Spectral collocation method for system of weakly singular Volterra integral equations
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Spectral collocation method for system of weakly singular Volterra integral equations Zhendong Gu1 Received: 1 October 2018 / Accepted: 2 May 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract Based on our previous research, we investigate spectral collocation method for system of weakly singular Volterra integral equations. The provided convergence analysis shows that global convergence order is related to regularity of the solution to this system, and the local convergence order on collocation points only depends on the regularity of kernel functions. Numerical experiments are carried out to confirm these theoretical results. Numerical methods are developed to solve nonlinear system of weakly singular Volterra integral equations and high-order weakly singular Volterra integro-differential equations. Keywords Spectral collocation method · System of weakly singular VIEs · Convergence analysis · Numerical experiments Mathematics Subject Classification (2010) 65M70 · 45D05
1 Introduction Systems of weakly singular Volterra integral equations (VIEs) appear for example in the spatial discretization of partial VIEs [8]. Weakly singular equations are widely applied in fractional calculus [9, 13, 14]. Many high-order weakly singular Volterra integro-differential equations (VIDEs) can be transformed to be system of weakly singular VIEs [3]. Waveform and time point relaxation methods were developed to solve large systems of nonlinear systems of weakly singular VIDEs [4, 15]. Discontinuous piecewise polynomial collocation methods were investigated for solving system of weakly singular VIEs of the first kind [12]. A hybrid collocation method Communicated by: Jan Hesthaven Zhendong Gu
[email protected]; [email protected]; [email protected] 1
School of Financial Mathematics and Statistic, Guangdong University of Finance, Guangzhou 510521, China
Z. Gu
[6] was developed for weakly singular VIEs. Collocation method was proposed to solve fractional differential equations involving non-singular kernel [1]. Spectral methods [5] are the numerical methods with high precision, which are widely used to solve Volterra-type integral equations [7, 18, 19]. In [2], spectral method was proposed to solve a system of fractional differential equations within a fractional derivative involving the Mittag-Leffler kernel. In our previous research, we have investigated the spectral collocation methods for weakly singular VIE with proportional delay [11], and the system of VIEs with smooth kernel functions [10]. Based on the findings, we investigate a spectral collocation method for a system of weakly singular VIEs in this paper. The system of weakly singular VIEs considered in this paper is y(t) = g(t) + V (y)(t), t ∈ [0, 1],
(1)
y(t) := [y1 (t), y2 (t), · · · , yM (t)]T ,
(2)
g(t) := [g1 (t), g2 (t), · · · , gM (t)] .
(3)
where T
where
means the transposed matrix of A. The integral operator V is defined as t V (y)(t) := [h(t − s) · K(t, s)]y(s)ds AT
0
⎡ := ⎣
M
V1q (yq )(t),
q=1
M q=1
V2q (yq )(
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