A priori and a posteriori error analysis for a system of singularly perturbed Volterra integro-differential equations

Abstract

In this work, we consider a system of Volterra integro-differential equations with initial conditions. The derivative term in equations is multiplied with distinct small positive parameters giving rise to overlapping layers. We propose a numerical scheme that avoids the extra condition on the problem’s data required by the scheme of Liang et al. (Comput Appl Math 39:255, 2020). We derive a priori and a posteriori error bounds for the proposed scheme and further rectify the shortcomings of a posteriori error estimation in Liang et al. (Comput Appl Math 39:255, 2020). Numerical results are presented in the form of graphs and tables that validate the proposed theory.

Appendix A. A posteriori error analysis for the discrete scheme of Liang et al. (2020)

The main aim of this appendix is to present the correct a posteriori error analysis for the discrete scheme of Liang et al. (2020). For this purpose, we first mention the discrete scheme considered in (Liang et al. 2020, Sect. 3) for the continuous problem (1.1)–(1.2):

One could obtain the first identity in the proof of (Liang et al. 2020, Theorem 1) only by using ${\mathcal {E}}D^{-}{\textbf{V}}_{i}+A_{i}{\textbf{V}}_{i}+\displaystyle \sum _{k=1}^{i}\int _{t_{k-1}}^{t_{k}}B(t_{i}, s){\textbf{V}}_{k}\,ds={\textbf{f}}_{i},$ which obviously does not hold true. The following lemma provides the correct a posteriori error estimation for the discrete scheme (A.1)–(A.2).

Theorem Appendix A.1

Let ${\textbf{v}}=(v_{1}, v_{2}(t))^{T}$ be the solution of the continuous problem (1.1)–(1.2), $\{{\textbf{V}}_{i}\}_{i=0}^{N}$ be the solution of the discrete problem (A.1)–(A.2) and $\mathbf {\widetilde{V}}$ be its piecewise linear interpolation vector as defined in (4.1). Then,

$$\begin{aligned} \Vert {\mathbf {\widetilde{V}}-{\textbf{v}}} \Vert _{\infty }\le C \max _{1\le i\le N}\tau _{i}\left( 1+\sum _{j=1}^{2}\vert {D^{-}V_{j,i}} \vert \right) . \end{aligned}$$

(A.3)

Proof

For $\forall t\in (t_{i-1}, t_{i})$, we have

$$\begin{aligned} {\mathcal {T}}\mathbf {\widetilde{V}}(t)-{\mathcal {T}}{\textbf{v}}(t)&={\mathcal {E}}[\mathbf {\widetilde{V}}(t)]'+A(t)\mathbf {\widetilde{V}}(t) +\int _{0}^{t}B(t, s)\mathbf {\widetilde{V}}(s)\,ds-{\textbf{f}}(t)\nonumber \\&={\mathcal {E}}D^{-}{\textbf{V}}_{i}+\Bigg (A_{i}+\int _{t_{i}}^{t}A'(s)\,ds\Bigg ) \Bigg ({\textbf{V}}_{i}+(t-t_{i})D^{-}{\textbf{V}}_{i}\Bigg )\nonumber \\&\quad +\sum _{k=1}^{i-1}\int _{t_{k-1}}^{t_{k}}B(t, s)\Big ({\textbf{V}}_{k} +(t-t_{k})D^{-}{\textbf{V}}_{k}\Big )\,ds \end{aligned}$$

(A.4)

$$\begin{aligned}&\quad +\int _{t_{i-1}}^{t}B(t, s)\Big ({\textbf{V}}_{i}+(t-t_{i})D^{-} {\textbf{V}}_{i}\Big )\,ds-\Big ({\textbf{f}}(t)-{\textbf{f}}_{i}+{\textbf{f}}_{i}\Big )\nonumber \\&={\mathcal {E}}D^{-}{\textbf{V}}_{i}+A_{i}{\textbf{V}}_{i}+\sum _{k=1}^{i} \int _{t_{k-1}}^{t_{k}}B(t_{i}, t_{k}){\textbf{V}}_{k}\,ds-{\textbf{f}}_{i}\nonumber \\&\quad +A_{i}(t-t_{i})D^{-}{\textbf{V}}_{i}+\int _{t_{i}}^{t}A'(s)\,ds\, {\textbf{V}}_{i} +(t-t_{i})\int _{t_{i}}^{t}A'(s)\,ds\, D^{-}{\textbf{V}}_{i}\nonumber \\&\quad +\sum _{k=1}^{i-1}\int _{t_{k-1}}^{t_{k}}\Big (B(t, s)-B(t_{i}, t_{k})\Big ) {\textbf{V}}_{k}\,ds-\int _{t_{i-1}}^{t_{i}}B(t_{i}, t_{i}){\textbf{V}}_{i}\,ds\nonumber \\&\quad +\sum _{k=1}^{i-1}\int _{t_{k-1}}^{t_{k}}B(t, s)(t-t_{k})D^{-}{\textbf{V}}_{k}\,ds +\int _{t_{i-1}}^{t}B(t, s)(t-t_{i})D^{-}{\textbf{V}}_{i}\,ds\nonumber \\&\quad +\int _{t_{i-1}}^{t}B(t, s){\textbf{V}}_{i}\,ds -\Big ({\textbf{f}}(t)-{\textbf{f}}_{i}\Big )\nonumber \\&=A_{i}(t-t_{i})D^{-}{\textbf{V}}_{i}+\int _{t_{i}}^{t}A'(s)\,ds\, {\textbf{V}}_{i}+(t-t_{i})\int _{t_{i}}^{t}A'(s)\,ds\, D^{-}{\textbf{V}}_{i}\nonumber \\&\quad +\sum _{k=1}^{i-1}\int _{t_{k-1}}^{t_{k}}\Big (B(t, s)-B(t_{i}, t_{k})\Big ) {\textbf{V}}_{k}\,ds-\int _{t_{i-1}}^{t_{i}}B(t_{i}, t_{i}){\textbf{V}}_{i}\,ds\nonumber \\&\quad +\sum _{k=1}^{i-1}\int _{t_{k-1}}^{t_{k}}B(t, s)(t-t_{k})D^{-} {\textbf{V}}_{k}\,ds+\int _{t_{i-1}}^{t}B(t, s)(t-t_{i})D^{-}{\textbf{V}}_{i}\,ds\nonumber \\&\quad +\int _{t_{i-1}}^{t}B(t, s){\textbf{V}}_{i}\,ds-\Big ({\textbf{f}}(t)-{\textbf{f}}_{i}\Big ), \end{aligned}$$

(A.5)

where we have used (A.1). Now, we evaluate each terms of (A.4) separately. By Taylor expansions of B(t, s) about $(t_{i}, t_{k})$ for some $(\rho , \varsigma )$ in between (t, s) and $(t_{i}, t_{k})$, it holds

$$\begin{aligned} \left| {\sum _{k=1}^{i-1}\int _{t_{k-1}}^{t_{k}}\Big (B(t, s)-B(t_{i}, t_{k})\Big ){\textbf{V}}_{k}\,ds}\right|&\le \sum _{k=1}^{i-1}\Bigg |\int _{t_{k-1}}^{t_{k}}\Big (B(t_{i}, t_{k})+(t-t_{i})\frac{\partial B}{\partial t}\bigg |_{(\rho , \varsigma )}\nonumber \\&~~~~~~~~+(s-t_{k})\frac{\partial B}{\partial s}\bigg |_{(\rho , \varsigma )}-B(t_{i}, t_{k})\Big ){\textbf{V}}_{k}\,ds\Bigg |\nonumber \\&\le \sum _{k=1}^{i-1}\Big (\tau _{i}{\bar{B}}_{t}+\tau _{k}{\bar{B}}_{s}\Big ) \vert {{\textbf{V}}_{k}} \vert \int _{t_{k-1}}^{t_{k}}\,ds\nonumber \\&\le C\max _{1\le k\le i}\tau _{k}\left( \sum _{k=1}^{i-1}\int _{t_{k-1}}^{t_{k}}\,ds\right) \begin{pmatrix} 1\\ 1 \end{pmatrix}\nonumber \\&\le C\max _{1\le k\le i}\tau _{k}\begin{pmatrix} 1\\ 1 \end{pmatrix}. \end{aligned}$$

(A.6)

Next, we use the arguments in Theorem 4.1 to obtain

$$\begin{aligned} \Bigg |\sum _{k=1}^{i-1}\int _{t_{k-1}}^{t_{k}}B(t, s)(t-t_{k})D^{-}{\textbf{V}}_{k}\,ds+\int _{t_{i-1}}^{t}B(t, s)(t-t_{i})D^{-}{\textbf{V}}_{i}\,ds\Bigg |&\le C\max _{1\le k\le i}\tau _{k}\begin{pmatrix} 1 &{} 1\\ 1 &{} 1 \end{pmatrix}\vert {D^{-}{\textbf{V}}_{k}} \vert . \end{aligned}$$

(A.7)

The rest of the terms can be estimated similar to Theorem 4.1. Thus, we obtain

$$\begin{aligned} \Vert {{\mathcal {T}}\mathbf {\widetilde{V}}(t)-{\mathcal {T}}{\textbf{v}}(t)} \Vert _{\infty }\le C\max _{1\le i\le N}\tau _{i}\left( 1+\sum _{j=1}^{2}\vert {D^{-}V_{j,i}} \vert \right) . \end{aligned}$$

Hence, using (2.2), we finally get the desired result (A.3). $\square$