convergence of the perturbed multistep collocation method is also investigated, which extends and includes the analysis of the multistep collocation method without perturbed terms. Some numerical experiments are given to illustrate the theoretical results. Keywords Integral-algebraic equations · Index 1 · Multistep collocation method · Existence and uniqueness · Convergence Mathematics Subject Classification 65R20
1 Introduction Volterra integral equations (VIEs) have rich applications in physics, biology, chemistry, etc. Generally, VIEs are divided into two classes: first-kind VIEs and second-kind VIEs. For first-kind VIEs, the unknown function only appears inside the integral, but for second-kind
Communicated by Antonio José Silva Neto. This work is supported by the National Nature Science Foundation of China (Nos. 11771128, 11101130), Fundamental Research Project of Shenzhen (JCYJ20190806143201649), Project (HIT.NSRIF.2020056) supported by Natural Scientific Research Innovation Foundation in Harbin Institute of Technology, and Research Start-Up Fund Foundation in Harbin Institute of Technology (20190019).
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Shijie Zhang [email protected] Tingting Zhang [email protected] Hui Liang [email protected]
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Research Center of Satellite Technology, Harbin Institute of Technology, Harbin 150080, People’s Republic of China
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School of Science, Harbin Institute of Technology, Shenzhen 518055, People’s Republic of China 0123456789().: V,-vol
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VIEs, the unknown function not only appears inside the integral, but also outside the integral. Coupled systems of integral-algebraic equations (IAEs) consisting of first-kind and secondkind VIEs also arise in many mathematical modeling processes (see, for example, Cannon 1984; Bartur et al. 1996; Kafarov et al. 1999; Gomilko 2003; Zenchuk 2008; Hadizadeh et al. 2011). As for differential-algebraic equations (DAEs), the index is very important to IAEs. Gear (1990) introduced differential index for IAEs, and Liang and Brunner (2013) defined the tractability index based on the ν-smoothing property of a Volterra integral operator and the matrix chain. According to the theory of Liang and Brunner (2013), a general linear index-1 IAEs can be decoupled into the following semi-explicit form: t y(t) = f (t) + K 11 (t, s)y(s) + K 12 (t, s)z(s) ds, 0 t K 21 (t, s)y(s) + K 22 (t, s)z(s) ds, (1) 0 = g(t) + 0
where t ∈ I := [0, T ], the data functions f , g, K i j (i, j = 1, 2) are sufficiently smooth, g(0) = 0, and |K 22 (t, t)| ≥ k0 > 0 for all t ∈ I . The a
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